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<!--Created on: 3/27/2026 12:40:09 PM-->
<!--Modified on: 3/27/2026 12:40:09 PM-->
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    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_StatementTable" xlink:label="us-gaap_StatementTable" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_StatementLineItems" xlink:to="us-gaap_StatementLineItems_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_FinancialDesignationPredecessorAndSuccessorFixedList" xlink:label="us-gaap_FinancialDesignationPredecessorAndSuccessorFixedList" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_FinancialDesignationPredecessorAndSuccessorFixedList_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Financial Designation, Predecessor and Successor [Fixed List]</label>
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    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_AssetsAbstract" xlink:to="us-gaap_AssetsAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_AssetsAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Assets [Abstract]</label>
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_CashAndCashEquivalentsAtCarryingValue_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cash and Cash Equivalents, at Carrying Value</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_CashAndCashEquivalentsAtCarryingValue_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cash and cash equivalents</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_AccountsReceivableNetCurrent" xlink:label="us-gaap_AccountsReceivableNetCurrent" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_AccountsReceivableNetCurrent" xlink:to="us-gaap_AccountsReceivableNetCurrent_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_AccountsReceivableNetCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Accounts Receivable, after Allowance for Credit Loss, Current</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_AccountsReceivableNetCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Accounts receivable, net</label>
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    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_InventoryNet" xlink:to="us-gaap_InventoryNet_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_InventoryNet_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Inventory, net</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:label="us-gaap_InventoryNet_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Total inventory</label>
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    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_PrepaidExpenseCurrent" xlink:to="us-gaap_PrepaidExpenseCurrent_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_OtherAssetsCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Other current assets</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_AssetsCurrent" xlink:label="us-gaap_AssetsCurrent" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_AssetsCurrent" xlink:to="us-gaap_AssetsCurrent_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:label="us-gaap_AssetsCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Total current assets</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_PropertyPlantAndEquipmentNet" xlink:label="us-gaap_PropertyPlantAndEquipmentNet" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_PropertyPlantAndEquipmentNet_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Property and equipment, net</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:label="us-gaap_PropertyPlantAndEquipmentNet_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Property and equipment, net</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_OperatingLeaseRightOfUseAsset" xlink:label="us-gaap_OperatingLeaseRightOfUseAsset" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_OperatingLeaseRightOfUseAsset" xlink:to="us-gaap_OperatingLeaseRightOfUseAsset_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_OperatingLeaseRightOfUseAsset_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Right-of-use assets, net</label>
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_IntangibleAssetsNetExcludingGoodwill_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Intangible assets, net</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:label="us-gaap_IntangibleAssetsNetExcludingGoodwill_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Total, Net Carrying Amount</label>
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    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_Goodwill" xlink:to="us-gaap_Goodwill_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_Goodwill_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Goodwill</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_Goodwill_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Goodwill, net</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:label="us-gaap_Goodwill_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Net Carrying Amount</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_OtherAssetsNoncurrent" xlink:label="us-gaap_OtherAssetsNoncurrent" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_OtherAssetsNoncurrent" xlink:to="us-gaap_OtherAssetsNoncurrent_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_OtherAssetsNoncurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Other Assets, Noncurrent</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_OtherAssetsNoncurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Other assets</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_Assets" xlink:label="us-gaap_Assets" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_Assets" xlink:to="us-gaap_Assets_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_Assets_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Assets</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:label="us-gaap_Assets_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Total assets</label>
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    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_LiabilitiesAndStockholdersEquityAbstract" xlink:to="us-gaap_LiabilitiesAndStockholdersEquityAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_LiabilitiesAndStockholdersEquityAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Liabilities and Equity [Abstract]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_LiabilitiesAndStockholdersEquityAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Liabilities and stockholders' equity</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_LiabilitiesCurrentAbstract" xlink:label="us-gaap_LiabilitiesCurrentAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_LiabilitiesCurrentAbstract" xlink:to="us-gaap_LiabilitiesCurrentAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_LiabilitiesCurrentAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Liabilities, Current [Abstract]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_LiabilitiesCurrentAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Current liabilities:</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_AccountsPayableCurrent" xlink:label="us-gaap_AccountsPayableCurrent" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_AccountsPayableCurrent" xlink:to="us-gaap_AccountsPayableCurrent_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_AccountsPayableCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Accounts Payable, Current</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_AccountsPayableCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Accounts payable</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_AccruedLiabilitiesCurrent" xlink:label="us-gaap_AccruedLiabilitiesCurrent" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_AccruedLiabilitiesCurrent" xlink:to="us-gaap_AccruedLiabilitiesCurrent_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_AccruedLiabilitiesCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Accrued Liabilities, Current</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_AccruedLiabilitiesCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Accrued liabilities</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:label="us-gaap_AccruedLiabilitiesCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Total accrued liabilities</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_OperatingLeaseLiabilityCurrent" xlink:label="us-gaap_OperatingLeaseLiabilityCurrent" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_OperatingLeaseLiabilityCurrent" xlink:to="us-gaap_OperatingLeaseLiabilityCurrent_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_OperatingLeaseLiabilityCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Operating Lease, Liability, Current</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_OperatingLeaseLiabilityCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Operating lease liabilities, current</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_AccruedOtherLiabilitiesCurrent" xlink:label="adgm_AccruedOtherLiabilitiesCurrent" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_AccruedOtherLiabilitiesCurrent" xlink:to="adgm_AccruedOtherLiabilitiesCurrent_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_AccruedOtherLiabilitiesCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amount of expenses incurred but not yet paid classified as other, due within one year or the normal operating cycle, if longer.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_AccruedOtherLiabilitiesCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Accrued Other Liabilities, Current</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_AccruedOtherLiabilitiesCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Other accrued liabilities</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_LiabilitiesCurrent" xlink:label="us-gaap_LiabilitiesCurrent" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_LiabilitiesCurrent" xlink:to="us-gaap_LiabilitiesCurrent_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_LiabilitiesCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Liabilities, Current</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:label="us-gaap_LiabilitiesCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Total current liabilities</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_OperatingLeaseLiabilityNoncurrent" xlink:label="us-gaap_OperatingLeaseLiabilityNoncurrent" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_OperatingLeaseLiabilityNoncurrent" xlink:to="us-gaap_OperatingLeaseLiabilityNoncurrent_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_OperatingLeaseLiabilityNoncurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Operating Lease, Liability, Noncurrent</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_OperatingLeaseLiabilityNoncurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Operating lease liabilities, long-term</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ConvertibleLongTermNotesPayable" xlink:label="us-gaap_ConvertibleLongTermNotesPayable" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ConvertibleLongTermNotesPayable" xlink:to="us-gaap_ConvertibleLongTermNotesPayable_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ConvertibleLongTermNotesPayable_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Convertible Notes Payable, Noncurrent</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ConvertibleLongTermNotesPayable_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Convertible notes payable, net</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_DerivativeLiabilitiesNoncurrent" xlink:label="us-gaap_DerivativeLiabilitiesNoncurrent" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_DerivativeLiabilitiesNoncurrent" xlink:to="us-gaap_DerivativeLiabilitiesNoncurrent_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_DerivativeLiabilitiesNoncurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Derivative Liability, Noncurrent</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_DerivativeLiabilitiesNoncurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Warrant liabilities</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_DeferredIncomeTaxLiabilitiesNet" xlink:label="us-gaap_DeferredIncomeTaxLiabilitiesNet" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_DeferredIncomeTaxLiabilitiesNet" xlink:to="us-gaap_DeferredIncomeTaxLiabilitiesNet_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_DeferredIncomeTaxLiabilitiesNet_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Deferred Income Tax Liabilities, Net</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_DeferredIncomeTaxLiabilitiesNet_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Deferred tax liabilities, net</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_OtherLiabilitiesNoncurrent" xlink:label="us-gaap_OtherLiabilitiesNoncurrent" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_OtherLiabilitiesNoncurrent" xlink:to="us-gaap_OtherLiabilitiesNoncurrent_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_OtherLiabilitiesNoncurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Other Liabilities, Noncurrent</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_OtherLiabilitiesNoncurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Other long-term liabilities</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_Liabilities" xlink:label="us-gaap_Liabilities" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_Liabilities" xlink:to="us-gaap_Liabilities_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_Liabilities_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Liabilities</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:label="us-gaap_Liabilities_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Total liabilities</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_CommitmentsAndContingencies" xlink:label="us-gaap_CommitmentsAndContingencies" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_CommitmentsAndContingencies" xlink:to="us-gaap_CommitmentsAndContingencies_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_CommitmentsAndContingencies_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Commitments and Contingencies</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_CommitmentsAndContingencies_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Commitments and contingencies</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_StockholdersEquityAbstract" xlink:label="us-gaap_StockholdersEquityAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_StockholdersEquityAbstract" xlink:to="us-gaap_StockholdersEquityAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_StockholdersEquityAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Equity, Attributable to Parent [Abstract]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_StockholdersEquityAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Stockholders' equity (deficit):</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_CommonStockValue" xlink:label="us-gaap_CommonStockValue" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_CommonStockValue" xlink:to="us-gaap_CommonStockValue_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_CommonStockValue_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Common Stock, Value, Issued</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_CommonStockValue_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Common stock, $0.0001 par value; 210,000,000 shares authorized at December 31, 2025 and December 31, 2024; 22,210,459 and 15,198,232 shares issued and outstanding at December 31, 2025 and December 31, 2024, respectively</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_AdditionalPaidInCapital" xlink:label="us-gaap_AdditionalPaidInCapital" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_AdditionalPaidInCapital" xlink:to="us-gaap_AdditionalPaidInCapital_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_AdditionalPaidInCapital_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Additional Paid in Capital</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_AdditionalPaidInCapital_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Additional paid-in capital</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_AccumulatedOtherComprehensiveIncomeLossNetOfTax" xlink:label="us-gaap_AccumulatedOtherComprehensiveIncomeLossNetOfTax" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_AccumulatedOtherComprehensiveIncomeLossNetOfTax" xlink:to="us-gaap_AccumulatedOtherComprehensiveIncomeLossNetOfTax_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_AccumulatedOtherComprehensiveIncomeLossNetOfTax_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Accumulated Other Comprehensive Income (Loss), Net of Tax</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_AccumulatedOtherComprehensiveIncomeLossNetOfTax_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Accumulated other comprehensive (loss) income</label>
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_StockholdersEquity_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Equity, Attributable to Parent</label>
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_LiabilitiesAndStockholdersEquity_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Liabilities and Equity</label>
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_CommonStockParOrStatedValuePerShare_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Common stock, par value (in dollars per share)</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_CommonStockParOrStatedValuePerShare_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Common stock par value</label>
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    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_CommonStockSharesAuthorized" xlink:to="us-gaap_CommonStockSharesAuthorized_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_CommonStockSharesAuthorized_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Common Stock, Shares Authorized</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_CommonStockSharesAuthorized_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Common stock, shares authorized</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_CommonStockSharesIssued" xlink:label="us-gaap_CommonStockSharesIssued" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_CommonStockSharesIssued" xlink:to="us-gaap_CommonStockSharesIssued_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_CommonStockSharesIssued_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Common Stock, Shares, Issued</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/periodEndLabel" xlink:label="us-gaap_CommonStockSharesIssued_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Balance (in shares)</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/periodStartLabel" xlink:label="us-gaap_CommonStockSharesIssued_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Balance (in shares)</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_CommonStockSharesIssued_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Common stock, shares issued</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_CommonStockSharesOutstanding" xlink:label="us-gaap_CommonStockSharesOutstanding" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_CommonStockSharesOutstanding" xlink:to="us-gaap_CommonStockSharesOutstanding_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_CommonStockSharesOutstanding_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Common Stock, Shares, Outstanding</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/periodEndLabel" xlink:label="us-gaap_CommonStockSharesOutstanding_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Successor's opening equity as of July 31, 2024 (Successor) (in shares)</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/periodStartLabel" xlink:label="us-gaap_CommonStockSharesOutstanding_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">ListCo closing equity as of July 30, 2024 (in Shares)</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_CommonStockSharesOutstanding_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Common stock, shares outstanding</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_CommonStockSharesOutstanding_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Total</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_StatementOfIncomeAndComprehensiveIncomeAbstract" xlink:label="us-gaap_StatementOfIncomeAndComprehensiveIncomeAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_StatementOfIncomeAndComprehensiveIncomeAbstract" xlink:to="us-gaap_StatementOfIncomeAndComprehensiveIncomeAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_StatementOfIncomeAndComprehensiveIncomeAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Consolidated Statements of Operations and Comprehensive (Loss) Income</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_FinancialInstrumentAxis" xlink:label="us-gaap_FinancialInstrumentAxis" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_FinancialInstrumentAxis" xlink:to="us-gaap_FinancialInstrumentAxis_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_FinancialInstrumentAxis_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Financial Instrument [Axis]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_TransfersAndServicingOfFinancialInstrumentsTypesOfFinancialInstrumentsDomain" xlink:label="us-gaap_TransfersAndServicingOfFinancialInstrumentsTypesOfFinancialInstrumentsDomain" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_TransfersAndServicingOfFinancialInstrumentsTypesOfFinancialInstrumentsDomain" xlink:to="us-gaap_TransfersAndServicingOfFinancialInstrumentsTypesOfFinancialInstrumentsDomain_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ConvertibleDebtSecuritiesMember" xlink:label="us-gaap_ConvertibleDebtSecuritiesMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ConvertibleDebtSecuritiesMember" xlink:to="us-gaap_ConvertibleDebtSecuritiesMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ConvertibleDebtSecuritiesMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Convertible Debt Securities [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ConvertibleDebtSecuritiesMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Convertible notes</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_WarrantMember" xlink:label="us-gaap_WarrantMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_WarrantMember" xlink:to="us-gaap_WarrantMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_WarrantMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Warrant [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_WarrantMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Common stock warrants</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_WarrantMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Warrant liabilities</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_RevenueFromContractWithCustomerExcludingAssessedTax" xlink:label="us-gaap_RevenueFromContractWithCustomerExcludingAssessedTax" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_RevenueFromContractWithCustomerExcludingAssessedTax_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Revenue from Contract with Customer, Excluding Assessed Tax</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_RevenueFromContractWithCustomerExcludingAssessedTax_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Revenue</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_CostsAndExpensesAbstract" xlink:label="us-gaap_CostsAndExpensesAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_CostsAndExpensesAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Costs and Expenses [Abstract]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_CostsAndExpensesAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cost of revenue and operating expenses:</label>
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    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_CostOfRevenue" xlink:to="us-gaap_CostOfRevenue_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_CostOfRevenue_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cost of Revenue</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_CostOfRevenue_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cost of revenue</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ResearchAndDevelopmentExpense" xlink:label="us-gaap_ResearchAndDevelopmentExpense" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ResearchAndDevelopmentExpense_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Research and Development Expense</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ResearchAndDevelopmentExpense_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Research and development</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_SellingGeneralAndAdministrativeExpense" xlink:label="us-gaap_SellingGeneralAndAdministrativeExpense" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_SellingGeneralAndAdministrativeExpense_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Selling, General and Administrative Expense</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_SellingGeneralAndAdministrativeExpense_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Selling, general, and administrative</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_GoodwillImpairmentLoss" xlink:label="us-gaap_GoodwillImpairmentLoss" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_GoodwillImpairmentLoss" xlink:to="us-gaap_GoodwillImpairmentLoss_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_GoodwillImpairmentLoss_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Goodwill, Impairment Loss</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_GoodwillImpairmentLoss_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Impairment of goodwill</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_GoodwillImpairmentLoss_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Impairment - Goodwill</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ImpairmentOfIntangibleAssetsExcludingGoodwill" xlink:label="us-gaap_ImpairmentOfIntangibleAssetsExcludingGoodwill" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ImpairmentOfIntangibleAssetsExcludingGoodwill_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Impairment of Intangible Assets (Excluding Goodwill)</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ImpairmentOfIntangibleAssetsExcludingGoodwill_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Impairment - Intangibles assets, net</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_ImpairmentOfIntangibleAssetsExcludingGoodwill_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Intangible assets impairment charge</label>
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_NonoperatingIncomeExpenseAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Other (expense) income:</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisLiabilityGainLossIncludedInEarnings" xlink:label="us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisLiabilityGainLossIncludedInEarnings" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisLiabilityGainLossIncludedInEarnings_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fair value measurement adjustments</label>
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    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_InvestmentIncomeInterest" xlink:label="us-gaap_InvestmentIncomeInterest" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_InvestmentIncomeInterest" xlink:to="us-gaap_InvestmentIncomeInterest_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_InvestmentIncomeInterest_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Investment Income, Interest</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_InvestmentIncomeInterest_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Interest income</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_OtherNonoperatingIncomeExpense" xlink:label="us-gaap_OtherNonoperatingIncomeExpense" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_OtherNonoperatingIncomeExpense" xlink:to="us-gaap_OtherNonoperatingIncomeExpense_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_OtherNonoperatingIncomeExpense_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Other Nonoperating Income (Expense)</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_OtherNonoperatingIncomeExpense_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Other (expense) income, net</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_NonoperatingIncomeExpense" xlink:label="us-gaap_NonoperatingIncomeExpense" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_NonoperatingIncomeExpense" xlink:to="us-gaap_NonoperatingIncomeExpense_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_NonoperatingIncomeExpense_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Nonoperating Income (Expense)</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_NonoperatingIncomeExpense_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Other income (expense), net</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:label="us-gaap_NonoperatingIncomeExpense_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Total other (loss) income, net</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_UndistributedEarnings" xlink:label="us-gaap_UndistributedEarnings" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_UndistributedEarnings" xlink:to="us-gaap_UndistributedEarnings_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_UndistributedEarnings_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Undistributed Earnings, Basic</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_UndistributedEarnings_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Net loss - Basic</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:label="us-gaap_UndistributedEarnings_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Net loss</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ComprehensiveIncomeNetOfTaxAbstract" xlink:label="us-gaap_ComprehensiveIncomeNetOfTaxAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ComprehensiveIncomeNetOfTaxAbstract" xlink:to="us-gaap_ComprehensiveIncomeNetOfTaxAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ComprehensiveIncomeNetOfTaxAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Comprehensive Income (Loss), Net of Tax, Attributable to Parent [Abstract]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ComprehensiveIncomeNetOfTaxAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Other comprehensive loss:</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_OtherComprehensiveIncomeLossForeignCurrencyTransactionAndTranslationAdjustmentNetOfTax" xlink:label="us-gaap_OtherComprehensiveIncomeLossForeignCurrencyTransactionAndTranslationAdjustmentNetOfTax" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_OtherComprehensiveIncomeLossForeignCurrencyTransactionAndTranslationAdjustmentNetOfTax" xlink:to="us-gaap_OtherComprehensiveIncomeLossForeignCurrencyTransactionAndTranslationAdjustmentNetOfTax_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_OtherComprehensiveIncomeLossForeignCurrencyTransactionAndTranslationAdjustmentNetOfTax_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Other Comprehensive Income (Loss), Foreign Currency Transaction and Translation Adjustment, Net of Tax</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_OtherComprehensiveIncomeLossForeignCurrencyTransactionAndTranslationAdjustmentNetOfTax_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Foreign currency translation adjustment</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ComprehensiveIncomeNetOfTax" xlink:label="us-gaap_ComprehensiveIncomeNetOfTax" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ComprehensiveIncomeNetOfTax" xlink:to="us-gaap_ComprehensiveIncomeNetOfTax_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ComprehensiveIncomeNetOfTax_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Comprehensive Income (Loss), Net of Tax, Attributable to Parent</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:label="us-gaap_ComprehensiveIncomeNetOfTax_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Comprehensive loss</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_EarningsPerShareBasic" xlink:label="us-gaap_EarningsPerShareBasic" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_EarningsPerShareBasic" xlink:to="us-gaap_EarningsPerShareBasic_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_EarningsPerShareBasic_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Earnings Per Share, Basic</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_EarningsPerShareBasic_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Basic net loss per share (in dollars per share)</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_EarningsPerShareBasic_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Net loss per share attributable to common stockholders - basic (in dollars per share)</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_EarningsPerShareDiluted" xlink:label="us-gaap_EarningsPerShareDiluted" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_EarningsPerShareDiluted" xlink:to="us-gaap_EarningsPerShareDiluted_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_EarningsPerShareDiluted_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Earnings Per Share, Diluted</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_EarningsPerShareDiluted_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Diluted net loss per share (in dollars per share)</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_EarningsPerShareDiluted_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Net loss per share attributable to common stockholders - diluted (in dollars per share)</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_WeightedAverageNumberOfSharesOutstandingBasic" xlink:label="us-gaap_WeightedAverageNumberOfSharesOutstandingBasic" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_WeightedAverageNumberOfSharesOutstandingBasic_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Weighted Average Number of Shares Outstanding, Basic</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_WeightedAverageNumberOfSharesOutstandingBasic_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Weighted-average shares outstanding - Basic (in shares)</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_WeightedAverageNumberOfSharesOutstandingBasic_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Weighted-average shares outstanding used in computing net loss per share attributable to common stockholders - basic (in shares)</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_WeightedAverageNumberOfDilutedSharesOutstanding" xlink:label="us-gaap_WeightedAverageNumberOfDilutedSharesOutstanding" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_WeightedAverageNumberOfDilutedSharesOutstanding_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Weighted Average Number of Shares Outstanding, Diluted</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_WeightedAverageNumberOfDilutedSharesOutstanding_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Weighted-average shares outstanding - Diluted (in shares)</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:label="us-gaap_WeightedAverageNumberOfDilutedSharesOutstanding_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Weighted-average shares outstanding - Diluted</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_WeightedAverageNumberOfDilutedSharesOutstanding_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Weighted-average shares outstanding used in computing net loss per share attributable to common stockholders - diluted (in shares)</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_StatementOfStockholdersEquityAbstract" xlink:label="us-gaap_StatementOfStockholdersEquityAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_StatementOfStockholdersEquityAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Consolidated Statements of Convertible Preferred Stock and Stockholders' Equity (Deficit)</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_StatementEquityComponentsAxis" xlink:label="us-gaap_StatementEquityComponentsAxis" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_EquityComponentDomain" xlink:label="us-gaap_EquityComponentDomain" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_CommonStockMember" xlink:label="us-gaap_CommonStockMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_CommonStockMember" xlink:to="us-gaap_CommonStockMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_CommonStockMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Common Stock</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_CommonStockMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Common Stock</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_AdditionalPaidInCapitalMember" xlink:label="us-gaap_AdditionalPaidInCapitalMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_AdditionalPaidInCapitalMember" xlink:to="us-gaap_AdditionalPaidInCapitalMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_AdditionalPaidInCapitalMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Additional Paid-in Capital [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_AdditionalPaidInCapitalMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Additional Paid-in Capital</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_RetainedEarningsMember" xlink:label="us-gaap_RetainedEarningsMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_RetainedEarningsMember" xlink:to="us-gaap_RetainedEarningsMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_RetainedEarningsMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Retained Earnings [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_RetainedEarningsMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Accumulated Deficit</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_AccumulatedOtherComprehensiveIncomeMember" xlink:label="us-gaap_AccumulatedOtherComprehensiveIncomeMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_AccumulatedOtherComprehensiveIncomeMember" xlink:to="us-gaap_AccumulatedOtherComprehensiveIncomeMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_AccumulatedOtherComprehensiveIncomeMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">AOCI Attributable to Parent [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_AccumulatedOtherComprehensiveIncomeMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Accumulated Other Comprehensive Income</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_IncreaseDecreaseInTemporaryEquityRollForward" xlink:label="us-gaap_IncreaseDecreaseInTemporaryEquityRollForward" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_IncreaseDecreaseInTemporaryEquityRollForward" xlink:to="us-gaap_IncreaseDecreaseInTemporaryEquityRollForward_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_IncreaseDecreaseInTemporaryEquityRollForward_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Increase (Decrease) in Temporary Equity [Roll Forward]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_IncreaseDecreaseInTemporaryEquityRollForward_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Increase (Decrease) in Convertible Preferred Stock</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_TemporaryEquityCarryingAmountAttributableToParent" xlink:label="us-gaap_TemporaryEquityCarryingAmountAttributableToParent" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_TemporaryEquityCarryingAmountAttributableToParent" xlink:to="us-gaap_TemporaryEquityCarryingAmountAttributableToParent_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_TemporaryEquityCarryingAmountAttributableToParent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Temporary Equity, Carrying Amount, Attributable to Parent</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/periodEndLabel" xlink:label="us-gaap_TemporaryEquityCarryingAmountAttributableToParent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Balance</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/periodStartLabel" xlink:label="us-gaap_TemporaryEquityCarryingAmountAttributableToParent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Balance</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_TemporaryEquityCarryingAmountAttributableToParent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Carrying Value</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_TemporaryEquitySharesIssued" xlink:label="us-gaap_TemporaryEquitySharesIssued" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_TemporaryEquitySharesIssued" xlink:to="us-gaap_TemporaryEquitySharesIssued_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_TemporaryEquitySharesIssued_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Temporary Equity, Shares Issued</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/periodEndLabel" xlink:label="us-gaap_TemporaryEquitySharesIssued_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Balance (in shares)</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/periodStartLabel" xlink:label="us-gaap_TemporaryEquitySharesIssued_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Balance (in shares)</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_TemporaryEquitySharesIssued_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Shares Issued</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_TemporaryEquityPrefundedWarrantsIssuedInExchangeOfExtinguishmentValue" xlink:label="adgm_TemporaryEquityPrefundedWarrantsIssuedInExchangeOfExtinguishmentValue" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_TemporaryEquityPrefundedWarrantsIssuedInExchangeOfExtinguishmentValue" xlink:to="adgm_TemporaryEquityPrefundedWarrantsIssuedInExchangeOfExtinguishmentValue_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_TemporaryEquityPrefundedWarrantsIssuedInExchangeOfExtinguishmentValue_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Value of pre-funded warrants issued in exchange for the extinguishment of convertible preferred stock classified as temporary equity</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_TemporaryEquityPrefundedWarrantsIssuedInExchangeOfExtinguishmentValue_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Temporary Equity, Prefunded Warrants Issued In Exchange Of Extinguishment, Value</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2009/role/negatedLabel" xlink:label="adgm_TemporaryEquityPrefundedWarrantsIssuedInExchangeOfExtinguishmentValue_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Exchange preferred stock for pre-funded warrants</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_TemporaryEquityPrefundedWarrantsIssuedInExchangeOfExtinguishmentShares" xlink:label="adgm_TemporaryEquityPrefundedWarrantsIssuedInExchangeOfExtinguishmentShares" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_TemporaryEquityPrefundedWarrantsIssuedInExchangeOfExtinguishmentShares" xlink:to="adgm_TemporaryEquityPrefundedWarrantsIssuedInExchangeOfExtinguishmentShares_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_TemporaryEquityPrefundedWarrantsIssuedInExchangeOfExtinguishmentShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The number of pre-funded warrants issued upon exchange for the extinguishment of convertible preferred stock classified as temporary equity.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_TemporaryEquityPrefundedWarrantsIssuedInExchangeOfExtinguishmentShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Temporary Equity, Prefunded Warrants Issued In Exchange Of Extinguishment, Shares</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_TemporaryEquityPrefundedWarrantsIssuedInExchangeOfExtinguishmentShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Exchange preferred stock for pre-funded warrants (shares)</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_IncreaseDecreaseInStockholdersEquityRollForward" xlink:label="us-gaap_IncreaseDecreaseInStockholdersEquityRollForward" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_IncreaseDecreaseInStockholdersEquityRollForward" xlink:to="us-gaap_IncreaseDecreaseInStockholdersEquityRollForward_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_IncreaseDecreaseInStockholdersEquityRollForward_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Increase (Decrease) in Stockholders' Equity [Roll Forward]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_IncreaseDecreaseInStockholdersEquityRollForward_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Increase (Decrease) in Stockholders' Deficit</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_OtherComprehensiveIncomeLossForeignCurrencyTransactionAndTranslationAdjustmentNetOfTax1" xlink:label="adgm_OtherComprehensiveIncomeLossForeignCurrencyTransactionAndTranslationAdjustmentNetOfTax1" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_OtherComprehensiveIncomeLossForeignCurrencyTransactionAndTranslationAdjustmentNetOfTax1" xlink:to="adgm_OtherComprehensiveIncomeLossForeignCurrencyTransactionAndTranslationAdjustmentNetOfTax1_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_OtherComprehensiveIncomeLossForeignCurrencyTransactionAndTranslationAdjustmentNetOfTax1_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amount after tax and reclassification adjustments of gain (loss) on foreign currency translation adjustments, foreign currency transactions designated and effective as economic hedges of a net investment in a foreign entity and intra-entity foreign currency transactions that are of a long-term-investment nature.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_OtherComprehensiveIncomeLossForeignCurrencyTransactionAndTranslationAdjustmentNetOfTax1_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Other Comprehensive Income Loss Foreign Currency Transaction And Translation Adjustment Net Of Tax 1</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_OtherComprehensiveIncomeLossForeignCurrencyTransactionAndTranslationAdjustmentNetOfTax1_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Foreign currency translation adjustment</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_StockIssuedDuringPeriodValueNewIssues" xlink:label="us-gaap_StockIssuedDuringPeriodValueNewIssues" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_StockIssuedDuringPeriodValueNewIssues" xlink:to="us-gaap_StockIssuedDuringPeriodValueNewIssues_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_StockIssuedDuringPeriodValueNewIssues_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Stock Issued During Period, Value, New Issues</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_StockIssuedDuringPeriodValueNewIssues_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Contribution of cash proceeds in PIPE Financing</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_StockIssuedDuringPeriodValueNewIssues_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Issuance of common shares, pre-funded warrants and warrants, net of issuance costs</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_StockIssuedDuringPeriodSharesNewIssues" xlink:label="us-gaap_StockIssuedDuringPeriodSharesNewIssues" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_StockIssuedDuringPeriodSharesNewIssues" xlink:to="us-gaap_StockIssuedDuringPeriodSharesNewIssues_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_StockIssuedDuringPeriodSharesNewIssues_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Stock Issued During Period, Shares, New Issues</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_StockIssuedDuringPeriodSharesNewIssues_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Contribution from PIPE Financing for cash</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_StockIssuedDuringPeriodSharesNewIssues_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Issuance of common shares, pre-funded warrants and warrants, net of issuance costs (in shares)</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_AdjustmentsToAdditionalPaidInCapitalWarrantIssued" xlink:label="us-gaap_AdjustmentsToAdditionalPaidInCapitalWarrantIssued" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_AdjustmentsToAdditionalPaidInCapitalWarrantIssued" xlink:to="us-gaap_AdjustmentsToAdditionalPaidInCapitalWarrantIssued_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_AdjustmentsToAdditionalPaidInCapitalWarrantIssued_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Adjustments to Additional Paid in Capital, Warrant Issued</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_AdjustmentsToAdditionalPaidInCapitalWarrantIssued_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Exchange preferred stock for pre-funded warrants</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_ExerciseOfPreFundedWarrants" xlink:label="adgm_ExerciseOfPreFundedWarrants" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_ExerciseOfPreFundedWarrants" xlink:to="adgm_ExerciseOfPreFundedWarrants_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_ExerciseOfPreFundedWarrants_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Value of stock issued as a result of the exercise of pre-funded warrants.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_ExerciseOfPreFundedWarrants_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Exercise of Pre-Funded Warrants</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_ExerciseOfPreFundedWarrants_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Exercise of pre-funded warrants</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_StockIssuedDuringPeriodExercisesOfPreFundedWarrants" xlink:label="adgm_StockIssuedDuringPeriodExercisesOfPreFundedWarrants" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_StockIssuedDuringPeriodExercisesOfPreFundedWarrants" xlink:to="adgm_StockIssuedDuringPeriodExercisesOfPreFundedWarrants_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_StockIssuedDuringPeriodExercisesOfPreFundedWarrants_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of shares issued during the period from exercise of pre-funded warrants.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_StockIssuedDuringPeriodExercisesOfPreFundedWarrants_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Stock Issued During Period Exercises Of Pre-Funded Warrants</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_StockIssuedDuringPeriodExercisesOfPreFundedWarrants_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Exercise of pre-funded warrants (in shares)</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_WaiverSharesIssuedValueDuringPeriod" xlink:label="adgm_WaiverSharesIssuedValueDuringPeriod" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_WaiverSharesIssuedValueDuringPeriod" xlink:to="adgm_WaiverSharesIssuedValueDuringPeriod_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_WaiverSharesIssuedValueDuringPeriod_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Value of number of waiver stocks issued during the period.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_WaiverSharesIssuedValueDuringPeriod_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Waiver Shares Issued, Value, During Period</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_WaiverSharesIssuedValueDuringPeriod_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Issuance of Waiver Shares</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_WaiverSharesIssuedDuringPeriod" xlink:label="adgm_WaiverSharesIssuedDuringPeriod" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_WaiverSharesIssuedDuringPeriod" xlink:to="adgm_WaiverSharesIssuedDuringPeriod_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_WaiverSharesIssuedDuringPeriod_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of waiver stocks issued during the period.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_WaiverSharesIssuedDuringPeriod_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Waiver Shares Issued During Period</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_WaiverSharesIssuedDuringPeriod_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Issuance of Waiver Shares (in shares)</label>
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    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_StockIssuedDuringPeriodValueStockOptionsExercised" xlink:to="us-gaap_StockIssuedDuringPeriodValueStockOptionsExercised_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_StockIssuedDuringPeriodValueStockOptionsExercised_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Stock Issued During Period, Value, Stock Options Exercised</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_StockIssuedDuringPeriodValueStockOptionsExercised_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Stock option exercises</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_StockIssuedDuringPeriodSharesStockOptionsExercised" xlink:label="us-gaap_StockIssuedDuringPeriodSharesStockOptionsExercised" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_StockIssuedDuringPeriodSharesStockOptionsExercised" xlink:to="us-gaap_StockIssuedDuringPeriodSharesStockOptionsExercised_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_StockIssuedDuringPeriodSharesStockOptionsExercised_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Share-Based Compensation Arrangement by Share-Based Payment Award, Options, Exercises in Period</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_StockIssuedDuringPeriodSharesStockOptionsExercised_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Stock option exercises (in shares)</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_AdjustmentsToAdditionalPaidInCapitalSharebasedCompensationRequisiteServicePeriodRecognitionValue" xlink:label="us-gaap_AdjustmentsToAdditionalPaidInCapitalSharebasedCompensationRequisiteServicePeriodRecognitionValue" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_AdjustmentsToAdditionalPaidInCapitalSharebasedCompensationRequisiteServicePeriodRecognitionValue" xlink:to="us-gaap_AdjustmentsToAdditionalPaidInCapitalSharebasedCompensationRequisiteServicePeriodRecognitionValue_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_AdjustmentsToAdditionalPaidInCapitalSharebasedCompensationRequisiteServicePeriodRecognitionValue_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">APIC, Share-Based Payment Arrangement, Increase for Cost Recognition</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_AdjustmentsToAdditionalPaidInCapitalSharebasedCompensationRequisiteServicePeriodRecognitionValue_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Stock-based compensation</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_StockIssuedDuringPeriodValueIssuanceOfEarnOutShares" xlink:label="adgm_StockIssuedDuringPeriodValueIssuanceOfEarnOutShares" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_StockIssuedDuringPeriodValueIssuanceOfEarnOutShares" xlink:to="adgm_StockIssuedDuringPeriodValueIssuanceOfEarnOutShares_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_StockIssuedDuringPeriodValueIssuanceOfEarnOutShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Value of stock issued as a result of issuance of earn out shares.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_StockIssuedDuringPeriodValueIssuanceOfEarnOutShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Stock Issued During Period, Value, Issuance Of Earn Out Shares</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_StockIssuedDuringPeriodValueIssuanceOfEarnOutShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Issuance of Sponsor Earnout</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_StockIssuedDuringPeriodSharesIssuanceOfEarnOutShares" xlink:label="adgm_StockIssuedDuringPeriodSharesIssuanceOfEarnOutShares" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_StockIssuedDuringPeriodSharesIssuanceOfEarnOutShares" xlink:to="adgm_StockIssuedDuringPeriodSharesIssuanceOfEarnOutShares_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_StockIssuedDuringPeriodSharesIssuanceOfEarnOutShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of shares issued during the period pursuant to issuance of earn out shares.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_StockIssuedDuringPeriodSharesIssuanceOfEarnOutShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Stock Issued During Period, Shares, Issuance Of Earn Out Shares</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_StockIssuedDuringPeriodSharesIssuanceOfEarnOutShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Issuance of earn-out shares (in shares)</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_NetIncomeLoss" xlink:label="us-gaap_NetIncomeLoss" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_NetIncomeLoss" xlink:to="us-gaap_NetIncomeLoss_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:label="us-gaap_NetIncomeLoss_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Net loss</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_NetIncomeLoss_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Net (loss) income</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_PaymentsOfStockIssuanceCosts" xlink:label="us-gaap_PaymentsOfStockIssuanceCosts" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_PaymentsOfStockIssuanceCosts" xlink:to="us-gaap_PaymentsOfStockIssuanceCosts_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_PaymentsOfStockIssuanceCosts_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Payments of Stock Issuance Costs</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_PaymentsOfStockIssuanceCosts_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Issuance of common shares, pre-funded warrants and warrants issuance costs</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_StatementOfCashFlowsAbstract" xlink:label="us-gaap_StatementOfCashFlowsAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_StatementOfCashFlowsAbstract" xlink:to="us-gaap_StatementOfCashFlowsAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_StatementOfCashFlowsAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Consolidated Statements of Cash Flows</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/srt/2024/elts/srt-2024.xsd#srt_RestatementAxis" xlink:label="srt_RestatementAxis" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="srt_RestatementAxis" xlink:to="srt_RestatementAxis_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="srt_RestatementAxis_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Revision of Prior Period [Axis]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/srt/2024/elts/srt-2024.xsd#srt_RestatementDomain" xlink:label="srt_RestatementDomain" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="srt_RestatementDomain" xlink:to="srt_RestatementDomain_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="srt_RestatementDomain_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Revision of Prior Period [Domain]</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_PriorToEffectOfBusinessCombinationMember" xlink:label="adgm_PriorToEffectOfBusinessCombinationMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_PriorToEffectOfBusinessCombinationMember" xlink:to="adgm_PriorToEffectOfBusinessCombinationMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_PriorToEffectOfBusinessCombinationMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The information pertaining to prior to effect of business combination.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_PriorToEffectOfBusinessCombinationMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Prior to Effect of Business Combination [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_PriorToEffectOfBusinessCombinationMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Prior to Effect of Business Combination</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_NetCashProvidedByUsedInOperatingActivitiesAbstract" xlink:label="us-gaap_NetCashProvidedByUsedInOperatingActivitiesAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_NetCashProvidedByUsedInOperatingActivitiesAbstract" xlink:to="us-gaap_NetCashProvidedByUsedInOperatingActivitiesAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_NetCashProvidedByUsedInOperatingActivitiesAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Net Cash Provided by (Used in) Operating Activities [Abstract]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_NetCashProvidedByUsedInOperatingActivitiesAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cash flows from operating activities:</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ProfitLoss" xlink:label="us-gaap_ProfitLoss" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ProfitLoss" xlink:to="us-gaap_ProfitLoss_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ProfitLoss_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Net Income (Loss), Including Portion Attributable to Noncontrolling Interest</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_ProfitLoss_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Net loss</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_AdjustmentsToReconcileNetIncomeLossToCashProvidedByUsedInOperatingActivitiesAbstract" xlink:label="us-gaap_AdjustmentsToReconcileNetIncomeLossToCashProvidedByUsedInOperatingActivitiesAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_AdjustmentsToReconcileNetIncomeLossToCashProvidedByUsedInOperatingActivitiesAbstract" xlink:to="us-gaap_AdjustmentsToReconcileNetIncomeLossToCashProvidedByUsedInOperatingActivitiesAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_AdjustmentsToReconcileNetIncomeLossToCashProvidedByUsedInOperatingActivitiesAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Adjustments to Reconcile Net Income (Loss) to Cash Provided by (Used in) Operating Activities [Abstract]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_AdjustmentsToReconcileNetIncomeLossToCashProvidedByUsedInOperatingActivitiesAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Adjustments to reconcile net loss to net cash used in operating activities:</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_DepreciationDepletionAndAmortization" xlink:label="us-gaap_DepreciationDepletionAndAmortization" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_DepreciationDepletionAndAmortization" xlink:to="us-gaap_DepreciationDepletionAndAmortization_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_DepreciationDepletionAndAmortization_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Depreciation, Depletion and Amortization</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_DepreciationDepletionAndAmortization_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Depreciation and amortization</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_OperatingLeaseExpense" xlink:label="us-gaap_OperatingLeaseExpense" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_OperatingLeaseExpense" xlink:to="us-gaap_OperatingLeaseExpense_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_OperatingLeaseExpense_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Operating Lease, Expense</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_OperatingLeaseExpense_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Non-cash operating lease expense</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ShareBasedCompensation" xlink:label="us-gaap_ShareBasedCompensation" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ShareBasedCompensation" xlink:to="us-gaap_ShareBasedCompensation_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ShareBasedCompensation_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Share-Based Payment Arrangement, Noncash Expense</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ShareBasedCompensation_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Stock-based compensation</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_InventoryWriteDown" xlink:label="us-gaap_InventoryWriteDown" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_InventoryWriteDown" xlink:to="us-gaap_InventoryWriteDown_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_InventoryWriteDown_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Inventory Write-down</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_InventoryWriteDown_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Provision for inventory impairment</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_AmortizationOfDebtDiscountPremium" xlink:label="us-gaap_AmortizationOfDebtDiscountPremium" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_AmortizationOfDebtDiscountPremium" xlink:to="us-gaap_AmortizationOfDebtDiscountPremium_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_AmortizationOfDebtDiscountPremium_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amortization of Debt Discount (Premium)</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_AmortizationOfDebtDiscountPremium_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amortization of term loan discount</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_GainLossOnSaleOfPropertyPlantEquipment" xlink:label="us-gaap_GainLossOnSaleOfPropertyPlantEquipment" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_GainLossOnSaleOfPropertyPlantEquipment" xlink:to="us-gaap_GainLossOnSaleOfPropertyPlantEquipment_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_GainLossOnSaleOfPropertyPlantEquipment_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Gain (Loss) on Disposition of Property Plant Equipment</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2009/role/negatedLabel" xlink:label="us-gaap_GainLossOnSaleOfPropertyPlantEquipment_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Loss on disposal of property and equipment</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_FairValueAdjustmentOfConvertibleNotes" xlink:label="adgm_FairValueAdjustmentOfConvertibleNotes" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_FairValueAdjustmentOfConvertibleNotes" xlink:to="adgm_FairValueAdjustmentOfConvertibleNotes_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_FairValueAdjustmentOfConvertibleNotes_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amount of expense (income) related to adjustment to fair value of convertible notes.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_FairValueAdjustmentOfConvertibleNotes_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fair Value Adjustment of Convertible Notes</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="adgm_FairValueAdjustmentOfConvertibleNotes_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Change in fair value of convertible notes payable</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_FairValueAdjustmentOfWarrants" xlink:label="us-gaap_FairValueAdjustmentOfWarrants" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_FairValueAdjustmentOfWarrants" xlink:to="us-gaap_FairValueAdjustmentOfWarrants_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_FairValueAdjustmentOfWarrants_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fair Value Adjustment of Warrants</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_FairValueAdjustmentOfWarrants_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Warrant liabilities fair value adjustment</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_FairValueAdjustmentOfWarrants_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Change in fair value of warrant liabilities</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_IncreaseDecreaseInOperatingCapitalAbstract" xlink:label="us-gaap_IncreaseDecreaseInOperatingCapitalAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_IncreaseDecreaseInOperatingCapitalAbstract" xlink:to="us-gaap_IncreaseDecreaseInOperatingCapitalAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_IncreaseDecreaseInOperatingCapitalAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Increase (Decrease) in Operating Capital [Abstract]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_IncreaseDecreaseInOperatingCapitalAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Net change in operating assets and liabilities:</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_IncreaseDecreaseInAccountsReceivable" xlink:label="us-gaap_IncreaseDecreaseInAccountsReceivable" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_IncreaseDecreaseInAccountsReceivable" xlink:to="us-gaap_IncreaseDecreaseInAccountsReceivable_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_IncreaseDecreaseInAccountsReceivable_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Increase (Decrease) in Accounts Receivable</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2009/role/negatedLabel" xlink:label="us-gaap_IncreaseDecreaseInAccountsReceivable_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Accounts receivable, net</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_IncreaseDecreaseInInventories" xlink:label="us-gaap_IncreaseDecreaseInInventories" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_IncreaseDecreaseInInventories" xlink:to="us-gaap_IncreaseDecreaseInInventories_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_IncreaseDecreaseInInventories_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Increase (Decrease) in Inventories</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2009/role/negatedLabel" xlink:label="us-gaap_IncreaseDecreaseInInventories_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Inventory, net</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_IncreaseDecreaseInPrepaidDeferredExpenseAndOtherAssets" xlink:label="us-gaap_IncreaseDecreaseInPrepaidDeferredExpenseAndOtherAssets" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_IncreaseDecreaseInPrepaidDeferredExpenseAndOtherAssets" xlink:to="us-gaap_IncreaseDecreaseInPrepaidDeferredExpenseAndOtherAssets_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_IncreaseDecreaseInPrepaidDeferredExpenseAndOtherAssets_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Increase (Decrease) in Prepaid Expense and Other Assets</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2009/role/negatedLabel" xlink:label="us-gaap_IncreaseDecreaseInPrepaidDeferredExpenseAndOtherAssets_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Prepaid expenses and other current assets</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_IncreaseDecreaseInAccountsPayable" xlink:label="us-gaap_IncreaseDecreaseInAccountsPayable" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_IncreaseDecreaseInAccountsPayable" xlink:to="us-gaap_IncreaseDecreaseInAccountsPayable_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_IncreaseDecreaseInAccountsPayable_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Increase (Decrease) in Accounts Payable</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_IncreaseDecreaseInAccountsPayable_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Accounts payable</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_IncreaseDecreaseInAccruedLiabilities" xlink:label="us-gaap_IncreaseDecreaseInAccruedLiabilities" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_IncreaseDecreaseInAccruedLiabilities" xlink:to="us-gaap_IncreaseDecreaseInAccruedLiabilities_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_IncreaseDecreaseInAccruedLiabilities_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Increase (Decrease) in Accrued Liabilities</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_IncreaseDecreaseInAccruedLiabilities_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Accrued liabilities</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_IncreaseDecreaseInAccruedTransactionCostsCurrent" xlink:label="adgm_IncreaseDecreaseInAccruedTransactionCostsCurrent" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_IncreaseDecreaseInAccruedTransactionCostsCurrent" xlink:to="adgm_IncreaseDecreaseInAccruedTransactionCostsCurrent_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_IncreaseDecreaseInAccruedTransactionCostsCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The increase (decrease) during the reporting period in the aggregate amount of business combination transaction costs incurred but not yet paid.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_IncreaseDecreaseInAccruedTransactionCostsCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Increase (Decrease) in Accrued Transaction Costs, Current</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_IncreaseDecreaseInAccruedTransactionCostsCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Accrued transaction costs</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_IncreaseDecreaseInOtherAccruedLiabilitiesCurrent" xlink:label="adgm_IncreaseDecreaseInOtherAccruedLiabilitiesCurrent" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_IncreaseDecreaseInOtherAccruedLiabilitiesCurrent" xlink:to="adgm_IncreaseDecreaseInOtherAccruedLiabilitiesCurrent_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_IncreaseDecreaseInOtherAccruedLiabilitiesCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The increase (decrease) during the reporting period in other expenses incurred but not yet paid.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_IncreaseDecreaseInOtherAccruedLiabilitiesCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Increase (Decrease) in Other Accrued Liabilities, Current</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_IncreaseDecreaseInOtherAccruedLiabilitiesCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Other accrued liabilities</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_IncreaseDecreaseInOperatingLeaseLiability" xlink:label="us-gaap_IncreaseDecreaseInOperatingLeaseLiability" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_IncreaseDecreaseInOperatingLeaseLiability" xlink:to="us-gaap_IncreaseDecreaseInOperatingLeaseLiability_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_IncreaseDecreaseInOperatingLeaseLiability_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Increase (Decrease) in Operating Lease Liability</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_IncreaseDecreaseInOperatingLeaseLiability_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Operating lease liabilities</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_IncreaseDecreaseInOtherNoncurrentLiabilities" xlink:label="us-gaap_IncreaseDecreaseInOtherNoncurrentLiabilities" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_IncreaseDecreaseInOtherNoncurrentLiabilities" xlink:to="us-gaap_IncreaseDecreaseInOtherNoncurrentLiabilities_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_IncreaseDecreaseInOtherNoncurrentLiabilities_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Increase (Decrease) in Other Noncurrent Liabilities</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_IncreaseDecreaseInOtherNoncurrentLiabilities_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Other long-term liabilities</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_IncreaseDecreaseInDeferredIncomeTaxes" xlink:label="us-gaap_IncreaseDecreaseInDeferredIncomeTaxes" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_IncreaseDecreaseInDeferredIncomeTaxes" xlink:to="us-gaap_IncreaseDecreaseInDeferredIncomeTaxes_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_IncreaseDecreaseInDeferredIncomeTaxes_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Increase (Decrease) in Deferred Income Taxes</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2009/role/negatedLabel" xlink:label="us-gaap_IncreaseDecreaseInDeferredIncomeTaxes_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Deferred taxes</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_NetCashProvidedByUsedInOperatingActivities" xlink:label="us-gaap_NetCashProvidedByUsedInOperatingActivities" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_NetCashProvidedByUsedInOperatingActivities" xlink:to="us-gaap_NetCashProvidedByUsedInOperatingActivities_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_NetCashProvidedByUsedInOperatingActivities_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Net Cash Provided by (Used in) Operating Activities</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_NetCashProvidedByUsedInOperatingActivities_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Net cash used in operating activities</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:label="us-gaap_NetCashProvidedByUsedInOperatingActivities_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Net cash used in operating activities</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_NetCashProvidedByUsedInInvestingActivitiesAbstract" xlink:label="us-gaap_NetCashProvidedByUsedInInvestingActivitiesAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_NetCashProvidedByUsedInInvestingActivitiesAbstract" xlink:to="us-gaap_NetCashProvidedByUsedInInvestingActivitiesAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_NetCashProvidedByUsedInInvestingActivitiesAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Net Cash Provided by (Used in) Investing Activities [Abstract]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_NetCashProvidedByUsedInInvestingActivitiesAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cash flows from investing activities:</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_PaymentsToAcquirePropertyPlantAndEquipment" xlink:label="us-gaap_PaymentsToAcquirePropertyPlantAndEquipment" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_PaymentsToAcquirePropertyPlantAndEquipment" xlink:to="us-gaap_PaymentsToAcquirePropertyPlantAndEquipment_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_PaymentsToAcquirePropertyPlantAndEquipment_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Payments to Acquire Property, Plant, and Equipment</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2009/role/negatedLabel" xlink:label="us-gaap_PaymentsToAcquirePropertyPlantAndEquipment_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Purchases of property and equipment</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_NetCashProvidedByUsedInInvestingActivities" xlink:label="us-gaap_NetCashProvidedByUsedInInvestingActivities" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_NetCashProvidedByUsedInInvestingActivities" xlink:to="us-gaap_NetCashProvidedByUsedInInvestingActivities_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_NetCashProvidedByUsedInInvestingActivities_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Net Cash Provided by (Used in) Investing Activities</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:label="us-gaap_NetCashProvidedByUsedInInvestingActivities_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Net cash used in investing activities</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_NetCashProvidedByUsedInFinancingActivitiesAbstract" xlink:label="us-gaap_NetCashProvidedByUsedInFinancingActivitiesAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_NetCashProvidedByUsedInFinancingActivitiesAbstract" xlink:to="us-gaap_NetCashProvidedByUsedInFinancingActivitiesAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_NetCashProvidedByUsedInFinancingActivitiesAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Net Cash Provided by (Used in) Financing Activities [Abstract]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_NetCashProvidedByUsedInFinancingActivitiesAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cash flows from financing activities:</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ProceedsFromIssuanceOfCommonStock" xlink:label="us-gaap_ProceedsFromIssuanceOfCommonStock" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ProceedsFromIssuanceOfCommonStock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Proceeds from Issuance of Common Stock</label>
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_NetCashProvidedByUsedInFinancingActivities_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Net Cash Provided by (Used in) Financing Activities</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:label="us-gaap_NetCashProvidedByUsedInFinancingActivities_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Net cash provided by financing activities</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_EffectOfExchangeRateOnCashCashEquivalentsRestrictedCashAndRestrictedCashEquivalentsIncludingDisposalGroupAndDiscontinuedOperations" xlink:label="us-gaap_EffectOfExchangeRateOnCashCashEquivalentsRestrictedCashAndRestrictedCashEquivalentsIncludingDisposalGroupAndDiscontinuedOperations" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalentsPeriodIncreaseDecreaseIncludingExchangeRateEffect" xlink:label="us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalentsPeriodIncreaseDecreaseIncludingExchangeRateEffect" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalentsPeriodIncreaseDecreaseIncludingExchangeRateEffect" xlink:to="us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalentsPeriodIncreaseDecreaseIncludingExchangeRateEffect_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalentsPeriodIncreaseDecreaseIncludingExchangeRateEffect_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cash, Cash Equivalents, Restricted Cash, and Restricted Cash Equivalents, Period Increase (Decrease), Including Exchange Rate Effect</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:label="us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalentsPeriodIncreaseDecreaseIncludingExchangeRateEffect_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Net change in cash and cash equivalents</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalents" xlink:label="us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalents" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalents" xlink:to="us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalents_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalents_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cash, Cash Equivalents, Restricted Cash, and Restricted Cash Equivalents</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/periodEndLabel" xlink:label="us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalents_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cash and cash equivalents, at end of period</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/periodStartLabel" xlink:label="us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalents_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cash and cash equivalents, at beginning of period</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_SupplementalCashFlowInformationAbstract" xlink:label="us-gaap_SupplementalCashFlowInformationAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_SupplementalCashFlowInformationAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Supplemental Cash Flow Information [Abstract]</label>
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    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_InterestPaidNet" xlink:to="us-gaap_InterestPaidNet_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_InterestPaidNet_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Interest Paid, Excluding Capitalized Interest, Operating Activities</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_InterestPaidNet_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cash paid for interest</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_NoncashInvestingAndFinancingItemsAbstract" xlink:label="us-gaap_NoncashInvestingAndFinancingItemsAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_NoncashInvestingAndFinancingItemsAbstract" xlink:to="us-gaap_NoncashInvestingAndFinancingItemsAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_NoncashInvestingAndFinancingItemsAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Noncash Investing and Financing Items [Abstract]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_NoncashInvestingAndFinancingItemsAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Supplemental disclosure of noncash investing and financing activities:</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_RightOfUseAssetObtainedInExchangeForOperatingLeaseLiability" xlink:label="us-gaap_RightOfUseAssetObtainedInExchangeForOperatingLeaseLiability" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_RightOfUseAssetObtainedInExchangeForOperatingLeaseLiability" xlink:to="us-gaap_RightOfUseAssetObtainedInExchangeForOperatingLeaseLiability_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_RightOfUseAssetObtainedInExchangeForOperatingLeaseLiability_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Right-of-Use Asset Obtained in Exchange for Operating Lease Liability</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2009/role/negatedLabel" xlink:label="us-gaap_RightOfUseAssetObtainedInExchangeForOperatingLeaseLiability_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Right-of-use assets obtained in exchange for lease liabilities</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_OperatingLeaseLiabilityRecordedForRightOfUseAssets" xlink:label="adgm_OperatingLeaseLiabilityRecordedForRightOfUseAssets" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_OperatingLeaseLiabilityRecordedForRightOfUseAssets" xlink:to="adgm_OperatingLeaseLiabilityRecordedForRightOfUseAssets_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_OperatingLeaseLiabilityRecordedForRightOfUseAssets_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amount of increase in the operating lease liability upon recognition of right of use assets.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_OperatingLeaseLiabilityRecordedForRightOfUseAssets_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Operating Lease Liability Recorded For Right Of Use Assets</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_OperatingLeaseLiabilityRecordedForRightOfUseAssets_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Lease liabilities recorded for operating lease right-of-use assets</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_PrefundedWarrantsIssuedInExchangeOfConvertiblePreferredStock" xlink:label="adgm_PrefundedWarrantsIssuedInExchangeOfConvertiblePreferredStock" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_PrefundedWarrantsIssuedInExchangeOfConvertiblePreferredStock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Value of warrants issued in exchange of convertible preferred stock classified as temporary equity in a noncash or part noncash transaction. Noncash is defined as transactions during a period that do not result in cash receipts or cash payments in the period. "Part noncash" refers to that portion of a transaction not resulting in cash receipts or cash payments in the period.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_PrefundedWarrantsIssuedInExchangeOfConvertiblePreferredStock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Prefunded Warrants, Issued In Exchange Of Convertible Preferred Stock</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_PrefundedWarrantsIssuedInExchangeOfConvertiblePreferredStock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Exchange preferred stock for pre-funded warrants</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_OrganizationConsolidationAndPresentationOfFinancialStatementsAbstract" xlink:label="us-gaap_OrganizationConsolidationAndPresentationOfFinancialStatementsAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_OrganizationConsolidationAndPresentationOfFinancialStatementsDisclosureTextBlock" xlink:label="us-gaap_OrganizationConsolidationAndPresentationOfFinancialStatementsDisclosureTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_AccountingPoliciesAbstract" xlink:label="us-gaap_AccountingPoliciesAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_AccountingPoliciesAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Summary of Significant Accounting Policies</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_SignificantAccountingPoliciesTextBlock" xlink:label="us-gaap_SignificantAccountingPoliciesTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_SignificantAccountingPoliciesTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Summary of Significant Accounting Policies</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_BusinessCombinationsAbstract" xlink:label="us-gaap_BusinessCombinationsAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_BusinessCombinationDisclosureTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Forward Merger</label>
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    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_FairValueDisclosuresTextBlock" xlink:label="us-gaap_FairValueDisclosuresTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_FairValueDisclosuresTextBlock" xlink:to="us-gaap_FairValueDisclosuresTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_FairValueDisclosuresTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fair Value Disclosures [Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_FairValueDisclosuresTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fair Value Measurements</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_InventoryDisclosureAbstract" xlink:label="us-gaap_InventoryDisclosureAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_InventoryDisclosureAbstract" xlink:to="us-gaap_InventoryDisclosureAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_InventoryDisclosureAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Inventory, net.</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_InventoryDisclosureTextBlock" xlink:label="us-gaap_InventoryDisclosureTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_InventoryDisclosureTextBlock" xlink:to="us-gaap_InventoryDisclosureTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_InventoryDisclosureTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Inventory Disclosure [Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_InventoryDisclosureTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Inventory, net</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_PropertyPlantAndEquipmentAbstract" xlink:label="us-gaap_PropertyPlantAndEquipmentAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_PropertyPlantAndEquipmentAbstract" xlink:to="us-gaap_PropertyPlantAndEquipmentAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_PropertyPlantAndEquipmentAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Property and Equipment</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_PropertyPlantAndEquipmentDisclosureTextBlock" xlink:label="us-gaap_PropertyPlantAndEquipmentDisclosureTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_PropertyPlantAndEquipmentDisclosureTextBlock" xlink:to="us-gaap_PropertyPlantAndEquipmentDisclosureTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_PropertyPlantAndEquipmentDisclosureTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Property, Plant and Equipment Disclosure [Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_PropertyPlantAndEquipmentDisclosureTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Property and Equipment</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_GoodwillAndIntangibleAssetsDisclosureAbstract" xlink:label="us-gaap_GoodwillAndIntangibleAssetsDisclosureAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_GoodwillAndIntangibleAssetsDisclosureAbstract" xlink:to="us-gaap_GoodwillAndIntangibleAssetsDisclosureAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_GoodwillAndIntangibleAssetsDisclosureAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Goodwill and Intangible Assets</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_GoodwillAndIntangibleAssetsDisclosureTextBlock" xlink:label="us-gaap_GoodwillAndIntangibleAssetsDisclosureTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_GoodwillAndIntangibleAssetsDisclosureTextBlock" xlink:to="us-gaap_GoodwillAndIntangibleAssetsDisclosureTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_GoodwillAndIntangibleAssetsDisclosureTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Goodwill and Intangible Assets Disclosure [Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_GoodwillAndIntangibleAssetsDisclosureTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Goodwill and Intangible Assets</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_AccruedLiabilitiesCurrentAbstract" xlink:label="us-gaap_AccruedLiabilitiesCurrentAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_AccruedLiabilitiesCurrentAbstract" xlink:to="us-gaap_AccruedLiabilitiesCurrentAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_AccruedLiabilitiesCurrentAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Accrued Liabilities</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_AccruedLiabilitiesDisclosureTextBlock" xlink:label="adgm_AccruedLiabilitiesDisclosureTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_AccruedLiabilitiesDisclosureTextBlock" xlink:to="adgm_AccruedLiabilitiesDisclosureTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_AccruedLiabilitiesDisclosureTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The entire disclosure for accrued liabilities at the end of the reporting period.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_AccruedLiabilitiesDisclosureTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Accrued Liabilities Disclosure [Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_AccruedLiabilitiesDisclosureTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Accrued Liabilities</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_DebtDisclosureAbstract" xlink:label="us-gaap_DebtDisclosureAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_DebtDisclosureAbstract" xlink:to="us-gaap_DebtDisclosureAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_DebtDisclosureAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Debt</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_DebtDisclosureTextBlock" xlink:label="us-gaap_DebtDisclosureTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_DebtDisclosureTextBlock" xlink:to="us-gaap_DebtDisclosureTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_DebtDisclosureTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Debt Disclosure [Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_DebtDisclosureTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Debt</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_WarrantsAndRightsNoteDisclosureAbstract" xlink:label="us-gaap_WarrantsAndRightsNoteDisclosureAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_WarrantsAndRightsNoteDisclosureAbstract" xlink:to="us-gaap_WarrantsAndRightsNoteDisclosureAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_WarrantsAndRightsNoteDisclosureAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Warrants</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_WarrantsAndRightsNoteDisclosureTextBlock" xlink:label="adgm_WarrantsAndRightsNoteDisclosureTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_WarrantsAndRightsNoteDisclosureTextBlock" xlink:to="adgm_WarrantsAndRightsNoteDisclosureTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_WarrantsAndRightsNoteDisclosureTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The entire disclosure of warrants and rights.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_WarrantsAndRightsNoteDisclosureTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Warrants and Rights Note Disclosure [Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_WarrantsAndRightsNoteDisclosureTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Warrants</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_LeasesAbstract" xlink:label="us-gaap_LeasesAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_LeasesAbstract" xlink:to="us-gaap_LeasesAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_LeasesAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Operating Leases</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_LesseeOperatingLeasesTextBlock" xlink:label="us-gaap_LesseeOperatingLeasesTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_LesseeOperatingLeasesTextBlock" xlink:to="us-gaap_LesseeOperatingLeasesTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_LesseeOperatingLeasesTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Lessee, Operating Leases [Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_LesseeOperatingLeasesTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Operating Leases</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_CommitmentsAndContingenciesDisclosureAbstract" xlink:label="us-gaap_CommitmentsAndContingenciesDisclosureAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_CommitmentsAndContingenciesDisclosureAbstract" xlink:to="us-gaap_CommitmentsAndContingenciesDisclosureAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_CommitmentsAndContingenciesDisclosureAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Commitments and Contingencies.</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_CommitmentsAndContingenciesDisclosureTextBlock" xlink:label="us-gaap_CommitmentsAndContingenciesDisclosureTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_CommitmentsAndContingenciesDisclosureTextBlock" xlink:to="us-gaap_CommitmentsAndContingenciesDisclosureTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_CommitmentsAndContingenciesDisclosureTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Commitments and Contingencies Disclosure [Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_CommitmentsAndContingenciesDisclosureTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Commitments and Contingencies</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_StockholderSEquityAndTemporaryEquityAbstract" xlink:label="adgm_StockholderSEquityAndTemporaryEquityAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_StockholderSEquityAndTemporaryEquityAbstract" xlink:to="adgm_StockholderSEquityAndTemporaryEquityAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_StockholderSEquityAndTemporaryEquityAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">No definition available.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_StockholderSEquityAndTemporaryEquityAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Mezzanine Equity and Stockholders' Equity (Deficit)</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_StockholderSEquityAndTemporaryEquityDisclosureTextBlock" xlink:label="adgm_StockholderSEquityAndTemporaryEquityDisclosureTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_StockholderSEquityAndTemporaryEquityDisclosureTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The entire disclosure for equity and temporary equity.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_StockholderSEquityAndTemporaryEquityDisclosureTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Stockholder's Equity And Temporary Equity Disclosure [Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_StockholderSEquityAndTemporaryEquityDisclosureTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Mezzanine Equity and Stockholders' Equity (Deficit)</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_DisclosureOfCompensationRelatedCostsSharebasedPaymentsAbstract" xlink:label="us-gaap_DisclosureOfCompensationRelatedCostsSharebasedPaymentsAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_DisclosureOfCompensationRelatedCostsSharebasedPaymentsAbstract" xlink:to="us-gaap_DisclosureOfCompensationRelatedCostsSharebasedPaymentsAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_DisclosureOfCompensationRelatedCostsSharebasedPaymentsAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Stock-Based Compensation</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_DisclosureOfCompensationRelatedCostsShareBasedPaymentsTextBlock" xlink:label="us-gaap_DisclosureOfCompensationRelatedCostsShareBasedPaymentsTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_DisclosureOfCompensationRelatedCostsShareBasedPaymentsTextBlock" xlink:to="us-gaap_DisclosureOfCompensationRelatedCostsShareBasedPaymentsTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_DisclosureOfCompensationRelatedCostsShareBasedPaymentsTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Share-Based Payment Arrangement [Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_DisclosureOfCompensationRelatedCostsShareBasedPaymentsTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Stock-Based Compensation</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_EarningsPerShareAbstract" xlink:label="us-gaap_EarningsPerShareAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_EarningsPerShareAbstract" xlink:to="us-gaap_EarningsPerShareAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_RelatedPartyTransactionsAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Related Party Transactions</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_RelatedPartyTransactionsDisclosureTextBlock" xlink:label="us-gaap_RelatedPartyTransactionsDisclosureTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_RelatedPartyTransactionsDisclosureTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Related Party Transactions</label>
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    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_SegmentReportingAbstract" xlink:to="us-gaap_SegmentReportingAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_SegmentReportingAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Segment Reporting</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_SegmentReportingDisclosureTextBlock" xlink:label="us-gaap_SegmentReportingDisclosureTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_SegmentReportingDisclosureTextBlock" xlink:to="us-gaap_SegmentReportingDisclosureTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_SegmentReportingDisclosureTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Segment Reporting Disclosure [Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_SegmentReportingDisclosureTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Segment Reporting</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_SubsequentEventsAbstract" xlink:label="us-gaap_SubsequentEventsAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_SubsequentEventsAbstract" xlink:to="us-gaap_SubsequentEventsAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_SubsequentEventsAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Subsequent Events</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_SubsequentEventsTextBlock" xlink:label="us-gaap_SubsequentEventsTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_SubsequentEventsTextBlock" xlink:to="us-gaap_SubsequentEventsTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_SubsequentEventsTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Subsequent Events [Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_SubsequentEventsTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Subsequent Events</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_BasisOfAccountingPolicyPolicyTextBlock" xlink:label="us-gaap_BasisOfAccountingPolicyPolicyTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_BasisOfAccountingPolicyPolicyTextBlock" xlink:to="us-gaap_BasisOfAccountingPolicyPolicyTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_BasisOfAccountingPolicyPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Basis of Accounting, Policy [Policy Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_BasisOfAccountingPolicyPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Basis of Presentation</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_EmergingGrowthCompanyPolicyTextBlock" xlink:label="adgm_EmergingGrowthCompanyPolicyTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_EmergingGrowthCompanyPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Disclosure of accounting policy for emerging growth company.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_EmergingGrowthCompanyPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Emerging Growth Company [Policy Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_EmergingGrowthCompanyPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Emerging Growth Company Status</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ConsolidationPolicyTextBlock" xlink:label="us-gaap_ConsolidationPolicyTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ConsolidationPolicyTextBlock" xlink:to="us-gaap_ConsolidationPolicyTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ConsolidationPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Consolidation, Policy [Policy Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ConsolidationPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Principles of Consolidation</label>
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    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_UseOfEstimates" xlink:to="us-gaap_UseOfEstimates_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_UseOfEstimates_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Use of Estimates, Policy [Policy Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_UseOfEstimates_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Use of Estimates and Assumptions</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_SegmentReportingPolicyPolicyTextBlock" xlink:label="us-gaap_SegmentReportingPolicyPolicyTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_SegmentReportingPolicyPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Segment Reporting, Policy [Policy Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_SegmentReportingPolicyPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Segments</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_CashAndCashEquivalentsPolicyTextBlock" xlink:label="us-gaap_CashAndCashEquivalentsPolicyTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_CashAndCashEquivalentsPolicyTextBlock" xlink:to="us-gaap_CashAndCashEquivalentsPolicyTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_CashAndCashEquivalentsPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cash and Cash Equivalents, Policy [Policy Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_CashAndCashEquivalentsPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cash and Cash Equivalents</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_ConcentrationsOfCreditRiskAndOffBalanceSheetRiskPolicyTextBlock" xlink:label="adgm_ConcentrationsOfCreditRiskAndOffBalanceSheetRiskPolicyTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_ConcentrationsOfCreditRiskAndOffBalanceSheetRiskPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Disclosure of accounting policy for credit risks and off balance sheet risks.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_ConcentrationsOfCreditRiskAndOffBalanceSheetRiskPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Concentrations of Credit Risk and Off-Balance Sheet Risk [Policy Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_ConcentrationsOfCreditRiskAndOffBalanceSheetRiskPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Concentrations of Credit Risk and Off-Balance Sheet Risk</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_RevenueFromContractWithCustomerPolicyTextBlock" xlink:label="us-gaap_RevenueFromContractWithCustomerPolicyTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_RevenueFromContractWithCustomerPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Revenue Recognition</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_InventoryPolicyTextBlock" xlink:label="us-gaap_InventoryPolicyTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_InventoryPolicyTextBlock" xlink:to="us-gaap_InventoryPolicyTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_InventoryPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Inventory, Policy [Policy Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_InventoryPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Inventory</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_PropertyPlantAndEquipmentPolicyTextBlock" xlink:label="us-gaap_PropertyPlantAndEquipmentPolicyTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_PropertyPlantAndEquipmentPolicyTextBlock" xlink:to="us-gaap_PropertyPlantAndEquipmentPolicyTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_PropertyPlantAndEquipmentPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Property, Plant and Equipment, Policy [Policy Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_PropertyPlantAndEquipmentPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Property and Equipment</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_GoodwillAndIntangibleAssetsIntangibleAssetsPolicy" xlink:label="us-gaap_GoodwillAndIntangibleAssetsIntangibleAssetsPolicy" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_GoodwillAndIntangibleAssetsIntangibleAssetsPolicy" xlink:to="us-gaap_GoodwillAndIntangibleAssetsIntangibleAssetsPolicy_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_GoodwillAndIntangibleAssetsIntangibleAssetsPolicy_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Goodwill and Intangible Assets, Intangible Assets, Policy [Policy Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_GoodwillAndIntangibleAssetsIntangibleAssetsPolicy_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Intangible Assets</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_GoodwillAndIntangibleAssetsGoodwillPolicy" xlink:label="us-gaap_GoodwillAndIntangibleAssetsGoodwillPolicy" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_GoodwillAndIntangibleAssetsGoodwillPolicy_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Goodwill and Intangible Assets, Goodwill, Policy [Policy Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_GoodwillAndIntangibleAssetsGoodwillPolicy_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Goodwill, net</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ConcentrationRiskCreditRisk" xlink:label="us-gaap_ConcentrationRiskCreditRisk" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ConcentrationRiskCreditRisk_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Concentration Risk, Credit Risk, Policy [Policy Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ConcentrationRiskCreditRisk_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Concentrations</label>
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    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ImpairmentOrDisposalOfLongLivedAssetsPolicyTextBlock" xlink:to="us-gaap_ImpairmentOrDisposalOfLongLivedAssetsPolicyTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ImpairmentOrDisposalOfLongLivedAssetsPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Impairment or Disposal of Long-Lived Assets, Policy [Policy Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ImpairmentOrDisposalOfLongLivedAssetsPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Impairment of Long-Lived Assets</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ForeignCurrencyTransactionsAndTranslationsPolicyTextBlock" xlink:label="us-gaap_ForeignCurrencyTransactionsAndTranslationsPolicyTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ForeignCurrencyTransactionsAndTranslationsPolicyTextBlock" xlink:to="us-gaap_ForeignCurrencyTransactionsAndTranslationsPolicyTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ForeignCurrencyTransactionsAndTranslationsPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Foreign Currency Translation and Transactions</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_LesseeLeasesPolicyTextBlock" xlink:label="us-gaap_LesseeLeasesPolicyTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_LesseeLeasesPolicyTextBlock" xlink:to="us-gaap_LesseeLeasesPolicyTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_LesseeLeasesPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Leases</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_CostOfSalesPolicyTextBlock" xlink:label="us-gaap_CostOfSalesPolicyTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_CostOfSalesPolicyTextBlock" xlink:to="us-gaap_CostOfSalesPolicyTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_CostOfSalesPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cost of Goods and Service [Policy Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_CostOfSalesPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cost of Revenue</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ResearchAndDevelopmentExpensePolicy" xlink:label="us-gaap_ResearchAndDevelopmentExpensePolicy" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ResearchAndDevelopmentExpensePolicy" xlink:to="us-gaap_ResearchAndDevelopmentExpensePolicy_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ResearchAndDevelopmentExpensePolicy_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Research and Development</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_SellingGeneralAndAdministrativeExpensesPolicyTextBlock" xlink:label="us-gaap_SellingGeneralAndAdministrativeExpensesPolicyTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_SellingGeneralAndAdministrativeExpensesPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Selling, General and Administrative</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_FairValueMeasurementPolicyPolicyTextBlock" xlink:label="us-gaap_FairValueMeasurementPolicyPolicyTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_FairValueMeasurementPolicyPolicyTextBlock" xlink:to="us-gaap_FairValueMeasurementPolicyPolicyTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_FairValueMeasurementPolicyPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fair Value Measurement, Policy [Policy Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_FairValueMeasurementPolicyPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fair Value Measurements</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_FairValueOfFinancialInstrumentsPolicy" xlink:label="us-gaap_FairValueOfFinancialInstrumentsPolicy" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_FairValueOfFinancialInstrumentsPolicy" xlink:to="us-gaap_FairValueOfFinancialInstrumentsPolicy_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_FairValueOfFinancialInstrumentsPolicy_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fair Value of Financial Instruments, Policy [Policy Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_FairValueOfFinancialInstrumentsPolicy_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fair Value Option for Convertible Notes</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_WarrantsPolicyTextBlock" xlink:label="adgm_WarrantsPolicyTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_WarrantsPolicyTextBlock" xlink:to="adgm_WarrantsPolicyTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_WarrantsPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Disclosure of accounting policy for warrants.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_WarrantsPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Warrants [Policy Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_WarrantsPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Warrants</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_DebtPolicyTextBlock" xlink:label="us-gaap_DebtPolicyTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_DebtPolicyTextBlock" xlink:to="us-gaap_DebtPolicyTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_DebtPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Debt, Policy [Policy Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_DebtPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Term Loan (Predecessor)</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_TemporaryEquityPolicyTextBlock" xlink:label="adgm_TemporaryEquityPolicyTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_TemporaryEquityPolicyTextBlock" xlink:to="adgm_TemporaryEquityPolicyTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_TemporaryEquityPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Disclosure of accounting policy for temporary equity.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_TemporaryEquityPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Temporary Equity [Policy Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_TemporaryEquityPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Convertible Preferred Stock (Predecessor)</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ShareBasedCompensationOptionAndIncentivePlansPolicy" xlink:label="us-gaap_ShareBasedCompensationOptionAndIncentivePlansPolicy" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ShareBasedCompensationOptionAndIncentivePlansPolicy" xlink:to="us-gaap_ShareBasedCompensationOptionAndIncentivePlansPolicy_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ShareBasedCompensationOptionAndIncentivePlansPolicy_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Share-Based Payment Arrangement [Policy Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ShareBasedCompensationOptionAndIncentivePlansPolicy_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Stock-Based Compensation</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_IncomeTaxPolicyTextBlock" xlink:label="us-gaap_IncomeTaxPolicyTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_IncomeTaxPolicyTextBlock" xlink:to="us-gaap_IncomeTaxPolicyTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_IncomeTaxPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Income Tax, Policy [Policy Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_IncomeTaxPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Income Taxes</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_NewAccountingPronouncementsPolicyPolicyTextBlock" xlink:label="us-gaap_NewAccountingPronouncementsPolicyPolicyTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_NewAccountingPronouncementsPolicyPolicyTextBlock" xlink:to="us-gaap_NewAccountingPronouncementsPolicyPolicyTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_NewAccountingPronouncementsPolicyPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">New Accounting Pronouncements, Policy [Policy Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_NewAccountingPronouncementsPolicyPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Recent Accounting Pronouncements</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ScheduleOfBusinessAcquisitionsByAcquisitionTextBlock" xlink:label="us-gaap_ScheduleOfBusinessAcquisitionsByAcquisitionTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ScheduleOfBusinessAcquisitionsByAcquisitionTextBlock" xlink:to="us-gaap_ScheduleOfBusinessAcquisitionsByAcquisitionTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ScheduleOfBusinessAcquisitionsByAcquisitionTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Schedule of Business Acquisitions, by Acquisition [Table Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ScheduleOfBusinessAcquisitionsByAcquisitionTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Summary of the purchase price calculation</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ScheduleOfRecognizedIdentifiedAssetsAcquiredAndLiabilitiesAssumedTableTextBlock" xlink:label="us-gaap_ScheduleOfRecognizedIdentifiedAssetsAcquiredAndLiabilitiesAssumedTableTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ScheduleOfRecognizedIdentifiedAssetsAcquiredAndLiabilitiesAssumedTableTextBlock" xlink:to="us-gaap_ScheduleOfRecognizedIdentifiedAssetsAcquiredAndLiabilitiesAssumedTableTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ScheduleOfRecognizedIdentifiedAssetsAcquiredAndLiabilitiesAssumedTableTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Schedule of Recognized Identified Assets Acquired and Liabilities Assumed [Table Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ScheduleOfRecognizedIdentifiedAssetsAcquiredAndLiabilitiesAssumedTableTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Schedule of allocation of purchase price</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_ScheduleOfNumberOfSharesOfCommonStockIssuedAndAmountsRecordedOnLineWithinStockholdersDeficitTableTextBlock" xlink:label="adgm_ScheduleOfNumberOfSharesOfCommonStockIssuedAndAmountsRecordedOnLineWithinStockholdersDeficitTableTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_ScheduleOfNumberOfSharesOfCommonStockIssuedAndAmountsRecordedOnLineWithinStockholdersDeficitTableTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Schedule Of Number Of Shares Of Common Stock Issued And Amounts Recorded On The Line Within Stockholders&amp;rsquo; Deficit [Table Text Block]</label>
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    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_FairValueByBalanceSheetGroupingTable" xlink:label="us-gaap_FairValueByBalanceSheetGroupingTable" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ClassOfWarrantOrRightAxis" xlink:label="us-gaap_ClassOfWarrantOrRightAxis" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ClassOfWarrantOrRightAxis_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class of Warrant or Right [Axis]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ClassOfWarrantOrRightDomain" xlink:label="us-gaap_ClassOfWarrantOrRightDomain" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ClassOfWarrantOrRightDomain_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class of Warrant or Right [Domain]</label>
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    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_ConvertWarrantsMember" xlink:to="adgm_ConvertWarrantsMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_ConvertWarrantsMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents Information pertaining to Convert Warrants.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_ConvertWarrantsMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Convert Warrants [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_ConvertWarrantsMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Convert Warrants</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_PipePreFundedWarrantsMember" xlink:label="adgm_PipePreFundedWarrantsMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_PipePreFundedWarrantsMember" xlink:to="adgm_PipePreFundedWarrantsMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_PipePreFundedWarrantsMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents information pertaining to PIPE Pre-funded warrants.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_PipePreFundedWarrantsMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">PIPE Pre-funded Warrants [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_PipePreFundedWarrantsMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">PIPE Pre-funded warrants</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_DebtInstrumentAxis" xlink:label="us-gaap_DebtInstrumentAxis" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_DebtInstrumentAxis" xlink:to="us-gaap_DebtInstrumentAxis_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_DebtInstrumentAxis_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Debt Instrument [Axis]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_DebtInstrumentNameDomain" xlink:label="us-gaap_DebtInstrumentNameDomain" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_DebtInstrumentNameDomain" xlink:to="us-gaap_DebtInstrumentNameDomain_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_DebtInstrumentNameDomain_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Debt Instrument, Name [Domain]</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_ConvertibleSecuritiesNotesMember" xlink:label="adgm_ConvertibleSecuritiesNotesMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_ConvertibleSecuritiesNotesMember" xlink:to="adgm_ConvertibleSecuritiesNotesMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_ConvertibleSecuritiesNotesMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents information pertaining to convertible securities notes.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_ConvertibleSecuritiesNotesMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Convertible Securities Notes [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_ConvertibleSecuritiesNotesMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Convertible Securities Notes</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_FairValueBalanceSheetGroupingFinancialStatementCaptionsLineItems" xlink:label="us-gaap_FairValueBalanceSheetGroupingFinancialStatementCaptionsLineItems" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_FairValueBalanceSheetGroupingFinancialStatementCaptionsLineItems" xlink:to="us-gaap_FairValueBalanceSheetGroupingFinancialStatementCaptionsLineItems_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_FairValueBalanceSheetGroupingFinancialStatementCaptionsLineItems_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fair Value, Balance Sheet Grouping, Financial Statement Captions [Line Items]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_FairValueBalanceSheetGroupingFinancialStatementCaptionsLineItems_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fair Value Measurements</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ScheduleOfFairValueAssetsAndLiabilitiesMeasuredOnRecurringBasisTableTextBlock" xlink:label="us-gaap_ScheduleOfFairValueAssetsAndLiabilitiesMeasuredOnRecurringBasisTableTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ScheduleOfFairValueAssetsAndLiabilitiesMeasuredOnRecurringBasisTableTextBlock" xlink:to="us-gaap_ScheduleOfFairValueAssetsAndLiabilitiesMeasuredOnRecurringBasisTableTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ScheduleOfFairValueAssetsAndLiabilitiesMeasuredOnRecurringBasisTableTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Schedule of Fair Value, Assets and Liabilities Measured on Recurring Basis [Table Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ScheduleOfFairValueAssetsAndLiabilitiesMeasuredOnRecurringBasisTableTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Schedule of Company's financial instruments at fair value based on the fair value hierarchy for each class of instrument</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_FairValueAssetsAndLiabilitiesMeasuredOnRecurringAndNonrecurringBasisValuationTechniquesTableTextBlock" xlink:label="us-gaap_FairValueAssetsAndLiabilitiesMeasuredOnRecurringAndNonrecurringBasisValuationTechniquesTableTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_FairValueAssetsAndLiabilitiesMeasuredOnRecurringAndNonrecurringBasisValuationTechniquesTableTextBlock" xlink:to="us-gaap_FairValueAssetsAndLiabilitiesMeasuredOnRecurringAndNonrecurringBasisValuationTechniquesTableTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_FairValueAssetsAndLiabilitiesMeasuredOnRecurringAndNonrecurringBasisValuationTechniquesTableTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fair Value Measurement Inputs and Valuation Techniques [Table Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_FairValueAssetsAndLiabilitiesMeasuredOnRecurringAndNonrecurringBasisValuationTechniquesTableTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Schedule of assumptions used in estimating fair value of convertible promissory notes and common stock warrant liabilities</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_FairValueLiabilitiesMeasuredOnRecurringBasisUnobservableInputReconciliationTextBlock" xlink:label="us-gaap_FairValueLiabilitiesMeasuredOnRecurringBasisUnobservableInputReconciliationTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_FairValueLiabilitiesMeasuredOnRecurringBasisUnobservableInputReconciliationTextBlock" xlink:to="us-gaap_FairValueLiabilitiesMeasuredOnRecurringBasisUnobservableInputReconciliationTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_FairValueLiabilitiesMeasuredOnRecurringBasisUnobservableInputReconciliationTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fair Value, Liabilities Measured on Recurring Basis, Unobservable Input Reconciliation [Table Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_FairValueLiabilitiesMeasuredOnRecurringBasisUnobservableInputReconciliationTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Schedule of presents changes in the Level 3 warrant liabilities measured at fair value</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ScheduleOfInventoryCurrentTableTextBlock" xlink:label="us-gaap_ScheduleOfInventoryCurrentTableTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ScheduleOfInventoryCurrentTableTextBlock" xlink:to="us-gaap_ScheduleOfInventoryCurrentTableTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ScheduleOfInventoryCurrentTableTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Schedule of Inventory, Current [Table Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ScheduleOfInventoryCurrentTableTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Schedule of components of inventory, net</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_PropertyPlantAndEquipmentTextBlock" xlink:label="us-gaap_PropertyPlantAndEquipmentTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_PropertyPlantAndEquipmentTextBlock" xlink:to="us-gaap_PropertyPlantAndEquipmentTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_PropertyPlantAndEquipmentTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Property, Plant and Equipment [Table Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_PropertyPlantAndEquipmentTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Schedule of components of property and equipment, net</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_ScheduleOfFiniteLivedAndIndefiniteLiveIntangibleAssetsTableTextBlock" xlink:label="adgm_ScheduleOfFiniteLivedAndIndefiniteLiveIntangibleAssetsTableTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_ScheduleOfFiniteLivedAndIndefiniteLiveIntangibleAssetsTableTextBlock" xlink:to="adgm_ScheduleOfFiniteLivedAndIndefiniteLiveIntangibleAssetsTableTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_ScheduleOfFiniteLivedAndIndefiniteLiveIntangibleAssetsTableTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Tabular disclosure of assets, excluding financial assets and goodwill, lacking physical substance and exist in perpetuity, by either major class or business segment.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_ScheduleOfFiniteLivedAndIndefiniteLiveIntangibleAssetsTableTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Schedule of Finite-Lived And Indefinite-Live Intangible Assets [Table Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_ScheduleOfFiniteLivedAndIndefiniteLiveIntangibleAssetsTableTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Schedule of intangible assets</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ScheduleOfIntangibleAssetsAndGoodwillTableTextBlock" xlink:label="us-gaap_ScheduleOfIntangibleAssetsAndGoodwillTableTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ScheduleOfIntangibleAssetsAndGoodwillTableTextBlock" xlink:to="us-gaap_ScheduleOfIntangibleAssetsAndGoodwillTableTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ScheduleOfIntangibleAssetsAndGoodwillTableTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Schedule of Intangible Assets and Goodwill [Table Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ScheduleOfIntangibleAssetsAndGoodwillTableTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Schedule of goodwill</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ScheduleOfAccruedLiabilitiesTableTextBlock" xlink:label="us-gaap_ScheduleOfAccruedLiabilitiesTableTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ScheduleOfAccruedLiabilitiesTableTextBlock" xlink:to="us-gaap_ScheduleOfAccruedLiabilitiesTableTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ScheduleOfAccruedLiabilitiesTableTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Schedule of Accrued Liabilities [Table Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ScheduleOfAccruedLiabilitiesTableTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Schedule of accrued liabilities</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ScheduleOfDebtTableTextBlock" xlink:label="us-gaap_ScheduleOfDebtTableTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ScheduleOfDebtTableTextBlock" xlink:to="us-gaap_ScheduleOfDebtTableTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ScheduleOfDebtTableTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Schedule of Debt [Table Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ScheduleOfDebtTableTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Schedule of debt</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_LeaseCostTableTextBlock" xlink:label="us-gaap_LeaseCostTableTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_LeaseCostTableTextBlock" xlink:to="us-gaap_LeaseCostTableTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_LeaseCostTableTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Lease, Cost [Table Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_LeaseCostTableTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Schedule of quantitative information of operating leases</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_LesseeOperatingLeaseLiabilityMaturityTableTextBlock" xlink:label="us-gaap_LesseeOperatingLeaseLiabilityMaturityTableTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_LesseeOperatingLeaseLiabilityMaturityTableTextBlock" xlink:to="us-gaap_LesseeOperatingLeaseLiabilityMaturityTableTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_LesseeOperatingLeaseLiabilityMaturityTableTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Lessee, Operating Lease, Liability, to be Paid, Maturity [Table Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_LesseeOperatingLeaseLiabilityMaturityTableTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Schedule of future minimum payments under the non-cancelable operating leases</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_TemporaryEquityTableTextBlock" xlink:label="us-gaap_TemporaryEquityTableTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_TemporaryEquityTableTextBlock" xlink:to="us-gaap_TemporaryEquityTableTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_TemporaryEquityTableTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Temporary Equity [Table Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_TemporaryEquityTableTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Schedule of issuance of preferred stock</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ScheduleOfStockholdersEquityTableTextBlock" xlink:label="us-gaap_ScheduleOfStockholdersEquityTableTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ScheduleOfStockholdersEquityTableTextBlock" xlink:to="us-gaap_ScheduleOfStockholdersEquityTableTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ScheduleOfStockholdersEquityTableTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Schedule of Stockholders Equity [Table Text Block]</label>
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    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ScheduleOfEmployeeServiceShareBasedCompensationAllocationOfRecognizedPeriodCostsTextBlock" xlink:to="us-gaap_ScheduleOfEmployeeServiceShareBasedCompensationAllocationOfRecognizedPeriodCostsTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <loc xlink:type="locator" xlink:href="https://xbrl.sec.gov/dei/2024/dei-2024.xsd#dei_LegalEntityAxis" xlink:label="dei_LegalEntityAxis" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="dei_LegalEntityAxis" xlink:to="dei_LegalEntityAxis_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="dei_LegalEntityAxis_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Legal Entity [Axis]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.sec.gov/dei/2024/dei-2024.xsd#dei_EntityDomain" xlink:label="dei_EntityDomain" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="dei_EntityDomain" xlink:to="dei_EntityDomain_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="dei_EntityDomain_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">All Entities [Member]</label>
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    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_EarningsPerShareBasicLineItems" xlink:to="us-gaap_EarningsPerShareBasicLineItems_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_EarningsPerShareBasicLineItems_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Earnings Per Share, Basic, by Common Class, Including Two Class Method [Line Items]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_EarningsPerShareBasicLineItems_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Loss Per Share ("LPS")</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ScheduleOfEarningsPerShareBasicAndDilutedTableTextBlock" xlink:label="us-gaap_ScheduleOfEarningsPerShareBasicAndDilutedTableTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ScheduleOfEarningsPerShareBasicAndDilutedTableTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Schedule of computation of basic and diluted net loss per share attributable to common stockholders</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ScheduleOfAntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareTextBlock" xlink:label="us-gaap_ScheduleOfAntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ScheduleOfAntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Schedule of Antidilutive Securities Excluded from Computation of Earnings Per Share [Table Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ScheduleOfAntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Schedule of potentially dilutive securities were excluded from the computation of diluted net loss per share</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ScheduleOfIncomeBeforeIncomeTaxDomesticAndForeignTableTextBlock" xlink:label="us-gaap_ScheduleOfIncomeBeforeIncomeTaxDomesticAndForeignTableTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ScheduleOfIncomeBeforeIncomeTaxDomesticAndForeignTableTextBlock" xlink:to="us-gaap_ScheduleOfIncomeBeforeIncomeTaxDomesticAndForeignTableTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ScheduleOfIncomeBeforeIncomeTaxDomesticAndForeignTableTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Summary of components of pretax loss from operations</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ScheduleOfEffectiveIncomeTaxRateReconciliationTableTextBlock" xlink:label="us-gaap_ScheduleOfEffectiveIncomeTaxRateReconciliationTableTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ScheduleOfEffectiveIncomeTaxRateReconciliationTableTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Schedule of Effective Income Tax Rate Reconciliation [Table Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ScheduleOfEffectiveIncomeTaxRateReconciliationTableTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Summary of reconciliation of income tax computed at the U.S. federal statutory tax rate to the total income tax expense</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ScheduleOfDeferredTaxAssetsAndLiabilitiesTableTextBlock" xlink:label="us-gaap_ScheduleOfDeferredTaxAssetsAndLiabilitiesTableTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ScheduleOfDeferredTaxAssetsAndLiabilitiesTableTextBlock" xlink:to="us-gaap_ScheduleOfDeferredTaxAssetsAndLiabilitiesTableTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ScheduleOfDeferredTaxAssetsAndLiabilitiesTableTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Schedule of Deferred Tax Assets and Liabilities [Table Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ScheduleOfDeferredTaxAssetsAndLiabilitiesTableTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Summary of components of deferred tax assets and liabilities</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ScheduleOfUnrecognizedTaxBenefitsRollForwardTableTextBlock" xlink:label="us-gaap_ScheduleOfUnrecognizedTaxBenefitsRollForwardTableTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ScheduleOfUnrecognizedTaxBenefitsRollForwardTableTextBlock" xlink:to="us-gaap_ScheduleOfUnrecognizedTaxBenefitsRollForwardTableTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ScheduleOfUnrecognizedTaxBenefitsRollForwardTableTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Schedule of Unrecognized Tax Benefits Roll Forward [Table Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ScheduleOfUnrecognizedTaxBenefitsRollForwardTableTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Summary of changes to unrecognized tax benefits</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ScheduleOfSegmentReportingInformationBySegmentTextBlock" xlink:label="us-gaap_ScheduleOfSegmentReportingInformationBySegmentTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ScheduleOfSegmentReportingInformationBySegmentTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Schedule of Segment Reporting Information, by Segment [Table Text Block]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_SubsidiaryOrEquityMethodInvesteeSaleOfStockBySubsidiaryOrEquityInvesteeTable" xlink:label="us-gaap_SubsidiaryOrEquityMethodInvesteeSaleOfStockBySubsidiaryOrEquityInvesteeTable" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_SubsidiaryOrEquityMethodInvesteeSaleOfStockBySubsidiaryOrEquityInvesteeTable_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Subsidiary or Equity Method Investee, Sale of Stock, Type [Table]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_SubsidiarySaleOfStockAxis" xlink:label="us-gaap_SubsidiarySaleOfStockAxis" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_SubsidiarySaleOfStockAxis" xlink:to="us-gaap_SubsidiarySaleOfStockAxis_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_SubsidiarySaleOfStockAxis_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Sale of Stock [Axis]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_SaleOfStockNameOfTransactionDomain" xlink:label="us-gaap_SaleOfStockNameOfTransactionDomain" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_SaleOfStockNameOfTransactionDomain_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Sale of Stock [Domain]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_PrivatePlacementMember" xlink:label="us-gaap_PrivatePlacementMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_PrivatePlacementMember" xlink:to="us-gaap_PrivatePlacementMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_PrivatePlacementMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Private Placement [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_PrivatePlacementMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Private Placement</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_SubsidiarySaleOfStockLineItems" xlink:label="us-gaap_SubsidiarySaleOfStockLineItems" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_SubsidiarySaleOfStockLineItems" xlink:to="us-gaap_SubsidiarySaleOfStockLineItems_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_SubsidiarySaleOfStockLineItems_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Subsidiary, Sale of Stock [Line Items]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_SubsidiarySaleOfStockLineItems_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Liquidity and Going Concern</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_PotentialProceedsFromExerciseOfWarrants" xlink:label="adgm_PotentialProceedsFromExerciseOfWarrants" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_PotentialProceedsFromExerciseOfWarrants" xlink:to="adgm_PotentialProceedsFromExerciseOfWarrants_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_PotentialProceedsFromExerciseOfWarrants_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The potential cash inflow from cash exercise of milestone warrants.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_PotentialProceedsFromExerciseOfWarrants_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Potential Proceeds from Exercise of Warrants</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_PotentialProceedsFromExerciseOfWarrants_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Potential gross proceeds from cash exercise of warrants</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_NumberOfReportableSegments" xlink:label="us-gaap_NumberOfReportableSegments" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_NumberOfReportableSegments" xlink:to="us-gaap_NumberOfReportableSegments_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_NumberOfReportableSegments_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of Reportable Segments</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_NumberOfReportableSegments_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of reporting segments</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_CashFDICInsuredAmount" xlink:label="us-gaap_CashFDICInsuredAmount" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_CashFDICInsuredAmount" xlink:to="us-gaap_CashFDICInsuredAmount_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_CashFDICInsuredAmount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cash, FDIC Insured Amount</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_CustomerPaymentPeriod" xlink:label="adgm_CustomerPaymentPeriod" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_CustomerPaymentPeriod" xlink:to="adgm_CustomerPaymentPeriod_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_CustomerPaymentPeriod_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The period within which customers are generally required to pay, in 'PnYnMnDTnHnMnS' format, for example, 'P1Y5M13D' represents the reported fact of one year, five months, and thirteen days.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_CustomerPaymentPeriod_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Customer Payment Period</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_CustomerPaymentPeriod_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Customer payment period</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ScheduleOfPropertyPlantAndEquipmentTable" xlink:label="us-gaap_ScheduleOfPropertyPlantAndEquipmentTable" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ScheduleOfPropertyPlantAndEquipmentTable" xlink:to="us-gaap_ScheduleOfPropertyPlantAndEquipmentTable_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ScheduleOfPropertyPlantAndEquipmentTable_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Property, Plant and Equipment [Table]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_PropertyPlantAndEquipmentByTypeAxis" xlink:label="us-gaap_PropertyPlantAndEquipmentByTypeAxis" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_PropertyPlantAndEquipmentByTypeAxis" xlink:to="us-gaap_PropertyPlantAndEquipmentByTypeAxis_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_PropertyPlantAndEquipmentByTypeAxis_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Long-Lived Tangible Asset [Axis]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_PropertyPlantAndEquipmentTypeDomain" xlink:label="us-gaap_PropertyPlantAndEquipmentTypeDomain" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_PropertyPlantAndEquipmentTypeDomain" xlink:to="us-gaap_PropertyPlantAndEquipmentTypeDomain_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_PropertyPlantAndEquipmentTypeDomain_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Long-Lived Tangible Asset [Domain]</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_ConsolesMember" xlink:label="adgm_ConsolesMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_ConsolesMember" xlink:to="adgm_ConsolesMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_ConsolesMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents information pertaining to consoles.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_ConsolesMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Consoles [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_ConsolesMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Consoles</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/srt/2024/elts/srt-2024.xsd#srt_RangeAxis" xlink:label="srt_RangeAxis" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="srt_RangeAxis" xlink:to="srt_RangeAxis_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="srt_RangeAxis_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Statistical Measurement [Axis]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/srt/2024/elts/srt-2024.xsd#srt_RangeMember" xlink:label="srt_RangeMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="srt_RangeMember" xlink:to="srt_RangeMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="srt_RangeMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Statistical Measurement [Domain]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/srt/2024/elts/srt-2024.xsd#srt_MaximumMember" xlink:label="srt_MaximumMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="srt_MaximumMember" xlink:to="srt_MaximumMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="srt_MaximumMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Maximum [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="srt_MaximumMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Maximum</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/srt/2024/elts/srt-2024.xsd#srt_MinimumMember" xlink:label="srt_MinimumMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="srt_MinimumMember" xlink:to="srt_MinimumMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="srt_MinimumMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Minimum [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="srt_MinimumMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Minimum</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_PropertyPlantAndEquipmentLineItems" xlink:label="us-gaap_PropertyPlantAndEquipmentLineItems" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_PropertyPlantAndEquipmentLineItems" xlink:to="us-gaap_PropertyPlantAndEquipmentLineItems_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_PropertyPlantAndEquipmentLineItems_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Property, Plant and Equipment [Line Items]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_PropertyPlantAndEquipmentLineItems_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Property and Equipment</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_PropertyPlantAndEquipmentUsefulLife" xlink:label="us-gaap_PropertyPlantAndEquipmentUsefulLife" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_PropertyPlantAndEquipmentUsefulLife" xlink:to="us-gaap_PropertyPlantAndEquipmentUsefulLife_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_PropertyPlantAndEquipmentUsefulLife_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Property, Plant and Equipment, Useful Life</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_PropertyPlantAndEquipmentUsefulLife_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Useful life</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ImpairmentOfIntangibleAssetsFinitelived" xlink:label="us-gaap_ImpairmentOfIntangibleAssetsFinitelived" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ImpairmentOfIntangibleAssetsFinitelived" xlink:to="us-gaap_ImpairmentOfIntangibleAssetsFinitelived_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ImpairmentOfIntangibleAssetsFinitelived_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Impairment of Intangible Assets, Finite-Lived</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ImpairmentOfIntangibleAssetsFinitelived_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Impairment of finite lived intangible assets</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ImpairmentOfIntangibleAssetFiniteLivedStatementOfIncomeOrComprehensiveIncomeExtensibleEnumeration" xlink:label="us-gaap_ImpairmentOfIntangibleAssetFiniteLivedStatementOfIncomeOrComprehensiveIncomeExtensibleEnumeration" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ImpairmentOfIntangibleAssetFiniteLivedStatementOfIncomeOrComprehensiveIncomeExtensibleEnumeration" xlink:to="us-gaap_ImpairmentOfIntangibleAssetFiniteLivedStatementOfIncomeOrComprehensiveIncomeExtensibleEnumeration_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ImpairmentOfIntangibleAssetFiniteLivedStatementOfIncomeOrComprehensiveIncomeExtensibleEnumeration_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Impairment, Intangible Asset, Finite-Lived, Statement of Income or Comprehensive Income [Extensible Enumeration]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ImpairmentOfIntangibleAssetsIndefinitelivedExcludingGoodwill" xlink:label="us-gaap_ImpairmentOfIntangibleAssetsIndefinitelivedExcludingGoodwill" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ImpairmentOfIntangibleAssetsIndefinitelivedExcludingGoodwill" xlink:to="us-gaap_ImpairmentOfIntangibleAssetsIndefinitelivedExcludingGoodwill_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ImpairmentOfIntangibleAssetsIndefinitelivedExcludingGoodwill_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Impairment of Intangible Assets, Indefinite-Lived (Excluding Goodwill)</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ImpairmentOfIntangibleAssetsIndefinitelivedExcludingGoodwill_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Impairment of indefinite live lived intangible assets</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ImpairmentOfIntangibleAssetIndefiniteLivedExcludingGoodwillStatementOfIncomeOrComprehensiveIncomeExtensibleEnumeration" xlink:label="us-gaap_ImpairmentOfIntangibleAssetIndefiniteLivedExcludingGoodwillStatementOfIncomeOrComprehensiveIncomeExtensibleEnumeration" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ImpairmentOfIntangibleAssetIndefiniteLivedExcludingGoodwillStatementOfIncomeOrComprehensiveIncomeExtensibleEnumeration" xlink:to="us-gaap_ImpairmentOfIntangibleAssetIndefiniteLivedExcludingGoodwillStatementOfIncomeOrComprehensiveIncomeExtensibleEnumeration_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ImpairmentOfIntangibleAssetIndefiniteLivedExcludingGoodwillStatementOfIncomeOrComprehensiveIncomeExtensibleEnumeration_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Impairment, Intangible Asset, Indefinite-Lived (Excluding Goodwill), Statement of Income or Comprehensive Income [Extensible Enumeration]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_NumberOfReportingUnits" xlink:label="us-gaap_NumberOfReportingUnits" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_NumberOfReportingUnits" xlink:to="us-gaap_NumberOfReportingUnits_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_NumberOfReportingUnits_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of Reporting Units</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_NumberOfReportingUnits_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of reporting unit</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ConcentrationRiskTable" xlink:label="us-gaap_ConcentrationRiskTable" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ConcentrationRiskTable" xlink:to="us-gaap_ConcentrationRiskTable_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ConcentrationRiskTable_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Concentration Risk [Table]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ConcentrationRiskByBenchmarkAxis" xlink:label="us-gaap_ConcentrationRiskByBenchmarkAxis" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ConcentrationRiskByBenchmarkAxis" xlink:to="us-gaap_ConcentrationRiskByBenchmarkAxis_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ConcentrationRiskByBenchmarkAxis_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Concentration Risk Benchmark [Axis]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ConcentrationRiskBenchmarkDomain" xlink:label="us-gaap_ConcentrationRiskBenchmarkDomain" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ConcentrationRiskBenchmarkDomain" xlink:to="us-gaap_ConcentrationRiskBenchmarkDomain_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ConcentrationRiskBenchmarkDomain_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Concentration Risk Benchmark [Domain]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_AccountsPayableMember" xlink:label="us-gaap_AccountsPayableMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_AccountsPayableMember" xlink:to="us-gaap_AccountsPayableMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_AccountsPayableMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Accounts Payable [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_AccountsPayableMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Accounts payable</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_CostOfGoodsProductLineMember" xlink:label="us-gaap_CostOfGoodsProductLineMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_CostOfGoodsProductLineMember" xlink:to="us-gaap_CostOfGoodsProductLineMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_CostOfGoodsProductLineMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cost of Goods and Service, Product and Service Benchmark [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_CostOfGoodsProductLineMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Expenditure</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ConcentrationRiskByTypeAxis" xlink:label="us-gaap_ConcentrationRiskByTypeAxis" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ConcentrationRiskByTypeAxis" xlink:to="us-gaap_ConcentrationRiskByTypeAxis_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ConcentrationRiskByTypeAxis_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Concentration Risk Type [Axis]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ConcentrationRiskTypeDomain" xlink:label="us-gaap_ConcentrationRiskTypeDomain" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ConcentrationRiskTypeDomain" xlink:to="us-gaap_ConcentrationRiskTypeDomain_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ConcentrationRiskTypeDomain_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Concentration Risk Type [Domain]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_SupplierConcentrationRiskMember" xlink:label="us-gaap_SupplierConcentrationRiskMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_SupplierConcentrationRiskMember" xlink:to="us-gaap_SupplierConcentrationRiskMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_SupplierConcentrationRiskMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Supplier Concentration Risk [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_SupplierConcentrationRiskMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Supplier concentration risk</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ShareBasedGoodsAndNonemployeeServicesTransactionBySupplierAxis" xlink:label="us-gaap_ShareBasedGoodsAndNonemployeeServicesTransactionBySupplierAxis" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ShareBasedGoodsAndNonemployeeServicesTransactionBySupplierAxis" xlink:to="us-gaap_ShareBasedGoodsAndNonemployeeServicesTransactionBySupplierAxis_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ShareBasedGoodsAndNonemployeeServicesTransactionBySupplierAxis_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Supplier [Axis]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ShareBasedGoodsAndNonemployeeServicesTransactionSupplierDomain" xlink:label="us-gaap_ShareBasedGoodsAndNonemployeeServicesTransactionSupplierDomain" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ShareBasedGoodsAndNonemployeeServicesTransactionSupplierDomain" xlink:to="us-gaap_ShareBasedGoodsAndNonemployeeServicesTransactionSupplierDomain_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ShareBasedGoodsAndNonemployeeServicesTransactionSupplierDomain_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Supplier [Domain]</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_TwoSuppliersMember" xlink:label="adgm_TwoSuppliersMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_TwoSuppliersMember" xlink:to="adgm_TwoSuppliersMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_TwoSuppliersMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents information pertaining to two suppliers.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_TwoSuppliersMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Two Suppliers [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_TwoSuppliersMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Two suppliers</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_FiveSuppliersMember" xlink:label="adgm_FiveSuppliersMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_FiveSuppliersMember" xlink:to="adgm_FiveSuppliersMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_FiveSuppliersMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents information pertaining to five suppliers.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_FiveSuppliersMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Five Suppliers [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_FiveSuppliersMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Five suppliers</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_TenSuppliersMember" xlink:label="adgm_TenSuppliersMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_TenSuppliersMember" xlink:to="adgm_TenSuppliersMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_TenSuppliersMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents information pertaining to ten suppliers.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_TenSuppliersMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Ten Suppliers [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_TenSuppliersMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Ten suppliers</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ConcentrationRiskLineItems" xlink:label="us-gaap_ConcentrationRiskLineItems" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ConcentrationRiskLineItems" xlink:to="us-gaap_ConcentrationRiskLineItems_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ConcentrationRiskLineItems_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Concentration Risk [Line Items]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ConcentrationRiskLineItems_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Concentrations</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ConcentrationRiskPercentage1" xlink:label="us-gaap_ConcentrationRiskPercentage1" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ConcentrationRiskPercentage1" xlink:to="us-gaap_ConcentrationRiskPercentage1_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ConcentrationRiskPercentage1_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Concentration Risk, Percentage</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ConcentrationRiskPercentage1_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Concentration risk percentage</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_ConcentrationRiskNumberOfSuppliers" xlink:label="adgm_ConcentrationRiskNumberOfSuppliers" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_ConcentrationRiskNumberOfSuppliers" xlink:to="adgm_ConcentrationRiskNumberOfSuppliers_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_ConcentrationRiskNumberOfSuppliers_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents the number of suppliers.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_ConcentrationRiskNumberOfSuppliers_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Concentration Risk, Number Of Suppliers</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_ConcentrationRiskNumberOfSuppliers_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of suppliers</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_MinimumProceedsFromIssuanceOfCommonStockForAutomaticConversionOfPreferredStock" xlink:label="adgm_MinimumProceedsFromIssuanceOfCommonStockForAutomaticConversionOfPreferredStock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_MinimumProceedsFromIssuanceOfCommonStockForAutomaticConversionOfPreferredStock" xlink:to="adgm_MinimumProceedsFromIssuanceOfCommonStockForAutomaticConversionOfPreferredStock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_MinimumProceedsFromIssuanceOfCommonStockForAutomaticConversionOfPreferredStock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents the minimum proceeds from issuance of common stock for automatic conversion of preferred stock.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_MinimumProceedsFromIssuanceOfCommonStockForAutomaticConversionOfPreferredStock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Minimum Proceeds from Issuance of Common Stock For Automatic Conversion of Preferred Stock</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_MinimumProceedsFromIssuanceOfCommonStockForAutomaticConversionOfPreferredStock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Aggregate gross proceeds</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_MinimumPublicOfferingSharePriceForAutomaticConversionOfPreferredStock" xlink:label="adgm_MinimumPublicOfferingSharePriceForAutomaticConversionOfPreferredStock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_MinimumPublicOfferingSharePriceForAutomaticConversionOfPreferredStock" xlink:to="adgm_MinimumPublicOfferingSharePriceForAutomaticConversionOfPreferredStock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_MinimumPublicOfferingSharePriceForAutomaticConversionOfPreferredStock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents the minimum public offering share price for automatic conversion of preferred stock.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_MinimumPublicOfferingSharePriceForAutomaticConversionOfPreferredStock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Minimum Public Offering Share Price For Automatic Conversion of Preferred Stock</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_MinimumPublicOfferingSharePriceForAutomaticConversionOfPreferredStock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Public offering price per share</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_IncomeTaxPenaltiesAndInterestAccrued" xlink:label="adgm_IncomeTaxPenaltiesAndInterestAccrued" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_IncomeTaxPenaltiesAndInterestAccrued" xlink:to="adgm_IncomeTaxPenaltiesAndInterestAccrued_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_IncomeTaxPenaltiesAndInterestAccrued_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amount accrued for interest and penalty on an underpayment of income taxes.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_IncomeTaxPenaltiesAndInterestAccrued_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Income Tax Penalties and Interest Accrued</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_IncomeTaxPenaltiesAndInterestAccrued_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Interest and penalties accrued</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_UnrecognizedTaxBenefitsIncomeTaxPenaltiesAndInterestExpense" xlink:label="us-gaap_UnrecognizedTaxBenefitsIncomeTaxPenaltiesAndInterestExpense" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_UnrecognizedTaxBenefitsIncomeTaxPenaltiesAndInterestExpense" xlink:to="us-gaap_UnrecognizedTaxBenefitsIncomeTaxPenaltiesAndInterestExpense_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_UnrecognizedTaxBenefitsIncomeTaxPenaltiesAndInterestExpense_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Unrecognized Tax Benefits, Income Tax Penalties and Interest Expense</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_UnrecognizedTaxBenefitsIncomeTaxPenaltiesAndInterestExpense_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Unrecognized tax benefits, income tax penalties and interest expense</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ScheduleOfBusinessAcquisitionsByAcquisitionTable" xlink:label="us-gaap_ScheduleOfBusinessAcquisitionsByAcquisitionTable" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ScheduleOfBusinessAcquisitionsByAcquisitionTable" xlink:to="us-gaap_ScheduleOfBusinessAcquisitionsByAcquisitionTable_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ScheduleOfBusinessAcquisitionsByAcquisitionTable_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Schedule of Business Acquisitions, by Acquisition [Table]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_BusinessAcquisitionLineItems" xlink:label="us-gaap_BusinessAcquisitionLineItems" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_BusinessAcquisitionLineItems" xlink:to="us-gaap_BusinessAcquisitionLineItems_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_BusinessAcquisitionLineItems_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Business Acquisition [Line Items]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_BusinessAcquisitionLineItems_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Forward Merger</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_TemporaryStockRedeemedOrCalledDuringPeriodShares" xlink:label="adgm_TemporaryStockRedeemedOrCalledDuringPeriodShares" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_TemporaryStockRedeemedOrCalledDuringPeriodShares" xlink:to="adgm_TemporaryStockRedeemedOrCalledDuringPeriodShares_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_TemporaryStockRedeemedOrCalledDuringPeriodShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of temporary stock bought back by the entity at the exercise price or redemption price.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_TemporaryStockRedeemedOrCalledDuringPeriodShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Temporary Stock Redeemed or Called During Period, Shares</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_TemporaryStockRedeemedOrCalledDuringPeriodShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of Class A ordinary shares redeemed</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_TemporaryEquityRedemptionPricePerShare" xlink:label="us-gaap_TemporaryEquityRedemptionPricePerShare" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_TemporaryEquityRedemptionPricePerShare" xlink:to="us-gaap_TemporaryEquityRedemptionPricePerShare_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_TemporaryEquityRedemptionPricePerShare_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Temporary Equity, Redemption Price Per Share</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_TemporaryEquityRedemptionPricePerShare_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Redemption price</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_TemporaryStockRedeemedOrCalledDuringPeriodValue" xlink:label="adgm_TemporaryStockRedeemedOrCalledDuringPeriodValue" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_TemporaryStockRedeemedOrCalledDuringPeriodValue" xlink:to="adgm_TemporaryStockRedeemedOrCalledDuringPeriodValue_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_TemporaryStockRedeemedOrCalledDuringPeriodValue_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Equity impact of the value of temporary stock bought back by the entity at the exercise price or redemption price.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_TemporaryStockRedeemedOrCalledDuringPeriodValue_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Temporary Stock Redeemed or Called During Period, Value</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_TemporaryStockRedeemedOrCalledDuringPeriodValue_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Redemption value</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_AryaSciencesAcquisitionCorpIvMember" xlink:label="adgm_AryaSciencesAcquisitionCorpIvMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_AryaSciencesAcquisitionCorpIvMember" xlink:to="adgm_AryaSciencesAcquisitionCorpIvMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_AryaSciencesAcquisitionCorpIvMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents information pertaining to ARYA Sciences Acquisition Corp IV.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_AryaSciencesAcquisitionCorpIvMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">ARYA Sciences Acquisition Corp IV [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_AryaSciencesAcquisitionCorpIvMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">ARYA Sciences Acquisition Corp IV</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_StatementClassOfStockAxis" xlink:label="us-gaap_StatementClassOfStockAxis" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_StatementClassOfStockAxis" xlink:to="us-gaap_StatementClassOfStockAxis_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_StatementClassOfStockAxis_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class of Stock [Axis]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ClassOfStockDomain" xlink:label="us-gaap_ClassOfStockDomain" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ClassOfStockDomain" xlink:to="us-gaap_ClassOfStockDomain_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ClassOfStockDomain_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class of Stock [Domain]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_CommonClassAMember" xlink:label="us-gaap_CommonClassAMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_CommonClassAMember" xlink:to="us-gaap_CommonClassAMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_CommonClassAMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Common Class A [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_CommonClassAMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class A ordinary share</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_CommonClassBMember" xlink:label="us-gaap_CommonClassBMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_CommonClassBMember" xlink:to="us-gaap_CommonClassBMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_CommonClassBMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Common Class B [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_CommonClassBMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class B ordinary share</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_RelatedPartyTransactionsByRelatedPartyAxis" xlink:label="us-gaap_RelatedPartyTransactionsByRelatedPartyAxis" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_RelatedPartyTransactionsByRelatedPartyAxis" xlink:to="us-gaap_RelatedPartyTransactionsByRelatedPartyAxis_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_RelatedPartyTransactionsByRelatedPartyAxis_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Related and Nonrelated Parties [Axis]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_RelatedPartyDomain" xlink:label="us-gaap_RelatedPartyDomain" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_RelatedPartyDomain" xlink:to="us-gaap_RelatedPartyDomain_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_RelatedPartyDomain_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Related and Nonrelated Parties [Domain]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_RelatedPartyMember" xlink:label="us-gaap_RelatedPartyMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_RelatedPartyMember" xlink:to="us-gaap_RelatedPartyMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_RelatedPartyMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Related Party [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_RelatedPartyMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Related Party</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_SponsorMember" xlink:label="adgm_SponsorMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_SponsorMember" xlink:to="adgm_SponsorMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_SponsorMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents information pertaining to sponsor.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_SponsorMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Sponsor [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_SponsorMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Sponsor</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_ConversionOfStockNumberOfSharesForEachShare" xlink:label="adgm_ConversionOfStockNumberOfSharesForEachShare" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_ConversionOfStockNumberOfSharesForEachShare" xlink:to="adgm_ConversionOfStockNumberOfSharesForEachShare_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_ConversionOfStockNumberOfSharesForEachShare_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of shares issued for each share upon conversion.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_ConversionOfStockNumberOfSharesForEachShare_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Conversion Of Stock, Number Of Shares For Each Share</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_ConversionOfStockNumberOfSharesForEachShare_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of share for each share</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_NumberOfSharesForfeited" xlink:label="adgm_NumberOfSharesForfeited" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_NumberOfSharesForfeited" xlink:to="adgm_NumberOfSharesForfeited_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_NumberOfSharesForfeited_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of shares forfeited</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_NumberOfSharesForfeited_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number Of Shares Forfeited</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_NumberOfSharesForfeited_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of shares forfeited</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_NumberOfCommonStockSharesIssuableSubjectedToVesting" xlink:label="adgm_NumberOfCommonStockSharesIssuableSubjectedToVesting" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_NumberOfCommonStockSharesIssuableSubjectedToVesting" xlink:to="adgm_NumberOfCommonStockSharesIssuableSubjectedToVesting_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_NumberOfCommonStockSharesIssuableSubjectedToVesting_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of common stock shares issuable that are subjected to share price trigger vesting.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_NumberOfCommonStockSharesIssuableSubjectedToVesting_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number Of Common Stock Shares Issuable, Subjected To Vesting</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_NumberOfCommonStockSharesIssuableSubjectedToVesting_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of common stock shares issuable that are subjected to vesting</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_ThresholdSharePriceForVestingOfShares" xlink:label="adgm_ThresholdSharePriceForVestingOfShares" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_ThresholdSharePriceForVestingOfShares" xlink:to="adgm_ThresholdSharePriceForVestingOfShares_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_ThresholdSharePriceForVestingOfShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold share price trigger for vesting of share.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_ThresholdSharePriceForVestingOfShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold Share Price For Vesting Of Shares</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_ThresholdSharePriceForVestingOfShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Share trigger price for vesting</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_ThresholdNumberOfTradingDaysToExceedOrMaintainTriggerForVestingOfShares" xlink:label="adgm_ThresholdNumberOfTradingDaysToExceedOrMaintainTriggerForVestingOfShares" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_ThresholdNumberOfTradingDaysToExceedOrMaintainTriggerForVestingOfShares" xlink:to="adgm_ThresholdNumberOfTradingDaysToExceedOrMaintainTriggerForVestingOfShares_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_ThresholdNumberOfTradingDaysToExceedOrMaintainTriggerForVestingOfShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold number of trading days to exceed or maintain the trigger price for vesting of shares.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_ThresholdNumberOfTradingDaysToExceedOrMaintainTriggerForVestingOfShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold Number Of Trading Days To Exceed Or Maintain Trigger For Vesting Of Shares</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_ThresholdNumberOfTradingDaysToExceedOrMaintainTriggerForVestingOfShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold number of trading days to exceed or maintain trigger price for vesting of shares</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_ThresholdNumberOfConsecutiveTradingDaysToExceedOrMaintainTriggerForVestingOfShares" xlink:label="adgm_ThresholdNumberOfConsecutiveTradingDaysToExceedOrMaintainTriggerForVestingOfShares" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_ThresholdNumberOfConsecutiveTradingDaysToExceedOrMaintainTriggerForVestingOfShares" xlink:to="adgm_ThresholdNumberOfConsecutiveTradingDaysToExceedOrMaintainTriggerForVestingOfShares_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_ThresholdNumberOfConsecutiveTradingDaysToExceedOrMaintainTriggerForVestingOfShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold number of consecutive trading days to exceed or maintain the trigger price for vesting of shares.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_ThresholdNumberOfConsecutiveTradingDaysToExceedOrMaintainTriggerForVestingOfShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold Number Of Consecutive Trading Days To Exceed Or Maintain Trigger For Vesting Of Shares</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_ThresholdNumberOfConsecutiveTradingDaysToExceedOrMaintainTriggerForVestingOfShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold number of consecutive trading days to exceed or maintain trigger price for vesting of shares</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_PreferredStockParOrStatedValuePerShare" xlink:label="us-gaap_PreferredStockParOrStatedValuePerShare" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_PreferredStockParOrStatedValuePerShare" xlink:to="us-gaap_PreferredStockParOrStatedValuePerShare_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_PreferredStockParOrStatedValuePerShare_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Preferred Stock, Par or Stated Value Per Share</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_PreferredStockParOrStatedValuePerShare_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Preferred stock, par value</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_PreferredStockConvertibleConversionRatio" xlink:label="us-gaap_PreferredStockConvertibleConversionRatio" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_PreferredStockConvertibleConversionRatio" xlink:to="us-gaap_PreferredStockConvertibleConversionRatio_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_PreferredStockConvertibleConversionRatio_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Preferred Stock, Convertible, Conversion Ratio</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_PreferredStockConvertibleConversionRatio_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Preferred stock conversion ratio</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_DebtConversionConvertedInstrumentAmount1" xlink:label="us-gaap_DebtConversionConvertedInstrumentAmount1" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_DebtConversionConvertedInstrumentAmount1" xlink:to="us-gaap_DebtConversionConvertedInstrumentAmount1_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_DebtConversionConvertedInstrumentAmount1_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Debt Conversion, Converted Instrument, Amount</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_DebtConversionConvertedInstrumentAmount1_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amount of convertible notes issued</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_ClassAryaSharesMember" xlink:label="adgm_ClassAryaSharesMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_ClassAryaSharesMember" xlink:to="adgm_ClassAryaSharesMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_ClassAryaSharesMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents information pertaining to Class A Arya Shares.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_ClassAryaSharesMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class A Arya Shares [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_ClassAryaSharesMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class A Arya Shares</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_BaseWarrantsMember" xlink:label="adgm_BaseWarrantsMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_BaseWarrantsMember" xlink:to="adgm_BaseWarrantsMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_BaseWarrantsMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents information pertaining to base warrants.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_BaseWarrantsMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Base Warrants [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_BaseWarrantsMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Base Warrants</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_InvestorMember" xlink:label="us-gaap_InvestorMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_InvestorMember" xlink:to="us-gaap_InvestorMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_InvestorMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Investor [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_InvestorMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">PIPE Investor</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_CommittedFinancingValue" xlink:label="adgm_CommittedFinancingValue" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_CommittedFinancingValue" xlink:to="adgm_CommittedFinancingValue_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_CommittedFinancingValue_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The amount of committed financing value.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_CommittedFinancingValue_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Committed Financing Value</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_CommittedFinancingValue_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">PIPE Financing</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_CommonStockCapitalSharesValueReservedForFutureIssuance" xlink:label="adgm_CommonStockCapitalSharesValueReservedForFutureIssuance" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_CommonStockCapitalSharesValueReservedForFutureIssuance" xlink:to="adgm_CommonStockCapitalSharesValueReservedForFutureIssuance_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_CommonStockCapitalSharesValueReservedForFutureIssuance_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Value of common shares reserved for future issuance.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_CommonStockCapitalSharesValueReservedForFutureIssuance_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Common Stock, Capital Shares Value Reserved for Future Issuance</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_CommonStockCapitalSharesValueReservedForFutureIssuance_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Value of common stock reserved for future issuance</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_StockIssuedDuringPeriodSharesInExchangeOfCommitmentToPurchaseSharesAndNotToRedeem" xlink:label="adgm_StockIssuedDuringPeriodSharesInExchangeOfCommitmentToPurchaseSharesAndNotToRedeem" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_StockIssuedDuringPeriodSharesInExchangeOfCommitmentToPurchaseSharesAndNotToRedeem" xlink:to="adgm_StockIssuedDuringPeriodSharesInExchangeOfCommitmentToPurchaseSharesAndNotToRedeem_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_StockIssuedDuringPeriodSharesInExchangeOfCommitmentToPurchaseSharesAndNotToRedeem_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of shares issued during the period in exchange of commitment to purchase shares and not to redeem those shares.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_StockIssuedDuringPeriodSharesInExchangeOfCommitmentToPurchaseSharesAndNotToRedeem_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Stock issued During Period, Shares In Exchange Of Commitment To Purchase Shares And Not To Redeem</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_StockIssuedDuringPeriodSharesInExchangeOfCommitmentToPurchaseSharesAndNotToRedeem_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of shares issued in exchange of commitment to purchase Class A shares of ARYA in the open market and not to redeem</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_ClassOfWarrantOrRightNumberOfWarrantsIssuedInExchangeOfCommitmentToPurchaseSharesAndNotToRedeem" xlink:label="adgm_ClassOfWarrantOrRightNumberOfWarrantsIssuedInExchangeOfCommitmentToPurchaseSharesAndNotToRedeem" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_ClassOfWarrantOrRightNumberOfWarrantsIssuedInExchangeOfCommitmentToPurchaseSharesAndNotToRedeem" xlink:to="adgm_ClassOfWarrantOrRightNumberOfWarrantsIssuedInExchangeOfCommitmentToPurchaseSharesAndNotToRedeem_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_ClassOfWarrantOrRightNumberOfWarrantsIssuedInExchangeOfCommitmentToPurchaseSharesAndNotToRedeem_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of warrants issued to purchase stock  in exchange of commitment to purchase shares and not to redeem those shares.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_ClassOfWarrantOrRightNumberOfWarrantsIssuedInExchangeOfCommitmentToPurchaseSharesAndNotToRedeem_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class Of Warrant Or Right Number Of Warrants Issued In Exchange Of Commitment To Purchase Shares And Not To Redeem</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_ClassOfWarrantOrRightNumberOfWarrantsIssuedInExchangeOfCommitmentToPurchaseSharesAndNotToRedeem_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of warrants issued in exchange of commitment to purchase Class A shares of ARYA in the open market and not to redeem</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_NumberOfSharesSubjectedToRedemption" xlink:label="adgm_NumberOfSharesSubjectedToRedemption" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_NumberOfSharesSubjectedToRedemption" xlink:to="adgm_NumberOfSharesSubjectedToRedemption_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_NumberOfSharesSubjectedToRedemption_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of shares that are subjected to redemption.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_NumberOfSharesSubjectedToRedemption_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number Of Shares Subjected To Redemption</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_StockIssuedDuringPeriodSharesInExchangeOfNonRedemptionOfShares" xlink:label="adgm_StockIssuedDuringPeriodSharesInExchangeOfNonRedemptionOfShares" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_StockIssuedDuringPeriodSharesInExchangeOfNonRedemptionOfShares" xlink:to="adgm_StockIssuedDuringPeriodSharesInExchangeOfNonRedemptionOfShares_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_StockIssuedDuringPeriodSharesInExchangeOfNonRedemptionOfShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of shares issued during the period in exchange of non-redemption of shares.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_StockIssuedDuringPeriodSharesInExchangeOfNonRedemptionOfShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Stock issued During Period, Shares, In Exchange Of Non-redemption Of Shares</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_StockIssuedDuringPeriodSharesInExchangeOfNonRedemptionOfShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of shares issued in exchange of non-redemption of shares</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_ClassOfWarrantOrRightNumberOfWarrantsIssuedInExchangeOfNonRedemptionOfShares" xlink:label="adgm_ClassOfWarrantOrRightNumberOfWarrantsIssuedInExchangeOfNonRedemptionOfShares" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_ClassOfWarrantOrRightNumberOfWarrantsIssuedInExchangeOfNonRedemptionOfShares" xlink:to="adgm_ClassOfWarrantOrRightNumberOfWarrantsIssuedInExchangeOfNonRedemptionOfShares_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_ClassOfWarrantOrRightNumberOfWarrantsIssuedInExchangeOfNonRedemptionOfShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of warrants issued to purchase stock  in exchange of non-redemption of shares.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_ClassOfWarrantOrRightNumberOfWarrantsIssuedInExchangeOfNonRedemptionOfShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class Of Warrant Or Right Number Of Warrants Issued In Exchange Of Non-redemption Of Shares</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_ClassOfWarrantOrRightNumberOfWarrantsIssuedInExchangeOfNonRedemptionOfShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of warrants issued in exchange of non-redemption of shares</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_CommonStockCapitalSharesReservedForFutureIssuance" xlink:label="us-gaap_CommonStockCapitalSharesReservedForFutureIssuance" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_CommonStockCapitalSharesReservedForFutureIssuance" xlink:to="us-gaap_CommonStockCapitalSharesReservedForFutureIssuance_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_CommonStockCapitalSharesReservedForFutureIssuance_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Common Stock, Capital Shares Reserved for Future Issuance</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_CommonStockCapitalSharesReservedForFutureIssuance_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Common stock reserved for future issuance under the 2024 Equity Incentive Plan</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:label="us-gaap_CommonStockCapitalSharesReservedForFutureIssuance_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Common stock reserved for future issuance</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_CommonStockCapitalSharesReservedForFutureIssuance_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Common stock reserved for future issuance</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_ClassOfWarrantOrRightReservedForFutureIssuance" xlink:label="adgm_ClassOfWarrantOrRightReservedForFutureIssuance" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_ClassOfWarrantOrRightReservedForFutureIssuance" xlink:to="adgm_ClassOfWarrantOrRightReservedForFutureIssuance_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_ClassOfWarrantOrRightReservedForFutureIssuance_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of warrants reserved for future issuance.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_ClassOfWarrantOrRightReservedForFutureIssuance_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class of Warrant or Right, Reserved For Future Issuance</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_ClassOfWarrantOrRightReservedForFutureIssuance_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Warrants reserved for future issuance</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_CashInvestment" xlink:label="adgm_CashInvestment" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_CashInvestment" xlink:to="adgm_CashInvestment_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_CashInvestment_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amount of cash investment.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_CashInvestment_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cash Investment</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_InvestmentInDebtForm" xlink:label="adgm_InvestmentInDebtForm" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_InvestmentInDebtForm" xlink:to="adgm_InvestmentInDebtForm_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_InvestmentInDebtForm_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amount of investment in the form of debt.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_InvestmentInDebtForm_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Investment In Debt Form</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_InvestmentInDebtForm_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Investment in the form of debt</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_InterestPayableCurrentAndNoncurrent" xlink:label="us-gaap_InterestPayableCurrentAndNoncurrent" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_InterestPayableCurrentAndNoncurrent" xlink:to="us-gaap_InterestPayableCurrentAndNoncurrent_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_InterestPayableCurrentAndNoncurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Interest Payable</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_InterestPayableCurrentAndNoncurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Accrued interest</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_AdditionalCashInvestment" xlink:label="adgm_AdditionalCashInvestment" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_AdditionalCashInvestment" xlink:to="adgm_AdditionalCashInvestment_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_AdditionalCashInvestment_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amount of additional cash investment.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_AdditionalCashInvestment_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Additional Cash Investment</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_StockIssuedDuringPeriodSharesInExchangeOfConvertibleDebtAndAdditionalCashInvestment" xlink:label="adgm_StockIssuedDuringPeriodSharesInExchangeOfConvertibleDebtAndAdditionalCashInvestment" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_StockIssuedDuringPeriodSharesInExchangeOfConvertibleDebtAndAdditionalCashInvestment" xlink:to="adgm_StockIssuedDuringPeriodSharesInExchangeOfConvertibleDebtAndAdditionalCashInvestment_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_StockIssuedDuringPeriodSharesInExchangeOfConvertibleDebtAndAdditionalCashInvestment_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of shares issued during the period in exchange of convertible debt and additional cash investment.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_StockIssuedDuringPeriodSharesInExchangeOfConvertibleDebtAndAdditionalCashInvestment_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Stock issued During Period, Shares, In Exchange Of Convertible Debt And Additional Cash Investment</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_StockIssuedDuringPeriodSharesInExchangeOfConvertibleDebtAndAdditionalCashInvestment_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of shares issued in exchange of convertible debt and additional cash investment</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_ClassOfWarrantOrRightNumberOfWarrantsIssuedInExchangeOfConvertibleDebtAndAdditionalCashInvestment" xlink:label="adgm_ClassOfWarrantOrRightNumberOfWarrantsIssuedInExchangeOfConvertibleDebtAndAdditionalCashInvestment" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_ClassOfWarrantOrRightNumberOfWarrantsIssuedInExchangeOfConvertibleDebtAndAdditionalCashInvestment" xlink:to="adgm_ClassOfWarrantOrRightNumberOfWarrantsIssuedInExchangeOfConvertibleDebtAndAdditionalCashInvestment_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_ClassOfWarrantOrRightNumberOfWarrantsIssuedInExchangeOfConvertibleDebtAndAdditionalCashInvestment_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of warrants issued to purchase stock  in exchange of convertible debt and additional cash investment.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_ClassOfWarrantOrRightNumberOfWarrantsIssuedInExchangeOfConvertibleDebtAndAdditionalCashInvestment_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class Of Warrant Or Right Number Of Warrants Issued In Exchange Of Convertible Debt And Additional Cash Investment</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_ClassOfWarrantOrRightNumberOfWarrantsIssuedInExchangeOfConvertibleDebtAndAdditionalCashInvestment_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of warrants issued in exchange of convertible debt and additional cash investment</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_DebtInstrumentFaceAmount" xlink:label="us-gaap_DebtInstrumentFaceAmount" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_DebtInstrumentFaceAmount" xlink:to="us-gaap_DebtInstrumentFaceAmount_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_DebtInstrumentFaceAmount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Debt Instrument, Face Amount</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_DebtInstrumentFaceAmount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Principal amount</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_DebtInstrumentFaceAmount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Debt instrument, face amount</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_DebtInstrumentInterestRateStatedPercentage" xlink:label="us-gaap_DebtInstrumentInterestRateStatedPercentage" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_DebtInstrumentInterestRateStatedPercentage" xlink:to="us-gaap_DebtInstrumentInterestRateStatedPercentage_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_DebtInstrumentInterestRateStatedPercentage_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Debt Instrument, Interest Rate, Stated Percentage</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_DebtInstrumentInterestRateStatedPercentage_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Interest rate</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_DebtInstrumentConvertibleConversionPrice1" xlink:label="us-gaap_DebtInstrumentConvertibleConversionPrice1" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_DebtInstrumentConvertibleConversionPrice1" xlink:to="us-gaap_DebtInstrumentConvertibleConversionPrice1_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_DebtInstrumentConvertibleConversionPrice1_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Debt Instrument, Convertible, Conversion Price</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_DebtInstrumentConvertibleConversionPrice1_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Conversion price (in USD per share)</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_DebtConversionConvertedInstrumentWarrantsOrOptionsIssued1" xlink:label="us-gaap_DebtConversionConvertedInstrumentWarrantsOrOptionsIssued1" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_DebtConversionConvertedInstrumentWarrantsOrOptionsIssued1" xlink:to="us-gaap_DebtConversionConvertedInstrumentWarrantsOrOptionsIssued1_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_DebtConversionConvertedInstrumentWarrantsOrOptionsIssued1_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Debt Conversion, Converted Instrument, Warrants or Options Issued</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_DebtConversionConvertedInstrumentWarrantsOrOptionsIssued1_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Warrants issued upon conversion of debt</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ClassOfWarrantOrRightExercisePriceOfWarrantsOrRights1" xlink:label="us-gaap_ClassOfWarrantOrRightExercisePriceOfWarrantsOrRights1" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ClassOfWarrantOrRightExercisePriceOfWarrantsOrRights1" xlink:to="us-gaap_ClassOfWarrantOrRightExercisePriceOfWarrantsOrRights1_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ClassOfWarrantOrRightExercisePriceOfWarrantsOrRights1_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class of Warrant or Right, Exercise Price of Warrants or Rights</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ClassOfWarrantOrRightExercisePriceOfWarrantsOrRights1_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Exercise price (in USD per share)</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_BusinessAcquisitionAxis" xlink:label="us-gaap_BusinessAcquisitionAxis" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_BusinessAcquisitionAxis" xlink:to="us-gaap_BusinessAcquisitionAxis_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_BusinessAcquisitionAxis_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Business Acquisition [Axis]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_BusinessAcquisitionAcquireeDomain" xlink:label="us-gaap_BusinessAcquisitionAcquireeDomain" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_BusinessAcquisitionAcquireeDomain" xlink:to="us-gaap_BusinessAcquisitionAcquireeDomain_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_BusinessAcquisitionAcquireeDomain_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Business Acquisition, Acquiree [Domain]</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_LegacyAdagioMember" xlink:label="adgm_LegacyAdagioMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_LegacyAdagioMember" xlink:to="adgm_LegacyAdagioMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_LegacyAdagioMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents information pertaining to Legacy Adagio.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_LegacyAdagioMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Legacy Adagio [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_LegacyAdagioMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Legacy Adagio</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_BusinessAcquisitionPercentageOfVotingInterestsAcquired" xlink:label="us-gaap_BusinessAcquisitionPercentageOfVotingInterestsAcquired" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_BusinessAcquisitionPercentageOfVotingInterestsAcquired" xlink:to="us-gaap_BusinessAcquisitionPercentageOfVotingInterestsAcquired_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_BusinessAcquisitionPercentageOfVotingInterestsAcquired_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Business Acquisition, Percentage of Voting Interests Acquired</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_BusinessAcquisitionPercentageOfVotingInterestsAcquired_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Voting rights acquired</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_BusinessAcquisitionEquityInterestsIssuedOrIssuableNumberOfSharesIssued" xlink:label="us-gaap_BusinessAcquisitionEquityInterestsIssuedOrIssuableNumberOfSharesIssued" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_BusinessAcquisitionEquityInterestsIssuedOrIssuableNumberOfSharesIssued" xlink:to="us-gaap_BusinessAcquisitionEquityInterestsIssuedOrIssuableNumberOfSharesIssued_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_BusinessAcquisitionEquityInterestsIssuedOrIssuableNumberOfSharesIssued_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Business Acquisition, Equity Interest Issued or Issuable, Number of Shares</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_BusinessAcquisitionEquityInterestsIssuedOrIssuableNumberOfSharesIssued_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of the Company's Common Stock issued</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsGrantsInPeriod" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsGrantsInPeriod" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsGrantsInPeriod" xlink:to="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsGrantsInPeriod_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsGrantsInPeriod_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Share-Based Compensation Arrangement by Share-Based Payment Award, Options, Grants in Period, Net of Forfeitures</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsGrantsInPeriod_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of replacement stock options granted to Legacy Adagio's option holders by the Company</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_BusinessCombinationNumberOfSharesAndOptionsIssued" xlink:label="adgm_BusinessCombinationNumberOfSharesAndOptionsIssued" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_BusinessCombinationNumberOfSharesAndOptionsIssued" xlink:to="adgm_BusinessCombinationNumberOfSharesAndOptionsIssued_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_BusinessCombinationNumberOfSharesAndOptionsIssued_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Total number of shares and options granted as part of business combination.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_BusinessCombinationNumberOfSharesAndOptionsIssued_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Business Combination, Number Of Shares And Options Issued</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:label="adgm_BusinessCombinationNumberOfSharesAndOptionsIssued_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Total shares and stock options</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_SharePrice" xlink:label="us-gaap_SharePrice" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_SharePrice" xlink:to="us-gaap_SharePrice_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_SharePrice_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Share Price</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_SharePrice_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Multiplied by the Company's Common Stock price at the Closing</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_BusinessCombinationConsiderationTransferredEquityInterestsIssuedAndIssuable" xlink:label="us-gaap_BusinessCombinationConsiderationTransferredEquityInterestsIssuedAndIssuable" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_BusinessCombinationConsiderationTransferredEquityInterestsIssuedAndIssuable" xlink:to="us-gaap_BusinessCombinationConsiderationTransferredEquityInterestsIssuedAndIssuable_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_BusinessCombinationConsiderationTransferredEquityInterestsIssuedAndIssuable_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Business Combination, Consideration Transferred, Equity Interests Issued and Issuable</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_BusinessCombinationConsiderationTransferredEquityInterestsIssuedAndIssuable_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Total</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_ClassOfWarrantOrRightNumberOfwarrantsIssuedInSettlementOfDebt" xlink:label="adgm_ClassOfWarrantOrRightNumberOfwarrantsIssuedInSettlementOfDebt" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_ClassOfWarrantOrRightNumberOfwarrantsIssuedInSettlementOfDebt" xlink:to="adgm_ClassOfWarrantOrRightNumberOfwarrantsIssuedInSettlementOfDebt_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_ClassOfWarrantOrRightNumberOfwarrantsIssuedInSettlementOfDebt_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of warrants issued as settlement of debt.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_ClassOfWarrantOrRightNumberOfwarrantsIssuedInSettlementOfDebt_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class of Warrant or Right, Number of Warrants Issued In Settlement Of Debt</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_ClassOfWarrantOrRightNumberOfwarrantsIssuedInSettlementOfDebt_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of PIPE Base Warrants issued in lieu of settling Bridge Financing Notes</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_ClassOfWarrantOrRightPriceOfWarrantsOrRights" xlink:label="adgm_ClassOfWarrantOrRightPriceOfWarrantsOrRights" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_ClassOfWarrantOrRightPriceOfWarrantsOrRights" xlink:to="adgm_ClassOfWarrantOrRightPriceOfWarrantsOrRights_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_ClassOfWarrantOrRightPriceOfWarrantsOrRights_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Price per share or per unit of warrants or rights outstanding.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_ClassOfWarrantOrRightPriceOfWarrantsOrRights_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class of Warrant or Right, Price of Warrants or Rights</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_ClassOfWarrantOrRightPriceOfWarrantsOrRights_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Multiplied by estimated value of PIPE Base Warrants at the Closing</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_BusinessCombinationConsiderationTransferredWarrantRightsIssued" xlink:label="adgm_BusinessCombinationConsiderationTransferredWarrantRightsIssued" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_BusinessCombinationConsiderationTransferredWarrantRightsIssued" xlink:to="adgm_BusinessCombinationConsiderationTransferredWarrantRightsIssued_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_BusinessCombinationConsiderationTransferredWarrantRightsIssued_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amount of fair value of warrants issued  in consideration for the business combination.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_BusinessCombinationConsiderationTransferredWarrantRightsIssued_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Business Combination, Consideration Transferred, Warrant Rights Issued</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_BusinessCombinationConsiderationTransferredWarrantRightsIssued_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Estimated fair value of PIPE Base Warrants issued in lieu of settling Bridge Financing Notes</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_BusinessCombinationConsiderationTransferred1" xlink:label="us-gaap_BusinessCombinationConsiderationTransferred1" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_BusinessCombinationConsiderationTransferred1" xlink:to="us-gaap_BusinessCombinationConsiderationTransferred1_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_BusinessCombinationConsiderationTransferred1_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Business Combination, Consideration Transferred</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:label="us-gaap_BusinessCombinationConsiderationTransferred1_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Total purchase price</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedAssetsAbstract" xlink:label="us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedAssetsAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedAssetsAbstract" xlink:to="us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedAssetsAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedAssetsAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Business Combination, Recognized Identifiable Assets Acquired and Liabilities Assumed, Assets [Abstract]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedAssetsAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Assets Acquired:</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedCashAndEquivalents" xlink:label="us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedCashAndEquivalents" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedCashAndEquivalents" xlink:to="us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedCashAndEquivalents_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedCashAndEquivalents_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Business Combination, Recognized Identifiable Assets Acquired and Liabilities Assumed, Cash and Equivalents</label>
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_StockholdersEquityOpeningBalance_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amount of equity (deficit) attributable to parent. Excludes
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_StockholdersEquityOpeningBalance_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Stockholders Equity, Opening Balance</label>
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    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_StockIssuedDuringPeriodSharesConversionOfConvertibleSecurities" xlink:label="us-gaap_StockIssuedDuringPeriodSharesConversionOfConvertibleSecurities" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_StockIssuedDuringPeriodSharesConversionOfConvertibleSecurities_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Conversion of ARYA convertible promissory Notes (in shares)</label>
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_StockIssuedDuringPeriodValueConversionOfConvertibleSecurities_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Stock Issued During Period, Value, Conversion of Convertible Securities</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_StockIssuedDuringPeriodValueConversionOfConvertibleSecurities_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Conversion of ARYA convertible promissory Notes</label>
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_StockIssuedDuringPeriodConversionOfSharesOfAcquireeEntityInProcessOfMerger_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Conversion of ARYA Class A ordinary shares and Class B ordinary shares (in shares)</label>
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_StockIssuedDuringPeriodSharesConversionOfTemporaryEquity_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Stock Issued During Period, Shares, Conversion of Temporary Equity</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_StockIssuedDuringPeriodSharesConversionOfTemporaryEquity_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Conversion of Class A ordinary shares subject to redemption (in shares)</label>
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_StockIssuedDuringPeriodValueConversionOfTemporaryEquity_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Conversion of Class A ordinary shares subject to redemption</label>
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_StockIssuedDuringPeriodSharesAcquisitions_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Shares issued for acquisition of Legacy Adagio</label>
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_StockIssuedDuringPeriodValueAcquisitions_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Stock Issued During Period, Value, Acquisitions</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_StockIssuedDuringPeriodValueAcquisitions_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Shares issued for acquisition of Legacy Adagio</label>
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_StockIssuedDuringPeriodValueAdditionalSharesAndReclassificationOfShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Stock Issued During Period, Value, Additional Shares And Reclassification Of Shares</label>
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_StockholdersEquityEndingBalance_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amount of equity (deficit) attributable to parent. Excludes temporary equity and equity attributable to
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    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_FairValueByFairValueHierarchyLevelAxis" xlink:label="us-gaap_FairValueByFairValueHierarchyLevelAxis" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_FairValueByFairValueHierarchyLevelAxis" xlink:to="us-gaap_FairValueByFairValueHierarchyLevelAxis_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_FairValueByFairValueHierarchyLevelAxis_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fair Value Hierarchy and NAV [Axis]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_FairValueMeasurementsFairValueHierarchyDomain" xlink:label="us-gaap_FairValueMeasurementsFairValueHierarchyDomain" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_FairValueMeasurementsFairValueHierarchyDomain" xlink:to="us-gaap_FairValueMeasurementsFairValueHierarchyDomain_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_FairValueMeasurementsFairValueHierarchyDomain_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fair Value Hierarchy and NAV [Domain]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_FairValueInputsLevel1Member" xlink:label="us-gaap_FairValueInputsLevel1Member" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_FairValueInputsLevel1Member" xlink:to="us-gaap_FairValueInputsLevel1Member_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_FairValueInputsLevel1Member_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fair Value, Inputs, Level 1 [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_FairValueInputsLevel1Member_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Level 1</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_FairValueInputsLevel3Member" xlink:label="us-gaap_FairValueInputsLevel3Member" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_FairValueInputsLevel3Member" xlink:to="us-gaap_FairValueInputsLevel3Member_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_FairValueInputsLevel3Member_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fair Value, Inputs, Level 3 [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_FairValueInputsLevel3Member_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Level 3</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_FairValueByMeasurementFrequencyAxis" xlink:label="us-gaap_FairValueByMeasurementFrequencyAxis" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_FairValueByMeasurementFrequencyAxis" xlink:to="us-gaap_FairValueByMeasurementFrequencyAxis_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_FairValueByMeasurementFrequencyAxis_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Measurement Frequency [Axis]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_FairValueByMeasurementFrequencyAxis_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Measurement Frequency [Axis]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_FairValueMeasurementFrequencyDomain" xlink:label="us-gaap_FairValueMeasurementFrequencyDomain" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_FairValueMeasurementFrequencyDomain" xlink:to="us-gaap_FairValueMeasurementFrequencyDomain_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_FairValueMeasurementFrequencyDomain_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Measurement Frequency [Domain]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_FairValueMeasurementFrequencyDomain_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Measurement Frequency [Domain]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_FairValueMeasurementsRecurringMember" xlink:label="us-gaap_FairValueMeasurementsRecurringMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_FairValueMeasurementsRecurringMember" xlink:to="us-gaap_FairValueMeasurementsRecurringMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_FairValueMeasurementsRecurringMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fair Value, Recurring [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_FairValueMeasurementsRecurringMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fair value measurements recurring basis</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_FinancialInstrumentsFinancialAssetsBalanceSheetGroupingsAbstract" xlink:label="us-gaap_FinancialInstrumentsFinancialAssetsBalanceSheetGroupingsAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_FinancialInstrumentsFinancialAssetsBalanceSheetGroupingsAbstract" xlink:to="us-gaap_FinancialInstrumentsFinancialAssetsBalanceSheetGroupingsAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_FinancialInstrumentsFinancialAssetsBalanceSheetGroupingsAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Financial Instruments, Financial Assets, Balance Sheet Groupings [Abstract]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_FinancialInstrumentsFinancialAssetsBalanceSheetGroupingsAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Assets:</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_CashAndCashEquivalentsFairValueDisclosure" xlink:label="us-gaap_CashAndCashEquivalentsFairValueDisclosure" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_CashAndCashEquivalentsFairValueDisclosure" xlink:to="us-gaap_CashAndCashEquivalentsFairValueDisclosure_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_CashAndCashEquivalentsFairValueDisclosure_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cash and Cash Equivalents, Fair Value Disclosure</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_CashAndCashEquivalentsFairValueDisclosure_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Money market account</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_FinancialInstrumentsFinancialLiabilitiesBalanceSheetGroupingsAbstract" xlink:label="us-gaap_FinancialInstrumentsFinancialLiabilitiesBalanceSheetGroupingsAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_FinancialInstrumentsFinancialLiabilitiesBalanceSheetGroupingsAbstract" xlink:to="us-gaap_FinancialInstrumentsFinancialLiabilitiesBalanceSheetGroupingsAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_FinancialInstrumentsFinancialLiabilitiesBalanceSheetGroupingsAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Financial Instruments, Financial Liabilities, Balance Sheet Groupings [Abstract]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_FinancialInstrumentsFinancialLiabilitiesBalanceSheetGroupingsAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Liabilities:</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ConvertibleDebtFairValueDisclosures" xlink:label="us-gaap_ConvertibleDebtFairValueDisclosures" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ConvertibleDebtFairValueDisclosures" xlink:to="us-gaap_ConvertibleDebtFairValueDisclosures_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ConvertibleDebtFairValueDisclosures_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Convertible Debt, Fair Value Disclosures</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ConvertibleDebtFairValueDisclosures_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Convertible notes payables</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_DerivativeLiabilities" xlink:label="us-gaap_DerivativeLiabilities" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_DerivativeLiabilities" xlink:to="us-gaap_DerivativeLiabilities_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_DerivativeLiabilities_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Derivative Liability</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_FairValueAssetLevel1ToLevel2TransfersAmount" xlink:label="adgm_FairValueAssetLevel1ToLevel2TransfersAmount" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_FairValueAssetLevel1ToLevel2TransfersAmount" xlink:to="adgm_FairValueAssetLevel1ToLevel2TransfersAmount_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_FairValueAssetLevel1ToLevel2TransfersAmount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amount of transfers of assets measured on a recurring basis out of Level 1 of the fair value hierarchy into Level 2.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_FairValueAssetLevel1ToLevel2TransfersAmount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fair Value, Asset, Level 1 to Level 2 Transfers, Amount</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_FairValueAssetLevel1ToLevel2TransfersAmount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fair value asset transfer level 1 to level 2</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_FairValueAssetLevel2ToLevel1TransfersAmount" xlink:label="adgm_FairValueAssetLevel2ToLevel1TransfersAmount" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_FairValueAssetLevel2ToLevel1TransfersAmount" xlink:to="adgm_FairValueAssetLevel2ToLevel1TransfersAmount_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_FairValueAssetLevel2ToLevel1TransfersAmount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amount of transfers of assets measured on a recurring basis out of Level 2 of the fair value hierarchy into Level 1.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_FairValueAssetLevel2ToLevel1TransfersAmount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fair Value, Asset, Level 2 to Level 1 Transfers, Amount</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_FairValueAssetLevel2ToLevel1TransfersAmount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fair value asset transfer level 2 to level 1</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisAssetTransfersNet" xlink:label="us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisAssetTransfersNet" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisAssetTransfersNet" xlink:to="us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisAssetTransfersNet_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisAssetTransfersNet_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fair Value, Measurement with Unobservable Inputs Reconciliation, Recurring Basis, Asset, Transfers, Net</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisAssetTransfersNet_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fair value asset transfer net, level 3</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_FairValueLiabilityLevel1ToLevel2TransfersAmount" xlink:label="adgm_FairValueLiabilityLevel1ToLevel2TransfersAmount" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_FairValueLiabilityLevel1ToLevel2TransfersAmount" xlink:to="adgm_FairValueLiabilityLevel1ToLevel2TransfersAmount_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_FairValueLiabilityLevel1ToLevel2TransfersAmount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amount of transfers of liabilities measured on a recurring basis out of Level 1 of the fair value hierarchy into Level 2.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_FairValueLiabilityLevel1ToLevel2TransfersAmount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fair Value, Liability, Level 1 to Level 2 Transfers, Amount</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_FairValueLiabilityLevel1ToLevel2TransfersAmount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fair value liability transfer level 1 to level 2</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_FairValueLiabilityLevel2ToLevel1TransfersAmount" xlink:label="adgm_FairValueLiabilityLevel2ToLevel1TransfersAmount" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_FairValueLiabilityLevel2ToLevel1TransfersAmount" xlink:to="adgm_FairValueLiabilityLevel2ToLevel1TransfersAmount_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_FairValueLiabilityLevel2ToLevel1TransfersAmount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amount of transfers of liabilities measured on a recurring basis out of Level 2 of the fair value hierarchy into Level 1.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_FairValueLiabilityLevel2ToLevel1TransfersAmount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fair Value, Liability Level 2 to Level 1 Transfers, Amount</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_FairValueLiabilityLevel2ToLevel1TransfersAmount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fair value liability transfer level 2 to level 1</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisLiabilityTransfersNet" xlink:label="us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisLiabilityTransfersNet" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisLiabilityTransfersNet" xlink:to="us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisLiabilityTransfersNet_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisLiabilityTransfersNet_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fair Value, Measurement with Unobservable Inputs Reconciliation, Recurring Basis, Liability, Transfers, Net</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisLiabilityTransfersNet_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fair value liability transfer net, level 3</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_FairValueAssetsAndLiabilitiesMeasuredOnRecurringAndNonrecurringBasisValuationTechniquesTable" xlink:label="us-gaap_FairValueAssetsAndLiabilitiesMeasuredOnRecurringAndNonrecurringBasisValuationTechniquesTable" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_FairValueAssetsAndLiabilitiesMeasuredOnRecurringAndNonrecurringBasisValuationTechniquesTable" xlink:to="us-gaap_FairValueAssetsAndLiabilitiesMeasuredOnRecurringAndNonrecurringBasisValuationTechniquesTable_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_FairValueAssetsAndLiabilitiesMeasuredOnRecurringAndNonrecurringBasisValuationTechniquesTable_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fair Value Measurement Inputs and Valuation Techniques [Table]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_FairValueAssetsAndLiabilitiesMeasuredOnRecurringAndNonrecurringBasisValuationTechniquesTable_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Schedule of valuation techniques</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_October2022ConvertibleNotesMember" xlink:label="adgm_October2022ConvertibleNotesMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_October2022ConvertibleNotesMember" xlink:to="adgm_October2022ConvertibleNotesMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_October2022ConvertibleNotesMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents the information pertaining to October 2022 convertible notes.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_October2022ConvertibleNotesMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">October 2022 Convertible Notes [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_October2022ConvertibleNotesMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">October 2022 Convertible Notes</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="adgm_October2022ConvertibleNotesMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">October 2022 Convertible Notes measured at fair value</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_April2023ConvertibleNotesMember" xlink:label="adgm_April2023ConvertibleNotesMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_April2023ConvertibleNotesMember" xlink:to="adgm_April2023ConvertibleNotesMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_April2023ConvertibleNotesMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents the information pertaining to April 2023 convertible notes.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_April2023ConvertibleNotesMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">April 2023 Convertible Notes [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_April2023ConvertibleNotesMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">April 2023 Convertible Notes</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="adgm_April2023ConvertibleNotesMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">April 2023 Convertible Notes measured at fair value</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_November2023ConvertibleNotesMember" xlink:label="adgm_November2023ConvertibleNotesMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_November2023ConvertibleNotesMember" xlink:to="adgm_November2023ConvertibleNotesMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_November2023ConvertibleNotesMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents the information pertaining to November 2023 convertible notes.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_November2023ConvertibleNotesMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">November 2023 Convertible Notes [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_November2023ConvertibleNotesMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">November 2023 Convertible Notes</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="adgm_November2023ConvertibleNotesMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">November 2023 Convertible Notes measured at fair value</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_February2024ConvertibleNotesMember" xlink:label="adgm_February2024ConvertibleNotesMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_February2024ConvertibleNotesMember" xlink:to="adgm_February2024ConvertibleNotesMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_February2024ConvertibleNotesMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents the information pertaining to February 2024 convertible notes.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_February2024ConvertibleNotesMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">February 2024 Convertible Notes [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_February2024ConvertibleNotesMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">February 2024 Convertible Notes</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_May2024ConvertibleNotesMember" xlink:label="adgm_May2024ConvertibleNotesMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_May2024ConvertibleNotesMember" xlink:to="adgm_May2024ConvertibleNotesMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_May2024ConvertibleNotesMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents information pertaining to May 2024 Convertible Notes.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_May2024ConvertibleNotesMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">May 2024 Convertible Notes [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_May2024ConvertibleNotesMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">May 2024 Convertible Notes</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_June2024ConvertibleNotesMember" xlink:label="adgm_June2024ConvertibleNotesMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_June2024ConvertibleNotesMember" xlink:to="adgm_June2024ConvertibleNotesMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_June2024ConvertibleNotesMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents information pertaining to June 2024 Convertible Notes.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_June2024ConvertibleNotesMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">June 2024 Convertible Notes [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_June2024ConvertibleNotesMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">June 2024 Convertible Notes</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_July2024ConvertibleNotesMember" xlink:label="adgm_July2024ConvertibleNotesMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_July2024ConvertibleNotesMember" xlink:to="adgm_July2024ConvertibleNotesMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_July2024ConvertibleNotesMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents information pertaining to July 2024 convertible notes.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_July2024ConvertibleNotesMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">July 2024 Convertible Notes [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_July2024ConvertibleNotesMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">July 2024 Convertible Notes</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_FairValueAssetsAndLiabilitiesMeasuredOnRecurringAndNonrecurringBasisValuationTechniquesLineItems" xlink:label="us-gaap_FairValueAssetsAndLiabilitiesMeasuredOnRecurringAndNonrecurringBasisValuationTechniquesLineItems" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_FairValueAssetsAndLiabilitiesMeasuredOnRecurringAndNonrecurringBasisValuationTechniquesLineItems" xlink:to="us-gaap_FairValueAssetsAndLiabilitiesMeasuredOnRecurringAndNonrecurringBasisValuationTechniquesLineItems_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_FairValueAssetsAndLiabilitiesMeasuredOnRecurringAndNonrecurringBasisValuationTechniquesLineItems_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fair Value Measurement Inputs and Valuation Techniques [Line Items]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_FairValueAssetsAndLiabilitiesMeasuredOnRecurringAndNonrecurringBasisValuationTechniquesLineItems_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fair Value Measurements</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_DebtInstrumentTerm" xlink:label="us-gaap_DebtInstrumentTerm" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_DebtInstrumentTerm" xlink:to="us-gaap_DebtInstrumentTerm_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_DebtInstrumentTerm_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Debt Instrument, Term</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_DebtInstrumentTerm_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Debt term</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_DebtConversionConvertedInstrumentSharesIssued1" xlink:label="us-gaap_DebtConversionConvertedInstrumentSharesIssued1" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_DebtConversionConvertedInstrumentSharesIssued1" xlink:to="us-gaap_DebtConversionConvertedInstrumentSharesIssued1_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_DebtConversionConvertedInstrumentSharesIssued1_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Debt Conversion, Converted Instrument, Shares Issued</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_DebtConversionConvertedInstrumentSharesIssued1_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of shares issued in conversion to shares</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_DebtConversionConvertedInstrumentSharesIssued1_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Warrants issued upon conversion of debt</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_DebtInstrumentMaximumAdditionalAmount" xlink:label="adgm_DebtInstrumentMaximumAdditionalAmount" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_DebtInstrumentMaximumAdditionalAmount" xlink:to="adgm_DebtInstrumentMaximumAdditionalAmount_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_DebtInstrumentMaximumAdditionalAmount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amount of maximum additional debt that can be taken.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_DebtInstrumentMaximumAdditionalAmount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Debt Instrument, Maximum Additional Amount</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_DebtInstrumentMaximumAdditionalAmount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Maximum additional debt</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ConvertibleNotesPayableCurrent" xlink:label="us-gaap_ConvertibleNotesPayableCurrent" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ConvertibleNotesPayableCurrent" xlink:to="us-gaap_ConvertibleNotesPayableCurrent_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ConvertibleNotesPayableCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Convertible Notes Payable, Current</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ConvertibleNotesPayableCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Convertible notes payable, current</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_FairValueLiabilitiesMeasuredOnRecurringBasisUnobservableInputReconciliationTable" xlink:label="us-gaap_FairValueLiabilitiesMeasuredOnRecurringBasisUnobservableInputReconciliationTable" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_FairValueLiabilitiesMeasuredOnRecurringBasisUnobservableInputReconciliationTable" xlink:to="us-gaap_FairValueLiabilitiesMeasuredOnRecurringBasisUnobservableInputReconciliationTable_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_FairValueLiabilitiesMeasuredOnRecurringBasisUnobservableInputReconciliationTable_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fair Value, Liabilities Measured on Recurring Basis, Unobservable Input Reconciliation [Table]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_FairValueLiabilitiesMeasuredOnRecurringBasisUnobservableInputReconciliationLineItems" xlink:label="us-gaap_FairValueLiabilitiesMeasuredOnRecurringBasisUnobservableInputReconciliationLineItems" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_FairValueLiabilitiesMeasuredOnRecurringBasisUnobservableInputReconciliationLineItems" xlink:to="us-gaap_FairValueLiabilitiesMeasuredOnRecurringBasisUnobservableInputReconciliationLineItems_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_FairValueLiabilitiesMeasuredOnRecurringBasisUnobservableInputReconciliationLineItems_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fair Value, Liabilities Measured on Recurring Basis, Unobservable Input Reconciliation [Line Items]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_FairValueLiabilitiesMeasuredOnRecurringBasisUnobservableInputReconciliationLineItems_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fair Value Measurements</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_FairValueLiabilitiesMeasuredOnRecurringBasisUnobservableInputReconciliationCalculationRollForward" xlink:label="us-gaap_FairValueLiabilitiesMeasuredOnRecurringBasisUnobservableInputReconciliationCalculationRollForward" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_FairValueLiabilitiesMeasuredOnRecurringBasisUnobservableInputReconciliationCalculationRollForward" xlink:to="us-gaap_FairValueLiabilitiesMeasuredOnRecurringBasisUnobservableInputReconciliationCalculationRollForward_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_FairValueLiabilitiesMeasuredOnRecurringBasisUnobservableInputReconciliationCalculationRollForward_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fair Value, Liabilities Measured on Recurring Basis, Unobservable Input Reconciliation, Calculation [Roll Forward]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_FairValueLiabilitiesMeasuredOnRecurringBasisUnobservableInputReconciliationCalculationRollForward_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Level 3 Reconciliation of Convertible promissory notes</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_FairValueLiabilitiesMeasuredOnRecurringBasisUnobservableInputReconciliationCalculationRollForward_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Changes in Level 3 Reconciliation</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationsRecurringBasisLiabilityValue" xlink:label="us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationsRecurringBasisLiabilityValue" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationsRecurringBasisLiabilityValue" xlink:to="us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationsRecurringBasisLiabilityValue_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationsRecurringBasisLiabilityValue_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fair Value, Measurement with Unobservable Inputs Reconciliation, Recurring Basis, Liability Value</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/periodEndLabel" xlink:label="us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationsRecurringBasisLiabilityValue_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Balance (end of year)</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/periodStartLabel" xlink:label="us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationsRecurringBasisLiabilityValue_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Balance (beginning of year)</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisLiabilityIssues" xlink:label="us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisLiabilityIssues" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisLiabilityIssues" xlink:to="us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisLiabilityIssues_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisLiabilityIssues_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fair Value, Measurement with Unobservable Inputs Reconciliation, Recurring Basis, Liability, Issuances</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisLiabilityIssues_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Additions</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisLiabilityIssues_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Accrued interest</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_DebtInstrumentTable" xlink:label="us-gaap_DebtInstrumentTable" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_DebtInstrumentTable" xlink:to="us-gaap_DebtInstrumentTable_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_DebtInstrumentTable_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Schedule of Long-Term Debt Instruments [Table]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_DebtInstrumentLineItems" xlink:label="us-gaap_DebtInstrumentLineItems" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_DebtInstrumentLineItems" xlink:to="us-gaap_DebtInstrumentLineItems_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_DebtInstrumentLineItems_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Debt Instrument [Line Items]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_DebtInstrumentLineItems_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Debt</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_MeasurementInputTypeAxis" xlink:label="us-gaap_MeasurementInputTypeAxis" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_MeasurementInputTypeAxis" xlink:to="us-gaap_MeasurementInputTypeAxis_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_MeasurementInputTypeAxis_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Measurement Input Type [Axis]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_MeasurementInputTypeDomain" xlink:label="us-gaap_MeasurementInputTypeDomain" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_MeasurementInputTypeDomain" xlink:to="us-gaap_MeasurementInputTypeDomain_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_MeasurementInputTypeDomain_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Measurement Input Type [Domain]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_MeasurementInputSharePriceMember" xlink:label="us-gaap_MeasurementInputSharePriceMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_MeasurementInputSharePriceMember" xlink:to="us-gaap_MeasurementInputSharePriceMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_MeasurementInputSharePriceMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Measurement Input, Share Price [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_MeasurementInputSharePriceMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Share price</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_MeasurementInputDiscountRateMember" xlink:label="us-gaap_MeasurementInputDiscountRateMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_MeasurementInputDiscountRateMember" xlink:to="us-gaap_MeasurementInputDiscountRateMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_MeasurementInputDiscountRateMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Measurement Input, Discount Rate [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_MeasurementInputDiscountRateMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Discount rate</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_MeasurementInputExpectedTermMember" xlink:label="us-gaap_MeasurementInputExpectedTermMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_MeasurementInputExpectedTermMember" xlink:to="us-gaap_MeasurementInputExpectedTermMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_MeasurementInputExpectedTermMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Measurement Input, Expected Term [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_MeasurementInputExpectedTermMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Expected Term (years)</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_MeasurementInputRiskFreeInterestRateMember" xlink:label="us-gaap_MeasurementInputRiskFreeInterestRateMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_MeasurementInputRiskFreeInterestRateMember" xlink:to="us-gaap_MeasurementInputRiskFreeInterestRateMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_MeasurementInputRiskFreeInterestRateMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Measurement Input, Risk Free Interest Rate [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_MeasurementInputRiskFreeInterestRateMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Risk-Free interest rate</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_DebtInstrumentMeasurementInput" xlink:label="us-gaap_DebtInstrumentMeasurementInput" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_DebtInstrumentMeasurementInput" xlink:to="us-gaap_DebtInstrumentMeasurementInput_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_DebtInstrumentMeasurementInput_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Debt Instrument, Measurement Input</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_DebtInstrumentMeasurementInput_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Promissory notes measurement input</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisLiabilitySettlements" xlink:label="us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisLiabilitySettlements" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisLiabilitySettlements" xlink:to="us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisLiabilitySettlements_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisLiabilitySettlements_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fair Value, Measurement with Unobservable Inputs Reconciliation, Recurring Basis, Liability, Settlements</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2009/role/negatedLabel" xlink:label="us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisLiabilitySettlements_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Exercises</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ClassOfWarrantOrRightTable" xlink:label="us-gaap_ClassOfWarrantOrRightTable" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ClassOfWarrantOrRightTable" xlink:to="us-gaap_ClassOfWarrantOrRightTable_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ClassOfWarrantOrRightTable_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class of Warrant or Right [Table]</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_PreFundedWarrantsMember" xlink:label="adgm_PreFundedWarrantsMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_PreFundedWarrantsMember" xlink:to="adgm_PreFundedWarrantsMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_PreFundedWarrantsMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents information pertaining to Pre-funded warrants.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_PreFundedWarrantsMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Pre-funded Warrants [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_PreFundedWarrantsMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Pre-funded Warrants</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_SeriesEPreFundedWarrantsMember" xlink:label="adgm_SeriesEPreFundedWarrantsMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_SeriesEPreFundedWarrantsMember" xlink:to="adgm_SeriesEPreFundedWarrantsMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_SeriesEPreFundedWarrantsMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents the information pertaining to Series E pre-funded warrants.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_SeriesEPreFundedWarrantsMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Series E Pre-funded Warrants [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_SeriesEPreFundedWarrantsMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Series E Pre-funded Warrants</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_SeriesEConvertiblePreferredStockMember" xlink:label="adgm_SeriesEConvertiblePreferredStockMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_SeriesEConvertiblePreferredStockMember" xlink:to="adgm_SeriesEConvertiblePreferredStockMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_SeriesEConvertiblePreferredStockMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Preferred stock that may be exchanged into common shares or other types of securities at the owner's option, classified as Series E.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_SeriesEConvertiblePreferredStockMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Series E Convertible Preferred Stock [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_SeriesEConvertiblePreferredStockMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Series E Preferred Stock Class</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="adgm_SeriesEConvertiblePreferredStockMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Series E Preferred Stock</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ClassOfWarrantOrRightLineItems" xlink:label="us-gaap_ClassOfWarrantOrRightLineItems" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ClassOfWarrantOrRightLineItems" xlink:to="us-gaap_ClassOfWarrantOrRightLineItems_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ClassOfWarrantOrRightLineItems_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class of Warrant or Right [Line Items]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ClassOfWarrantOrRightLineItems_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Warrants</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_TemporaryEquityEstimatedFairValueConversionMultipleFactor" xlink:label="adgm_TemporaryEquityEstimatedFairValueConversionMultipleFactor" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_TemporaryEquityEstimatedFairValueConversionMultipleFactor" xlink:to="adgm_TemporaryEquityEstimatedFairValueConversionMultipleFactor_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_TemporaryEquityEstimatedFairValueConversionMultipleFactor_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The conversion factor applied to the fair value of common stock to determine the fair value of the temporary equity.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_TemporaryEquityEstimatedFairValueConversionMultipleFactor_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Temporary Equity, Estimated Fair Value, Conversion Multiple Factor</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_TemporaryEquityEstimatedFairValueConversionMultipleFactor_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Conversion factor</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_WarrantsAndRightsOutstanding" xlink:label="us-gaap_WarrantsAndRightsOutstanding" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_WarrantsAndRightsOutstanding" xlink:to="us-gaap_WarrantsAndRightsOutstanding_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_WarrantsAndRightsOutstanding_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Warrants and Rights Outstanding</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_WarrantsAndRightsOutstanding_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Warrant liabilities</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_ClassOfWarrantOrRightNumberOfWarrantsIssued" xlink:label="adgm_ClassOfWarrantOrRightNumberOfWarrantsIssued" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_ClassOfWarrantOrRightNumberOfWarrantsIssued" xlink:to="adgm_ClassOfWarrantOrRightNumberOfWarrantsIssued_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_ClassOfWarrantOrRightNumberOfWarrantsIssued_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of warrants issued to purchase stock.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_ClassOfWarrantOrRightNumberOfWarrantsIssued_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class Of Warrant Or Right, Number Of Warrants Issued</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_ClassOfWarrantOrRightNumberOfWarrantsIssued_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of warrants issued</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ClassOfWarrantOrRightNumberOfSecuritiesCalledByEachWarrantOrRight" xlink:label="us-gaap_ClassOfWarrantOrRightNumberOfSecuritiesCalledByEachWarrantOrRight" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ClassOfWarrantOrRightNumberOfSecuritiesCalledByEachWarrantOrRight" xlink:to="us-gaap_ClassOfWarrantOrRightNumberOfSecuritiesCalledByEachWarrantOrRight_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ClassOfWarrantOrRightNumberOfSecuritiesCalledByEachWarrantOrRight_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class of Warrant or Right, Number of Securities Called by Each Warrant or Right</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ClassOfWarrantOrRightNumberOfSecuritiesCalledByEachWarrantOrRight_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Shares called by per each warrant</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_MeasurementInputPriceVolatilityMember" xlink:label="us-gaap_MeasurementInputPriceVolatilityMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_MeasurementInputPriceVolatilityMember" xlink:to="us-gaap_MeasurementInputPriceVolatilityMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_MeasurementInputPriceVolatilityMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Measurement Input, Price Volatility [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_MeasurementInputPriceVolatilityMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Volatility</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_MeasurementInputExpectedDividendRateMember" xlink:label="us-gaap_MeasurementInputExpectedDividendRateMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_MeasurementInputExpectedDividendRateMember" xlink:to="us-gaap_MeasurementInputExpectedDividendRateMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_MeasurementInputExpectedDividendRateMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Measurement Input, Expected Dividend Rate [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_MeasurementInputExpectedDividendRateMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Expected dividend yield</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_WarrantsAndRightsOutstandingMeasurementInput" xlink:label="us-gaap_WarrantsAndRightsOutstandingMeasurementInput" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_WarrantsAndRightsOutstandingMeasurementInput" xlink:to="us-gaap_WarrantsAndRightsOutstandingMeasurementInput_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_WarrantsAndRightsOutstandingMeasurementInput_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Warrants and Rights Outstanding, Measurement Input</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_WarrantsAndRightsOutstandingMeasurementInput_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Warrants measurement inputs</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_WarrantsExercised" xlink:label="adgm_WarrantsExercised" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_WarrantsExercised" xlink:to="adgm_WarrantsExercised_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_WarrantsExercised_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents the number of warrants exercised during the reported period.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_WarrantsExercised_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Warrants Exercised</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_WarrantsExercised_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Warrants exercised (in shares)</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_SharesIssuedDuringPeriodWarrantsExercised" xlink:label="adgm_SharesIssuedDuringPeriodWarrantsExercised" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_SharesIssuedDuringPeriodWarrantsExercised" xlink:to="adgm_SharesIssuedDuringPeriodWarrantsExercised_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_SharesIssuedDuringPeriodWarrantsExercised_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents the number of shares issued upon exercise of warrants.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_SharesIssuedDuringPeriodWarrantsExercised_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Shares Issued During Period, Warrants Exercised</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_SharesIssuedDuringPeriodWarrantsExercised_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Share issued during period, warrants exercised</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="adgm_SharesIssuedDuringPeriodWarrantsExercised_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of shares issued on the exercise of warrants</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_InventoryRawMaterialsNetOfReserves" xlink:label="us-gaap_InventoryRawMaterialsNetOfReserves" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_InventoryRawMaterialsNetOfReserves" xlink:to="us-gaap_InventoryRawMaterialsNetOfReserves_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_InventoryRawMaterialsNetOfReserves_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Inventory, Raw Materials, Net of Reserves</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_InventoryRawMaterialsNetOfReserves_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Raw materials</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_InventoryWorkInProcessNetOfReserves" xlink:label="us-gaap_InventoryWorkInProcessNetOfReserves" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_InventoryWorkInProcessNetOfReserves" xlink:to="us-gaap_InventoryWorkInProcessNetOfReserves_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_InventoryWorkInProcessNetOfReserves_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Inventory, Work in Process, Net of Reserves</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_InventoryWorkInProcessNetOfReserves_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Work-in-Process</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_InventoryFinishedGoodsNetOfReserves" xlink:label="us-gaap_InventoryFinishedGoodsNetOfReserves" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_InventoryFinishedGoodsNetOfReserves" xlink:to="us-gaap_InventoryFinishedGoodsNetOfReserves_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_InventoryFinishedGoodsNetOfReserves_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Inventory, Finished Goods, Net of Reserves</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_InventoryFinishedGoodsNetOfReserves_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Finished goods</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_InventoryAdjustments" xlink:label="us-gaap_InventoryAdjustments" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_InventoryAdjustments" xlink:to="us-gaap_InventoryAdjustments_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_InventoryAdjustments_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Inventory Adjustments</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_InventoryAdjustments_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Inventory scrap value</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_TechnologyEquipmentMember" xlink:label="us-gaap_TechnologyEquipmentMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_TechnologyEquipmentMember" xlink:to="us-gaap_TechnologyEquipmentMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_TechnologyEquipmentMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Technology Equipment [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_TechnologyEquipmentMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Consoles</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_MachineryAndEquipmentMember" xlink:label="us-gaap_MachineryAndEquipmentMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_MachineryAndEquipmentMember" xlink:to="us-gaap_MachineryAndEquipmentMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_MachineryAndEquipmentMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Machinery and Equipment [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_MachineryAndEquipmentMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Machinery and equipment</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_LeaseholdImprovementsMember" xlink:label="us-gaap_LeaseholdImprovementsMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_LeaseholdImprovementsMember" xlink:to="us-gaap_LeaseholdImprovementsMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_LeaseholdImprovementsMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Leasehold Improvements [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_LeaseholdImprovementsMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Leasehold improvements</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ToolsDiesAndMoldsMember" xlink:label="us-gaap_ToolsDiesAndMoldsMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ToolsDiesAndMoldsMember" xlink:to="us-gaap_ToolsDiesAndMoldsMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ToolsDiesAndMoldsMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Tools, Dies and Molds [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ToolsDiesAndMoldsMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Tools and molds</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ComputerEquipmentMember" xlink:label="us-gaap_ComputerEquipmentMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ComputerEquipmentMember" xlink:to="us-gaap_ComputerEquipmentMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ComputerEquipmentMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Computer Equipment [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ComputerEquipmentMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Computer equipment</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_EquipmentMember" xlink:label="us-gaap_EquipmentMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_EquipmentMember" xlink:to="us-gaap_EquipmentMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_EquipmentMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Equipment [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_EquipmentMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Demo equipment</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_FurnitureAndFixturesMember" xlink:label="us-gaap_FurnitureAndFixturesMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_FurnitureAndFixturesMember" xlink:to="us-gaap_FurnitureAndFixturesMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_FurnitureAndFixturesMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Furniture and Fixtures [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_FurnitureAndFixturesMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Furniture and fixtures</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_VehiclesMember" xlink:label="us-gaap_VehiclesMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_VehiclesMember" xlink:to="us-gaap_VehiclesMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_VehiclesMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Vehicles [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_VehiclesMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Vehicles</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_PropertyPlantAndEquipmentGross" xlink:label="us-gaap_PropertyPlantAndEquipmentGross" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_PropertyPlantAndEquipmentGross" xlink:to="us-gaap_PropertyPlantAndEquipmentGross_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_PropertyPlantAndEquipmentGross_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Property, Plant and Equipment, Gross</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_PropertyPlantAndEquipmentGross_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Total property, plant, and equipment</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_AccumulatedDepreciationDepletionAndAmortizationPropertyPlantAndEquipment" xlink:label="us-gaap_AccumulatedDepreciationDepletionAndAmortizationPropertyPlantAndEquipment" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_AccumulatedDepreciationDepletionAndAmortizationPropertyPlantAndEquipment" xlink:to="us-gaap_AccumulatedDepreciationDepletionAndAmortizationPropertyPlantAndEquipment_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_AccumulatedDepreciationDepletionAndAmortizationPropertyPlantAndEquipment_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Accumulated Depreciation, Depletion and Amortization, Property, Plant, and Equipment</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2009/role/negatedLabel" xlink:label="us-gaap_AccumulatedDepreciationDepletionAndAmortizationPropertyPlantAndEquipment_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Less: accumulated depreciation</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_Depreciation" xlink:label="us-gaap_Depreciation" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_Depreciation" xlink:to="us-gaap_Depreciation_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_Depreciation_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Depreciation</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_Depreciation_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Depreciation expense</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_IndefiniteLivedIntangibleAssetsBeforeImpairmentExcludingGoodwill" xlink:label="adgm_IndefiniteLivedIntangibleAssetsBeforeImpairmentExcludingGoodwill" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_IndefiniteLivedIntangibleAssetsBeforeImpairmentExcludingGoodwill" xlink:to="adgm_IndefiniteLivedIntangibleAssetsBeforeImpairmentExcludingGoodwill_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_IndefiniteLivedIntangibleAssetsBeforeImpairmentExcludingGoodwill_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amount of assets before impairment, excluding financial assets and goodwill, lacking physical substance and having a projected indefinite period of benefit.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_IndefiniteLivedIntangibleAssetsBeforeImpairmentExcludingGoodwill_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Indefinite-Lived Intangible Assets, Before Impairment (Excluding Goodwill)</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_IndefiniteLivedIntangibleAssetsBeforeImpairmentExcludingGoodwill_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">IPR&amp;D, Gross Carrying Amount</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_IndefiniteLivedIntangibleAssetsImpairmentExcludingGoodwill" xlink:label="adgm_IndefiniteLivedIntangibleAssetsImpairmentExcludingGoodwill" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_IndefiniteLivedIntangibleAssetsImpairmentExcludingGoodwill" xlink:to="adgm_IndefiniteLivedIntangibleAssetsImpairmentExcludingGoodwill_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_IndefiniteLivedIntangibleAssetsImpairmentExcludingGoodwill_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amount of accumulated impairment of assets before impairment, excluding financial assets and goodwill, lacking physical substance and having a projected indefinite period of benefit.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_IndefiniteLivedIntangibleAssetsImpairmentExcludingGoodwill_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Indefinite-Lived Intangible Assets, Impairment (Excluding Goodwill)</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2009/role/negatedLabel" xlink:label="adgm_IndefiniteLivedIntangibleAssetsImpairmentExcludingGoodwill_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">IPR&amp;D, Impairment</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_IndefiniteLivedIntangibleAssetsExcludingGoodwill" xlink:label="us-gaap_IndefiniteLivedIntangibleAssetsExcludingGoodwill" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_IndefiniteLivedIntangibleAssetsExcludingGoodwill" xlink:to="us-gaap_IndefiniteLivedIntangibleAssetsExcludingGoodwill_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_IndefiniteLivedIntangibleAssetsExcludingGoodwill_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Indefinite-Lived Intangible Assets (Excluding Goodwill)</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:label="us-gaap_IndefiniteLivedIntangibleAssetsExcludingGoodwill_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">IPR&amp;D, Net Carrying Amount</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_FiniteLivedIntangibleAssetsGross" xlink:label="us-gaap_FiniteLivedIntangibleAssetsGross" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_FiniteLivedIntangibleAssetsGross" xlink:to="us-gaap_FiniteLivedIntangibleAssetsGross_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_FiniteLivedIntangibleAssetsGross_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Finite-Lived Intangible Assets, Gross</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_FiniteLivedIntangibleAssetsGross_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Developed technology, Gross Carrying Amount</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_FiniteLivedIntangibleAssetsAccumulatedAmortization" xlink:label="us-gaap_FiniteLivedIntangibleAssetsAccumulatedAmortization" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_FiniteLivedIntangibleAssetsAccumulatedAmortization" xlink:to="us-gaap_FiniteLivedIntangibleAssetsAccumulatedAmortization_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_FiniteLivedIntangibleAssetsAccumulatedAmortization_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Finite-Lived Intangible Assets, Accumulated Amortization</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2009/role/negatedLabel" xlink:label="us-gaap_FiniteLivedIntangibleAssetsAccumulatedAmortization_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Developed technology, Accumulated Amortization</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_FiniteLivedIntangibleAssetsImpairmentExcludingGoodwill" xlink:label="adgm_FiniteLivedIntangibleAssetsImpairmentExcludingGoodwill" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_FiniteLivedIntangibleAssetsImpairmentExcludingGoodwill" xlink:to="adgm_FiniteLivedIntangibleAssetsImpairmentExcludingGoodwill_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_FiniteLivedIntangibleAssetsImpairmentExcludingGoodwill_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amount of accumulated impairment of assets before impairment, excluding financial assets and goodwill, lacking physical substance and having a projected finite period of benefit.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_FiniteLivedIntangibleAssetsImpairmentExcludingGoodwill_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Finite-Lived Intangible Assets, Impairment (Excluding Goodwill)</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2009/role/negatedLabel" xlink:label="adgm_FiniteLivedIntangibleAssetsImpairmentExcludingGoodwill_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Developed technology, Impairment</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_FiniteLivedIntangibleAssetsNet" xlink:label="us-gaap_FiniteLivedIntangibleAssetsNet" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_FiniteLivedIntangibleAssetsNet" xlink:to="us-gaap_FiniteLivedIntangibleAssetsNet_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_FiniteLivedIntangibleAssetsNet_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Finite-Lived Intangible Assets, Net</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_FiniteLivedIntangibleAssetsNet_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Developed technology, Net Carrying Amount</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_IntangibleAssetsGrossExcludingGoodwill" xlink:label="us-gaap_IntangibleAssetsGrossExcludingGoodwill" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_IntangibleAssetsGrossExcludingGoodwill" xlink:to="us-gaap_IntangibleAssetsGrossExcludingGoodwill_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_IntangibleAssetsGrossExcludingGoodwill_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Intangible Assets, Gross (Excluding Goodwill)</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:label="us-gaap_IntangibleAssetsGrossExcludingGoodwill_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Total, Gross Carrying Amount</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_IntangibleAssetsAccumulatedAmortization" xlink:label="adgm_IntangibleAssetsAccumulatedAmortization" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_IntangibleAssetsAccumulatedAmortization" xlink:to="adgm_IntangibleAssetsAccumulatedAmortization_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_IntangibleAssetsAccumulatedAmortization_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Accumulated amount of amortization of assets, excluding financial assets and goodwill, lacking physical substance.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_IntangibleAssetsAccumulatedAmortization_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Intangible Assets, Accumulated Amortization</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2009/role/negatedTotalLabel" xlink:label="adgm_IntangibleAssetsAccumulatedAmortization_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Total, Accumulated Amortization</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_IntangibleAssetsImpairmentExcludingGoodwill" xlink:label="adgm_IntangibleAssetsImpairmentExcludingGoodwill" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_IntangibleAssetsImpairmentExcludingGoodwill" xlink:to="adgm_IntangibleAssetsImpairmentExcludingGoodwill_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_IntangibleAssetsImpairmentExcludingGoodwill_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amount of accumulated impairment of assets before impairment, excluding financial assets and goodwill, lacking physical substance.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_IntangibleAssetsImpairmentExcludingGoodwill_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Intangible Assets, Impairment (Excluding Goodwill)</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2009/role/negatedTotalLabel" xlink:label="adgm_IntangibleAssetsImpairmentExcludingGoodwill_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Total, Impairment</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_AmortizationOfIntangibleAssets" xlink:label="us-gaap_AmortizationOfIntangibleAssets" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_AmortizationOfIntangibleAssets" xlink:to="us-gaap_AmortizationOfIntangibleAssets_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_AmortizationOfIntangibleAssets_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amortization of Intangible Assets</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_AmortizationOfIntangibleAssets_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amortization of intangible assets</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_GoodwillGross" xlink:label="us-gaap_GoodwillGross" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_GoodwillGross" xlink:to="us-gaap_GoodwillGross_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_GoodwillGross_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Goodwill, Gross</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_GoodwillGross_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Goodwill Amount</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_GoodwillImpairedAccumulatedImpairmentLoss" xlink:label="us-gaap_GoodwillImpairedAccumulatedImpairmentLoss" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_GoodwillImpairedAccumulatedImpairmentLoss" xlink:to="us-gaap_GoodwillImpairedAccumulatedImpairmentLoss_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_GoodwillImpairedAccumulatedImpairmentLoss_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Goodwill, Impaired, Accumulated Impairment Loss</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2009/role/negatedLabel" xlink:label="us-gaap_GoodwillImpairedAccumulatedImpairmentLoss_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Impairment</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_AccruedProfessionalFeesCurrent" xlink:label="us-gaap_AccruedProfessionalFeesCurrent" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_AccruedProfessionalFeesCurrent" xlink:to="us-gaap_AccruedProfessionalFeesCurrent_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_AccruedProfessionalFeesCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Accrued Professional Fees, Current</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_AccruedProfessionalFeesCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Accrued deferred professional fees</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_EmployeeRelatedLiabilitiesCurrent" xlink:label="us-gaap_EmployeeRelatedLiabilitiesCurrent" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_EmployeeRelatedLiabilitiesCurrent" xlink:to="us-gaap_EmployeeRelatedLiabilitiesCurrent_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_EmployeeRelatedLiabilitiesCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Employee-related Liabilities, Current</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_EmployeeRelatedLiabilitiesCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Compensation and related expenses</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_ResearchAndDevelopmentExpenseAccrualCurrent" xlink:label="adgm_ResearchAndDevelopmentExpenseAccrualCurrent" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_ResearchAndDevelopmentExpenseAccrualCurrent" xlink:to="adgm_ResearchAndDevelopmentExpenseAccrualCurrent_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_ResearchAndDevelopmentExpenseAccrualCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Total of the carrying values as of the balance sheet date of obligations incurred through that date and payable for obligations related to research and development. Used to reflect the current portion of the liabilities (due within one year or within the normal operating cycle if longer).</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_ResearchAndDevelopmentExpenseAccrualCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Research And Development Expense Accrual, Current</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_ResearchAndDevelopmentExpenseAccrualCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Research and development expenses</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_OtherAccruedLiabilitiesCurrent" xlink:label="us-gaap_OtherAccruedLiabilitiesCurrent" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_OtherAccruedLiabilitiesCurrent" xlink:to="us-gaap_OtherAccruedLiabilitiesCurrent_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_OtherAccruedLiabilitiesCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Other Accrued Liabilities, Current</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_OtherAccruedLiabilitiesCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Other</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_LongTermDebt" xlink:label="us-gaap_LongTermDebt" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_LongTermDebt" xlink:to="us-gaap_LongTermDebt_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_LongTermDebt_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Long-Term Debt</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_LongTermDebt_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Total outstanding debt</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_DebtInstrumentConvertibleThresholdPercentageOfStockPriceTrigger" xlink:label="us-gaap_DebtInstrumentConvertibleThresholdPercentageOfStockPriceTrigger" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_DebtInstrumentConvertibleThresholdPercentageOfStockPriceTrigger" xlink:to="us-gaap_DebtInstrumentConvertibleThresholdPercentageOfStockPriceTrigger_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_DebtInstrumentConvertibleThresholdPercentageOfStockPriceTrigger_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Debt Instrument, Convertible, Threshold Percentage of Stock Price Trigger</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_DebtInstrumentConvertibleThresholdPercentageOfStockPriceTrigger_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Percentage of effective price of common stock sold to trigger conversion</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_DebtConversionConvertedInstrumentConvertibleAmount" xlink:label="adgm_DebtConversionConvertedInstrumentConvertibleAmount" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_DebtConversionConvertedInstrumentConvertibleAmount" xlink:to="adgm_DebtConversionConvertedInstrumentConvertibleAmount_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_DebtConversionConvertedInstrumentConvertibleAmount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The amount of the debt issued to convert another series of debt.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_DebtConversionConvertedInstrumentConvertibleAmount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Debt Conversion, Converted Instrument, Convertible Amount</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_DebtConversionConvertedInstrumentConvertibleAmount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Debt issued to replace a convertible amount</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_DebtInstrumentCarryingAmount" xlink:label="us-gaap_DebtInstrumentCarryingAmount" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_DebtInstrumentCarryingAmount" xlink:to="us-gaap_DebtInstrumentCarryingAmount_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_DebtInstrumentCarryingAmount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Long-Term Debt, Gross</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_DebtInstrumentCarryingAmount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Outstanding notes</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_InterestExpenseDebt" xlink:label="us-gaap_InterestExpenseDebt" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_InterestExpenseDebt" xlink:to="us-gaap_InterestExpenseDebt_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_InterestExpenseDebt_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Interest Expense, Debt</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_InterestExpenseDebt_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Interest</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_BusinessCombinationNumberOfSharesOfSuccessorIssuedToPredecessorShares" xlink:label="adgm_BusinessCombinationNumberOfSharesOfSuccessorIssuedToPredecessorShares" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_BusinessCombinationNumberOfSharesOfSuccessorIssuedToPredecessorShares" xlink:to="adgm_BusinessCombinationNumberOfSharesOfSuccessorIssuedToPredecessorShares_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_BusinessCombinationNumberOfSharesOfSuccessorIssuedToPredecessorShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents the number of shares of successor shares in exchange for predecessor shares company upon consummation of business combination.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_BusinessCombinationNumberOfSharesOfSuccessorIssuedToPredecessorShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Business Combination, Number of Shares of Successor Issued to Predecessor Shares</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_BusinessCombinationNumberOfSharesOfSuccessorIssuedToPredecessorShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Commons stock issued to convert Legacy Adagio shares</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_NumberOfSharesConverted" xlink:label="adgm_NumberOfSharesConverted" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_NumberOfSharesConverted" xlink:to="adgm_NumberOfSharesConverted_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_NumberOfSharesConverted_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents the number of shares converted upon consummation of business combination.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_NumberOfSharesConverted_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of Shares Converted</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_NumberOfSharesConverted_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Shares converted</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_DebtInstrumentMaximumBorrowingCapacity" xlink:label="adgm_DebtInstrumentMaximumBorrowingCapacity" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_DebtInstrumentMaximumBorrowingCapacity" xlink:to="adgm_DebtInstrumentMaximumBorrowingCapacity_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_DebtInstrumentMaximumBorrowingCapacity_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Maximum amount of debt to be issued.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_DebtInstrumentMaximumBorrowingCapacity_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Debt Instrument, Maximum Borrowing Capacity</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_DebtInstrumentMaximumBorrowingCapacity_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Right to issue additional convertible promissory notes</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_DelayedDrawCommitmentMember" xlink:label="adgm_DelayedDrawCommitmentMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_DelayedDrawCommitmentMember" xlink:to="adgm_DelayedDrawCommitmentMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_DelayedDrawCommitmentMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents Information pertaining to delayed draw commitment.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_DelayedDrawCommitmentMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Delayed Draw Commitment [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_DelayedDrawCommitmentMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Delayed Draw Commitment</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_DebtConversionOriginalDebtAmount1" xlink:label="us-gaap_DebtConversionOriginalDebtAmount1" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_DebtConversionOriginalDebtAmount1" xlink:to="us-gaap_DebtConversionOriginalDebtAmount1_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_DebtConversionOriginalDebtAmount1_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Debt Conversion, Original Debt, Amount</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_DebtConversionOriginalDebtAmount1_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Debt amount</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_SiliconValleyBankTermLoanMember" xlink:label="adgm_SiliconValleyBankTermLoanMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_SiliconValleyBankTermLoanMember" xlink:to="adgm_SiliconValleyBankTermLoanMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_SiliconValleyBankTermLoanMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents Information pertaining to Silicon Valley Bank Term Loan</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_SiliconValleyBankTermLoanMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Silicon Valley Bank Term Loan [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_SiliconValleyBankTermLoanMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Silicon Valley Bank Term Loan</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="adgm_SiliconValleyBankTermLoanMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">SVB term loan</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_SiliconValleyBankInitialTermLoanMember" xlink:label="adgm_SiliconValleyBankInitialTermLoanMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_SiliconValleyBankInitialTermLoanMember" xlink:to="adgm_SiliconValleyBankInitialTermLoanMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_SiliconValleyBankInitialTermLoanMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents information pertaining to silicon valley initial bank term loan (SVB term loan)</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_SiliconValleyBankInitialTermLoanMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Silicon Valley Bank Initial Term Loan [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_SiliconValleyBankInitialTermLoanMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">SVB Initial Term Loan</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_SiliconValleyBankSubsequentTermLoanMember" xlink:label="adgm_SiliconValleyBankSubsequentTermLoanMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_SiliconValleyBankSubsequentTermLoanMember" xlink:to="adgm_SiliconValleyBankSubsequentTermLoanMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_SiliconValleyBankSubsequentTermLoanMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents information pertaining to silicon valley subsequent bank term loan (SVB term loan)</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_SiliconValleyBankSubsequentTermLoanMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Silicon Valley Bank Subsequent Term Loan [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_SiliconValleyBankSubsequentTermLoanMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">SVB Subsequent Term Loan</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_InitialWarrantsMember" xlink:label="adgm_InitialWarrantsMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_InitialWarrantsMember" xlink:to="adgm_InitialWarrantsMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_InitialWarrantsMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents information pertaining to initial warrants.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_InitialWarrantsMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Initial Warrants [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_InitialWarrantsMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Initial warrants</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_AdditionalWarrantsMember" xlink:label="adgm_AdditionalWarrantsMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_AdditionalWarrantsMember" xlink:to="adgm_AdditionalWarrantsMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_AdditionalWarrantsMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents information pertaining to additional warrants.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_AdditionalWarrantsMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Additional Warrants [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_AdditionalWarrantsMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Additional warrants</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_DebtInstrumentInterestRateThresholdPercentage" xlink:label="adgm_DebtInstrumentInterestRateThresholdPercentage" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_DebtInstrumentInterestRateThresholdPercentage" xlink:to="adgm_DebtInstrumentInterestRateThresholdPercentage_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_DebtInstrumentInterestRateThresholdPercentage_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold interest rate for funds borrowed, under the debt agreement.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_DebtInstrumentInterestRateThresholdPercentage_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Debt Instrument, Interest Rate, Threshold, Percentage</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_DebtInstrumentInterestRateThresholdPercentage_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Interest rate</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_DebtInstrumentBasisSpreadOnVariableRate1" xlink:label="us-gaap_DebtInstrumentBasisSpreadOnVariableRate1" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_DebtInstrumentBasisSpreadOnVariableRate1" xlink:to="us-gaap_DebtInstrumentBasisSpreadOnVariableRate1_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_DebtInstrumentBasisSpreadOnVariableRate1_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Debt Instrument, Basis Spread on Variable Rate</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_DebtInstrumentBasisSpreadOnVariableRate1_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Variable interest rate</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_DebtInstrumentVariableInterestRateTypeExtensibleEnumeration" xlink:label="us-gaap_DebtInstrumentVariableInterestRateTypeExtensibleEnumeration" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_DebtInstrumentVariableInterestRateTypeExtensibleEnumeration" xlink:to="us-gaap_DebtInstrumentVariableInterestRateTypeExtensibleEnumeration_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_DebtInstrumentVariableInterestRateTypeExtensibleEnumeration_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Debt Instrument, Variable Interest Rate, Type [Extensible Enumeration]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_DebtInstrumentVariableInterestRateTypeExtensibleEnumeration_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Debt Instrument, Variable Interest Rate, Type [Extensible Enumeration]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ClassOfWarrantOrRightNumberOfSecuritiesCalledByWarrantsOrRights" xlink:label="us-gaap_ClassOfWarrantOrRightNumberOfSecuritiesCalledByWarrantsOrRights" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ClassOfWarrantOrRightNumberOfSecuritiesCalledByWarrantsOrRights" xlink:to="us-gaap_ClassOfWarrantOrRightNumberOfSecuritiesCalledByWarrantsOrRights_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ClassOfWarrantOrRightNumberOfSecuritiesCalledByWarrantsOrRights_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class of Warrant or Right, Number of Securities Called by Warrants or Rights</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ClassOfWarrantOrRightNumberOfSecuritiesCalledByWarrantsOrRights_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Shares issuable in the exercise of warrants</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_DebtInstrumentUnamortizedDiscount" xlink:label="us-gaap_DebtInstrumentUnamortizedDiscount" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_DebtInstrumentUnamortizedDiscount" xlink:to="us-gaap_DebtInstrumentUnamortizedDiscount_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_DebtInstrumentUnamortizedDiscount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Debt Instrument, Unamortized Discount</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_DebtInstrumentUnamortizedDiscount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Unamortized debt discount</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_DebtInstrumentMinimumRequiredEquityFinancing" xlink:label="adgm_DebtInstrumentMinimumRequiredEquityFinancing" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_DebtInstrumentMinimumRequiredEquityFinancing" xlink:to="adgm_DebtInstrumentMinimumRequiredEquityFinancing_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_DebtInstrumentMinimumRequiredEquityFinancing_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Minimum Amount of cash inflow from the issuance of common stock, preferred stock, treasury stock, stock options, and other types of equity required to satisfy the milestone condition of term loan.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_DebtInstrumentMinimumRequiredEquityFinancing_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Debt Instrument, Minimum Required Equity Financing</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_DebtInstrumentMinimumRequiredEquityFinancing_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Minimum required equity financing</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_LineOfCreditFacilityRemainingBorrowingCapacity" xlink:label="us-gaap_LineOfCreditFacilityRemainingBorrowingCapacity" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_LineOfCreditFacilityRemainingBorrowingCapacity" xlink:to="us-gaap_LineOfCreditFacilityRemainingBorrowingCapacity_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_LineOfCreditFacilityRemainingBorrowingCapacity_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Line of Credit Facility, Remaining Borrowing Capacity</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_LineOfCreditFacilityRemainingBorrowingCapacity_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amount not drawn</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_LoansPayableToBank" xlink:label="us-gaap_LoansPayableToBank" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_LoansPayableToBank" xlink:to="us-gaap_LoansPayableToBank_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_LoansPayableToBank_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Loans Payable to Bank</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_LoansPayableToBank_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Outstanding principal</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_SvbWarrantsMember" xlink:label="adgm_SvbWarrantsMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_SvbWarrantsMember" xlink:to="adgm_SvbWarrantsMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_SvbWarrantsMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents information pertaining to SVB Warrants issued in conjunction with the Loan and Security Agreement (LSA).</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_SvbWarrantsMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Svb Warrants [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_SvbWarrantsMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">SVB Warrants</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_ClassOfWarrantOrRightExchangedInShares" xlink:label="adgm_ClassOfWarrantOrRightExchangedInShares" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_ClassOfWarrantOrRightExchangedInShares" xlink:to="adgm_ClassOfWarrantOrRightExchangedInShares_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_ClassOfWarrantOrRightExchangedInShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents the number of warrants exchanged into shares of common stock.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_ClassOfWarrantOrRightExchangedInShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class of Warrant or Right, Exchanged in Shares</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_ClassOfWarrantOrRightExchangedInShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class of warrant or right, exchanged in shares</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ClassOfWarrantOrRightOutstanding" xlink:label="us-gaap_ClassOfWarrantOrRightOutstanding" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ClassOfWarrantOrRightOutstanding" xlink:to="us-gaap_ClassOfWarrantOrRightOutstanding_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ClassOfWarrantOrRightOutstanding_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class of Warrant or Right, Outstanding</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ClassOfWarrantOrRightOutstanding_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Warrants</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_ClassOfWarrantOrRightOutstanding_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Warrants outstanding (in shares)</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_PrivateInvestmentInPublicEquityMember" xlink:label="adgm_PrivateInvestmentInPublicEquityMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_PrivateInvestmentInPublicEquityMember" xlink:to="adgm_PrivateInvestmentInPublicEquityMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_PrivateInvestmentInPublicEquityMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Private Investment in Public Equity, is a method where private investors purchase shares of a publicly traded company, often at a discount, to provide the company with capital, bypassing the traditional public offering process.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_PrivateInvestmentInPublicEquityMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Private Investment in Public Equity [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_PrivateInvestmentInPublicEquityMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">PIPE Financing</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_MeasurementInputVolatilityHaircutMember" xlink:label="adgm_MeasurementInputVolatilityHaircutMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_MeasurementInputVolatilityHaircutMember" xlink:to="adgm_MeasurementInputVolatilityHaircutMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_MeasurementInputVolatilityHaircutMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Measurement input using volatility haircut.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_MeasurementInputVolatilityHaircutMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Measurement Input, Volatility Haircut [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_MeasurementInputVolatilityHaircutMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Measurement Input, Volatility Haircut</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_ClassOfWarrantOrRightNumberOfWarrantsIssuedInExchangeOfNonRedeemableShares" xlink:label="adgm_ClassOfWarrantOrRightNumberOfWarrantsIssuedInExchangeOfNonRedeemableShares" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_ClassOfWarrantOrRightNumberOfWarrantsIssuedInExchangeOfNonRedeemableShares" xlink:to="adgm_ClassOfWarrantOrRightNumberOfWarrantsIssuedInExchangeOfNonRedeemableShares_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_ClassOfWarrantOrRightNumberOfWarrantsIssuedInExchangeOfNonRedeemableShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of warrants issued in exchange of non-redeemable shares.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_ClassOfWarrantOrRightNumberOfWarrantsIssuedInExchangeOfNonRedeemableShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class Of Warrant Or Right, Number Of Warrants Issued In Exchange of Non-redeemable Shares</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_ClassOfWarrantOrRightNumberOfWarrantsIssuedInExchangeOfNonRedeemableShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of warrants issued in exchange of non-redeemable shares</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_SharesIssuesDuringPeriodExchangeOfNonRedeemableShares" xlink:label="adgm_SharesIssuesDuringPeriodExchangeOfNonRedeemableShares" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_SharesIssuesDuringPeriodExchangeOfNonRedeemableShares" xlink:to="adgm_SharesIssuesDuringPeriodExchangeOfNonRedeemableShares_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_SharesIssuesDuringPeriodExchangeOfNonRedeemableShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of shares issued in exchange of non-redeemable shares.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_SharesIssuesDuringPeriodExchangeOfNonRedeemableShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Shares Issues During Period, Exchange of Non-redeemable Shares</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_SharesIssuesDuringPeriodExchangeOfNonRedeemableShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of shares issued in exchange of non-redeemable shares</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_NumberOfSharesExchangedForEntitySCommonStock" xlink:label="adgm_NumberOfSharesExchangedForEntitySCommonStock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_NumberOfSharesExchangedForEntitySCommonStock" xlink:to="adgm_NumberOfSharesExchangedForEntitySCommonStock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_NumberOfSharesExchangedForEntitySCommonStock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of shares exchanged for entity's common stock in a merger transaction.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_NumberOfSharesExchangedForEntitySCommonStock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of Shares Exchanged For Entity's Common Stock</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_NumberOfSharesExchangedForEntitySCommonStock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of shares exchanged for common stock</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_WarrantsFairValuePerUnit" xlink:label="adgm_WarrantsFairValuePerUnit" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_WarrantsFairValuePerUnit" xlink:to="adgm_WarrantsFairValuePerUnit_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_WarrantsFairValuePerUnit_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The fair value of each warrant as calculated on certain assumptions.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_WarrantsFairValuePerUnit_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Warrants, Fair Value Per Unit</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_WarrantsFairValuePerUnit_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Warrant fair value (in dollar per unit)</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_PIPEPreFundedWarrants2025Member" xlink:label="adgm_PIPEPreFundedWarrants2025Member" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_PIPEPreFundedWarrants2025Member" xlink:to="adgm_PIPEPreFundedWarrants2025Member_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_PIPEPreFundedWarrants2025Member_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Security that gives the holder the right to purchase shares of stock in accordance with the terms of the instrument, usually upon payment of a specified amount, issued in 2025 in relation to a PIPE.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_PIPEPreFundedWarrants2025Member_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">P I P E Pre-Funded Warrants, 2025 [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_PIPEPreFundedWarrants2025Member_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">2025 PIPE Pre-Funded Warrants</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_CommonStockWarrantsMilestoneMember" xlink:label="adgm_CommonStockWarrantsMilestoneMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_CommonStockWarrantsMilestoneMember" xlink:to="adgm_CommonStockWarrantsMilestoneMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_CommonStockWarrantsMilestoneMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents information pertaining to common stock milestone warrants.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_CommonStockWarrantsMilestoneMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Common Stock Warrants - Milestone [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_CommonStockWarrantsMilestoneMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Milestone Warrants</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_CommonStockWarrantsTrancheMember" xlink:label="adgm_CommonStockWarrantsTrancheMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_CommonStockWarrantsTrancheMember" xlink:to="adgm_CommonStockWarrantsTrancheMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_CommonStockWarrantsTrancheMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents information pertaining to common stock Tranche A warrants</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_CommonStockWarrantsTrancheMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Common Stock Warrants - Tranche A [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_CommonStockWarrantsTrancheMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Tranche A Warrants</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_CommonStockWarrantsTrancheBMember" xlink:label="adgm_CommonStockWarrantsTrancheBMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_CommonStockWarrantsTrancheBMember" xlink:to="adgm_CommonStockWarrantsTrancheBMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_CommonStockWarrantsTrancheBMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents information pertaining to common stock Tranche B warrants.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_CommonStockWarrantsTrancheBMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Common Stock Warrants - Tranche B [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_CommonStockWarrantsTrancheBMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Tranche B Warrants</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_CommonStockWarrantsTrancheCMember" xlink:label="adgm_CommonStockWarrantsTrancheCMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_CommonStockWarrantsTrancheCMember" xlink:to="adgm_CommonStockWarrantsTrancheCMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_CommonStockWarrantsTrancheCMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents information pertaining to common stock Tranche C warrants.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_CommonStockWarrantsTrancheCMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Common Stock Warrants - Tranche C [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_CommonStockWarrantsTrancheCMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Tranche C Warrants</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_SaleOfStockNumberOfSharesIssuedInTransaction" xlink:label="us-gaap_SaleOfStockNumberOfSharesIssuedInTransaction" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_SaleOfStockNumberOfSharesIssuedInTransaction" xlink:to="us-gaap_SaleOfStockNumberOfSharesIssuedInTransaction_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_SaleOfStockNumberOfSharesIssuedInTransaction_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Sale of Stock, Number of Shares Issued in Transaction</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_SaleOfStockNumberOfSharesIssuedInTransaction_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of shares issued</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_SaleOfStockNumberOfShareWarrantsIssuedInTransaction" xlink:label="adgm_SaleOfStockNumberOfShareWarrantsIssuedInTransaction" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_SaleOfStockNumberOfShareWarrantsIssuedInTransaction" xlink:to="adgm_SaleOfStockNumberOfShareWarrantsIssuedInTransaction_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_SaleOfStockNumberOfShareWarrantsIssuedInTransaction_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The number of share warrants issued or sold by the subsidiary or equity method investee per stock transaction.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_SaleOfStockNumberOfShareWarrantsIssuedInTransaction_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Sale of Stock, Number of Share Warrants Issued in Transaction</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_SaleOfStockNumberOfShareWarrantsIssuedInTransaction_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Warrants issued (in shares)</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_SaleOfStockAndPreFundedWarrantNumberOfWarrantsAccompanyingEachShare" xlink:label="adgm_SaleOfStockAndPreFundedWarrantNumberOfWarrantsAccompanyingEachShare" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_SaleOfStockAndPreFundedWarrantNumberOfWarrantsAccompanyingEachShare" xlink:to="adgm_SaleOfStockAndPreFundedWarrantNumberOfWarrantsAccompanyingEachShare_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_SaleOfStockAndPreFundedWarrantNumberOfWarrantsAccompanyingEachShare_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The number of warrants accompanying each share and pre-funded warrant sold in a transaction.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_SaleOfStockAndPreFundedWarrantNumberOfWarrantsAccompanyingEachShare_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Sale of Stock and Pre-funded Warrant, Number of Warrants Accompanying Each Share</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_SaleOfStockAndPreFundedWarrantNumberOfWarrantsAccompanyingEachShare_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of warrant accompanying each share and each pre-funded warrant sold.</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_SharesAndMilestoneWarrantsIssuedCombinedPricePerShare" xlink:label="adgm_SharesAndMilestoneWarrantsIssuedCombinedPricePerShare" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_SharesAndMilestoneWarrantsIssuedCombinedPricePerShare" xlink:to="adgm_SharesAndMilestoneWarrantsIssuedCombinedPricePerShare_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_SharesAndMilestoneWarrantsIssuedCombinedPricePerShare_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Per share amount of shares and milestone warrant issued.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_SharesAndMilestoneWarrantsIssuedCombinedPricePerShare_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Shares and Milestone Warrants Issued, Combined Price Per Share</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_SharesAndMilestoneWarrantsIssuedCombinedPricePerShare_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Purchase price of each Share (in USD per share)</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_SharesAndWarrantsIssuedAllocatedPriceToEachMilestoneWarrant" xlink:label="adgm_SharesAndWarrantsIssuedAllocatedPriceToEachMilestoneWarrant" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_SharesAndWarrantsIssuedAllocatedPriceToEachMilestoneWarrant" xlink:to="adgm_SharesAndWarrantsIssuedAllocatedPriceToEachMilestoneWarrant_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_SharesAndWarrantsIssuedAllocatedPriceToEachMilestoneWarrant_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Per share amount allocated to each warrant in combined issuance of shares and warrant.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_SharesAndWarrantsIssuedAllocatedPriceToEachMilestoneWarrant_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Shares and Warrants Issued, Allocated Price to Each Milestone Warrant</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_SharesAndWarrantsIssuedAllocatedPriceToEachMilestoneWarrant_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Purchase price of each Share allocated to the Milestone Warrants (in USD per share)</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_SharesAndWarrantsIssuedCombinedPriceOfEachPreFundedAndMilestoneWarrant" xlink:label="adgm_SharesAndWarrantsIssuedCombinedPriceOfEachPreFundedAndMilestoneWarrant" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_SharesAndWarrantsIssuedCombinedPriceOfEachPreFundedAndMilestoneWarrant" xlink:to="adgm_SharesAndWarrantsIssuedCombinedPriceOfEachPreFundedAndMilestoneWarrant_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_SharesAndWarrantsIssuedCombinedPriceOfEachPreFundedAndMilestoneWarrant_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Per combine share amount allocated to each pre-funded and milestone warrant in combined issuance of shares and warrant.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_SharesAndWarrantsIssuedCombinedPriceOfEachPreFundedAndMilestoneWarrant_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Shares and Warrants Issued, Combined Price Of Each Pre-Funded and Milestone Warrant</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_SharesAndWarrantsIssuedCombinedPriceOfEachPreFundedAndMilestoneWarrant_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Combined purchase price of each Pre-Funded Warrant and Milestone Warrants (in USD per share)</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_WarrantsAndRightsOutstandingTerm" xlink:label="us-gaap_WarrantsAndRightsOutstandingTerm" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_WarrantsAndRightsOutstandingTerm" xlink:to="us-gaap_WarrantsAndRightsOutstandingTerm_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_WarrantsAndRightsOutstandingTerm_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Warrants and Rights Outstanding, Term</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_WarrantsAndRightsOutstandingTermFollowingResultsOfClinicalTrial" xlink:label="adgm_WarrantsAndRightsOutstandingTermFollowingResultsOfClinicalTrial" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_WarrantsAndRightsOutstandingTermFollowingResultsOfClinicalTrial" xlink:to="adgm_WarrantsAndRightsOutstandingTermFollowingResultsOfClinicalTrial_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_WarrantsAndRightsOutstandingTermFollowingResultsOfClinicalTrial_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Period between results of clinical trials and expiration of outstanding warrant and right embodying unconditional obligation requiring redemption by transferring asset at specified or determinable date or upon event certain to occur, in 'PnYnMnDTnHnMnS' format, for example, 'P1Y5M13D' represents reported fact of one year, five months, and thirteen days.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_WarrantsAndRightsOutstandingTermFollowingResultsOfClinicalTrial_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Warrants and Rights Outstanding, Term Following Results of Clinical Trial</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_WarrantsAndRightsOutstandingTermFollowingResultsOfClinicalTrial_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Term following clinical trial announcements</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_WarrantsAndRightsOutstandingTermFollowingRegulatoryApproval" xlink:label="adgm_WarrantsAndRightsOutstandingTermFollowingRegulatoryApproval" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_WarrantsAndRightsOutstandingTermFollowingRegulatoryApproval" xlink:to="adgm_WarrantsAndRightsOutstandingTermFollowingRegulatoryApproval_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_WarrantsAndRightsOutstandingTermFollowingRegulatoryApproval_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Period between regulatory approval of products  and expiration of outstanding warrant and right embodying unconditional obligation requiring redemption by transferring asset at specified or determinable date or upon event certain to occur, in 'PnYnMnDTnHnMnS' format, for example, 'P1Y5M13D' represents reported fact of one year, five months, and thirteen days.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_WarrantsAndRightsOutstandingTermFollowingRegulatoryApproval_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Warrants and Rights Outstanding, Term Following Regulatory Approval</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_WarrantsAndRightsOutstandingTermFollowingRegulatoryApproval_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Term following announcement of FDA approval of product</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_WarrantsExerciseMaximumOwnershipOfCommonStockThreshold" xlink:label="adgm_WarrantsExerciseMaximumOwnershipOfCommonStockThreshold" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_WarrantsExerciseMaximumOwnershipOfCommonStockThreshold" xlink:to="adgm_WarrantsExerciseMaximumOwnershipOfCommonStockThreshold_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_WarrantsExerciseMaximumOwnershipOfCommonStockThreshold_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Percentage of ownership of common stock of after the exercise warrants.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_WarrantsExerciseMaximumOwnershipOfCommonStockThreshold_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Warrants Exercise, Maximum Ownership Of Common Stock Threshold</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_WarrantsExerciseMaximumOwnershipOfCommonStockThreshold_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Maximum ownership of common stock by warrant holder after exercise (as a percent)</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_WarrantsExerciseMaximumOwnershipOfCommonStockThresholdAtHoldersOption" xlink:label="adgm_WarrantsExerciseMaximumOwnershipOfCommonStockThresholdAtHoldersOption" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_WarrantsExerciseMaximumOwnershipOfCommonStockThresholdAtHoldersOption" xlink:to="adgm_WarrantsExerciseMaximumOwnershipOfCommonStockThresholdAtHoldersOption_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_WarrantsExerciseMaximumOwnershipOfCommonStockThresholdAtHoldersOption_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Percentage of ownership of common stock of after the exercise warrants at the option of the holders.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_WarrantsExerciseMaximumOwnershipOfCommonStockThresholdAtHoldersOption_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Warrants Exercise, Maximum Ownership Of Common Stock Threshold, At Holders Option</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_WarrantsExerciseMaximumOwnershipOfCommonStockThresholdAtHoldersOption_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Maximum ownership of common stock by warrant holder after exercise, at the holder's option (as a percent)</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_WarrantsExerciseMaximumOwnershipOfCommonStockChangeThresholdAtHoldersOption" xlink:label="adgm_WarrantsExerciseMaximumOwnershipOfCommonStockChangeThresholdAtHoldersOption" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_WarrantsExerciseMaximumOwnershipOfCommonStockChangeThresholdAtHoldersOption" xlink:to="adgm_WarrantsExerciseMaximumOwnershipOfCommonStockChangeThresholdAtHoldersOption_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_WarrantsExerciseMaximumOwnershipOfCommonStockChangeThresholdAtHoldersOption_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The maximum percentage of ownership of common stock that threshold can be changed to after the exercise warrants at the option of the holders.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_WarrantsExerciseMaximumOwnershipOfCommonStockChangeThresholdAtHoldersOption_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Warrants Exercise, Maximum Ownership Of Common Stock Change Threshold, At Holders Option</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_WarrantsExerciseMaximumOwnershipOfCommonStockChangeThresholdAtHoldersOption_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Maximum ownership of common stock threshold can be changed to by warrant holder after exercise, at the holder's option (as a percent)</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_CommonStockOwnershipChangeThresholdAtHoldersOptionNotificationPeriod" xlink:label="adgm_CommonStockOwnershipChangeThresholdAtHoldersOptionNotificationPeriod" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_CommonStockOwnershipChangeThresholdAtHoldersOptionNotificationPeriod" xlink:to="adgm_CommonStockOwnershipChangeThresholdAtHoldersOptionNotificationPeriod_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_CommonStockOwnershipChangeThresholdAtHoldersOptionNotificationPeriod_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The notification period to change maximum percentage of ownership of common stock after the exercise warrants at the option of the holders.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_CommonStockOwnershipChangeThresholdAtHoldersOptionNotificationPeriod_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Common Stock Ownership Change Threshold, At Holders Option, Notification Period</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_CommonStockOwnershipChangeThresholdAtHoldersOptionNotificationPeriod_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Notification period of the change in ownership in commons stock threshold of warrant holders after exercies</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_SharesIssuedPricePerShare" xlink:label="us-gaap_SharesIssuedPricePerShare" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_SharesIssuedPricePerShare" xlink:to="us-gaap_SharesIssuedPricePerShare_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_SharesIssuedPricePerShare_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Shares Issued, Price Per Share</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_SharesIssuedPricePerShare_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Original issue price</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_SharesIssuedPricePerShare_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Price per share (in USD per share)</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_LesseeOperatingLeaseOptionToExtend" xlink:label="us-gaap_LesseeOperatingLeaseOptionToExtend" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_LesseeOperatingLeaseOptionToExtend" xlink:to="us-gaap_LesseeOperatingLeaseOptionToExtend_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_LesseeOperatingLeaseOptionToExtend_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Lessee, Operating Lease, Option to Extend</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_LesseeOperatingLeaseOptionToExtend_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Option to extend the lease term</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_LesseeOperatingLeaseRenewalTerm" xlink:label="us-gaap_LesseeOperatingLeaseRenewalTerm" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_LesseeOperatingLeaseRenewalTerm" xlink:to="us-gaap_LesseeOperatingLeaseRenewalTerm_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_LesseeOperatingLeaseRenewalTerm_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Lessee, Operating Lease, Renewal Term</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_LesseeOperatingLeaseRenewalTerm_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Operating lease renewal term</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_LeaseCostAbstract" xlink:label="us-gaap_LeaseCostAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_LeaseCostAbstract" xlink:to="us-gaap_LeaseCostAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_LeaseCostAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Lease, Cost [Abstract]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_LeaseCostAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Quantitative information of operating leases</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_OperatingLeasePayments" xlink:label="us-gaap_OperatingLeasePayments" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_OperatingLeasePayments" xlink:to="us-gaap_OperatingLeasePayments_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_OperatingLeasePayments_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Operating Lease, Payments</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_OperatingLeasePayments_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Operating cash flows paid for operating leases</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_OperatingLeaseWeightedAverageRemainingLeaseTerm1" xlink:label="us-gaap_OperatingLeaseWeightedAverageRemainingLeaseTerm1" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_OperatingLeaseWeightedAverageRemainingLeaseTerm1" xlink:to="us-gaap_OperatingLeaseWeightedAverageRemainingLeaseTerm1_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_OperatingLeaseWeightedAverageRemainingLeaseTerm1_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Operating Lease, Weighted Average Remaining Lease Term</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_OperatingLeaseWeightedAverageRemainingLeaseTerm1_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Weighted average remaining lease term (years)</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_OperatingLeaseWeightedAverageDiscountRatePercent" xlink:label="us-gaap_OperatingLeaseWeightedAverageDiscountRatePercent" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_OperatingLeaseWeightedAverageDiscountRatePercent" xlink:to="us-gaap_OperatingLeaseWeightedAverageDiscountRatePercent_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_OperatingLeaseWeightedAverageDiscountRatePercent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Operating Lease, Weighted Average Discount Rate, Percent</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_OperatingLeaseWeightedAverageDiscountRatePercent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Weighted average discount rate</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueNextTwelveMonths" xlink:label="us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueNextTwelveMonths" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueNextTwelveMonths" xlink:to="us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueNextTwelveMonths_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueNextTwelveMonths_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Lessee, Operating Lease, Liability, to be Paid, Year One</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueNextTwelveMonths_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Year ending December 31, 2026</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearTwo" xlink:label="us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearTwo" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearTwo" xlink:to="us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearTwo_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearTwo_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Lessee, Operating Lease, Liability, to be Paid, Year Two</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearTwo_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Year ending December 31, 2027</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearThree" xlink:label="us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearThree" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearThree" xlink:to="us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearThree_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearThree_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Lessee, Operating Lease, Liability, to be Paid, Year Three</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearThree_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Year ending December 31, 2028</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearFour" xlink:label="us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearFour" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearFour" xlink:to="us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearFour_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearFour_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Lessee, Operating Lease, Liability, to be Paid, Year Four</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearFour_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Year ending December 31, 2029</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearFive" xlink:label="us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearFive" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearFive" xlink:to="us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearFive_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearFive_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Lessee, Operating Lease, Liability, to be Paid, Year Five</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearFive_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Year ending December 31, 2030</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_LesseeOperatingLeaseLiabilityPaymentsDue" xlink:label="us-gaap_LesseeOperatingLeaseLiabilityPaymentsDue" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_LesseeOperatingLeaseLiabilityPaymentsDue" xlink:to="us-gaap_LesseeOperatingLeaseLiabilityPaymentsDue_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_LesseeOperatingLeaseLiabilityPaymentsDue_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Lessee, Operating Lease, Liability, to be Paid</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:label="us-gaap_LesseeOperatingLeaseLiabilityPaymentsDue_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Total undiscounted future cash flows</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_LesseeOperatingLeaseLiabilityUndiscountedExcessAmount" xlink:label="us-gaap_LesseeOperatingLeaseLiabilityUndiscountedExcessAmount" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_LesseeOperatingLeaseLiabilityUndiscountedExcessAmount" xlink:to="us-gaap_LesseeOperatingLeaseLiabilityUndiscountedExcessAmount_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_LesseeOperatingLeaseLiabilityUndiscountedExcessAmount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Lessee, Operating Lease, Liability, Undiscounted Excess Amount</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2009/role/negatedLabel" xlink:label="us-gaap_LesseeOperatingLeaseLiabilityUndiscountedExcessAmount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Less: imputed interest</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_OperatingLeaseLiability" xlink:label="us-gaap_OperatingLeaseLiability" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_OperatingLeaseLiability" xlink:to="us-gaap_OperatingLeaseLiability_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_OperatingLeaseLiability_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Operating Lease, Liability</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_OperatingLeaseLiability_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Total operating lease liability</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_SharesAuthorizedNumberOfClass" xlink:label="adgm_SharesAuthorizedNumberOfClass" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_SharesAuthorizedNumberOfClass" xlink:to="adgm_SharesAuthorizedNumberOfClass_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_SharesAuthorizedNumberOfClass_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The number of class of shares authorized per articles of incorporation.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_SharesAuthorizedNumberOfClass_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Shares Authorized, Number of Class</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_SharesAuthorizedNumberOfClass_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of shares authorized</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_SharesAuthorized" xlink:label="adgm_SharesAuthorized" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_SharesAuthorized" xlink:to="adgm_SharesAuthorized_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_SharesAuthorized_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The maximum number of shares permitted to be issued by an entity's charter and bylaws.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_SharesAuthorized_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Shares Authorized</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_SharesAuthorized_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Shares authorized</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_TemporaryEquitySharesAuthorized" xlink:label="us-gaap_TemporaryEquitySharesAuthorized" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_TemporaryEquitySharesAuthorized" xlink:to="us-gaap_TemporaryEquitySharesAuthorized_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_TemporaryEquitySharesAuthorized_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Temporary Equity, Shares Authorized</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_TemporaryEquitySharesAuthorized_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Convertible preferred stock, shares authorized</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_TemporaryEquitySharesAuthorized_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of Shares Authorized</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_TemporaryEquityByClassOfStockTable" xlink:label="us-gaap_TemporaryEquityByClassOfStockTable" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_TemporaryEquityByClassOfStockTable" xlink:to="us-gaap_TemporaryEquityByClassOfStockTable_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_TemporaryEquityByClassOfStockTable_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Temporary Equity, by Class of Stock [Table]</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_SeriesaConvertiblePreferredStockMember" xlink:label="adgm_SeriesaConvertiblePreferredStockMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_SeriesaConvertiblePreferredStockMember" xlink:to="adgm_SeriesaConvertiblePreferredStockMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_SeriesaConvertiblePreferredStockMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Preferred stock that may be exchanged into common shares or other types of securities at the owner's option, classified as Series A.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_SeriesaConvertiblePreferredStockMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">SeriesA Convertible Preferred Stock [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_SeriesaConvertiblePreferredStockMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Series A Preferred Stock Class</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_SeriesBConvertiblePreferredStockMember" xlink:label="adgm_SeriesBConvertiblePreferredStockMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_SeriesBConvertiblePreferredStockMember" xlink:to="adgm_SeriesBConvertiblePreferredStockMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_SeriesBConvertiblePreferredStockMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Preferred stock that may be exchanged into common shares or other types of securities at the owner's option, classified as Series B.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_SeriesBConvertiblePreferredStockMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Series B Convertible Preferred Stock [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_SeriesBConvertiblePreferredStockMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Series B Preferred Stock Class</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_SeriesCConvertiblePreferredStockMember" xlink:label="adgm_SeriesCConvertiblePreferredStockMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_SeriesCConvertiblePreferredStockMember" xlink:to="adgm_SeriesCConvertiblePreferredStockMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_SeriesCConvertiblePreferredStockMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Preferred stock that may be exchanged into common shares or other types of securities at the owner's option, classified as Series C.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_SeriesCConvertiblePreferredStockMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Series C Convertible Preferred Stock [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_SeriesCConvertiblePreferredStockMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Series C Preferred Stock Class</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_SeriesDConvertiblePreferredStockMember" xlink:label="adgm_SeriesDConvertiblePreferredStockMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_SeriesDConvertiblePreferredStockMember" xlink:to="adgm_SeriesDConvertiblePreferredStockMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_SeriesDConvertiblePreferredStockMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Preferred stock that may be exchanged into common shares or other types of securities at the owner's option, classified as Series D.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_SeriesDConvertiblePreferredStockMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Series D Convertible Preferred Stock [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_SeriesDConvertiblePreferredStockMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Series D Preferred Stock Class</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_TemporaryEquityLineItems" xlink:label="us-gaap_TemporaryEquityLineItems" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_TemporaryEquityLineItems" xlink:to="us-gaap_TemporaryEquityLineItems_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_TemporaryEquityLineItems_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Temporary Equity [Line Items]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_TemporaryEquityLineItems_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Mezzanine Equity and Stockholders' Equity (Deficit)</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_TemporaryEquityNumberOfSharesExtinguished" xlink:label="adgm_TemporaryEquityNumberOfSharesExtinguished" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_TemporaryEquityNumberOfSharesExtinguished" xlink:to="adgm_TemporaryEquityNumberOfSharesExtinguished_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_TemporaryEquityNumberOfSharesExtinguished_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of shares of original convertible preferred stock classified as temporary equity extinguished.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_TemporaryEquityNumberOfSharesExtinguished_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Temporary Equity, Number Of Shares Extinguished</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_TemporaryEquityNumberOfSharesExtinguished_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of preferred stocks extinguished</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_TemporaryEquitySharesOutstanding" xlink:label="us-gaap_TemporaryEquitySharesOutstanding" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_TemporaryEquitySharesOutstanding" xlink:to="us-gaap_TemporaryEquitySharesOutstanding_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_TemporaryEquitySharesOutstanding_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Temporary Equity, Shares Outstanding</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_TemporaryEquitySharesOutstanding_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Convertible preferred stock, shares outstanding</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_TemporaryEquitySharesOutstanding_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Conversion of preferred stock</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_PreferredStockConvertibleConversionPrice" xlink:label="us-gaap_PreferredStockConvertibleConversionPrice" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_PreferredStockConvertibleConversionPrice" xlink:to="us-gaap_PreferredStockConvertibleConversionPrice_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_PreferredStockConvertibleConversionPrice_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Preferred Stock, Convertible, Conversion Price</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_PreferredStockConvertibleConversionPrice_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Conversion Price Per Share</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_PreferredStockConvertibleConversionPrice_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Initial conversion price</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_PreferredStockConvertibleSharesIssuable" xlink:label="us-gaap_PreferredStockConvertibleSharesIssuable" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_PreferredStockConvertibleSharesIssuable" xlink:to="us-gaap_PreferredStockConvertibleSharesIssuable_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_PreferredStockConvertibleSharesIssuable_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Preferred Stock, Convertible, Shares Issuable</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_PreferredStockConvertibleSharesIssuable_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of Common Stock Equivalent Shares</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_PreferredStockConvertibleSharesIssuable_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Conversion of preferred stock</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_TemporaryEquityLiquidationPreference" xlink:label="us-gaap_TemporaryEquityLiquidationPreference" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_TemporaryEquityLiquidationPreference" xlink:to="us-gaap_TemporaryEquityLiquidationPreference_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_TemporaryEquityLiquidationPreference_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Temporary Equity, Liquidation Preference</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_TemporaryEquityLiquidationPreference_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Liquidation Preference</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ConversionOfStockSharesConverted1" xlink:label="us-gaap_ConversionOfStockSharesConverted1" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ConversionOfStockSharesConverted1" xlink:to="us-gaap_ConversionOfStockSharesConverted1_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ConversionOfStockSharesConverted1_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Conversion of Stock, Shares Converted</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ConversionOfStockSharesConverted1_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Convertible preferred stock</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ConversionOfStockSharesIssued1" xlink:label="us-gaap_ConversionOfStockSharesIssued1" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ConversionOfStockSharesIssued1" xlink:to="us-gaap_ConversionOfStockSharesIssued1_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ConversionOfStockSharesIssued1_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Conversion of Stock, Shares Issued</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ConversionOfStockSharesIssued1_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Convertible preferred stock issued</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_BusinessCombinationSharesIssuedInConversionOfAcquireeShares" xlink:label="adgm_BusinessCombinationSharesIssuedInConversionOfAcquireeShares" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_BusinessCombinationSharesIssuedInConversionOfAcquireeShares" xlink:to="adgm_BusinessCombinationSharesIssuedInConversionOfAcquireeShares_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_BusinessCombinationSharesIssuedInConversionOfAcquireeShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of shares of equity interests issued to convert the acquiree's shares.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_BusinessCombinationSharesIssuedInConversionOfAcquireeShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Business Combination, Shares Issued In Conversion Of Acquiree Shares</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_BusinessCombinationSharesIssuedInConversionOfAcquireeShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Conversion of Legacy shares to the Company's shares</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_PreferredStockTransactionsInPeriod" xlink:label="adgm_PreferredStockTransactionsInPeriod" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_PreferredStockTransactionsInPeriod" xlink:to="adgm_PreferredStockTransactionsInPeriod_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_PreferredStockTransactionsInPeriod_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The preferred stock transactions reported during the period.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_PreferredStockTransactionsInPeriod_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Preferred Stock Transactions In Period</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_PreferredStockTransactionsInPeriod_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Preferred stock transactions</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_PreferredStockDividendRatePercentage" xlink:label="us-gaap_PreferredStockDividendRatePercentage" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_PreferredStockDividendRatePercentage" xlink:to="us-gaap_PreferredStockDividendRatePercentage_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_PreferredStockDividendRatePercentage_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Preferred Stock, Dividend Rate, Percentage</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_PreferredStockDividendRatePercentage_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Preferred stock dividends (in percent)</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_ShareDividendToDate" xlink:label="adgm_ShareDividendToDate" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_ShareDividendToDate" xlink:to="adgm_ShareDividendToDate_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_ShareDividendToDate_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The amount of dividend declared to date for each share of outstanding.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_ShareDividendToDate_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Share Dividend To Date</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_ShareDividendToDate_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Dividend declared as of reporting date (in USD per share)</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_ThresholdMinimumNumberOfCommonStockConvertibleFromConvertiblePreferredStockOutstandingForVoting" xlink:label="adgm_ThresholdMinimumNumberOfCommonStockConvertibleFromConvertiblePreferredStockOutstandingForVoting" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_ThresholdMinimumNumberOfCommonStockConvertibleFromConvertiblePreferredStockOutstandingForVoting" xlink:to="adgm_ThresholdMinimumNumberOfCommonStockConvertibleFromConvertiblePreferredStockOutstandingForVoting_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_ThresholdMinimumNumberOfCommonStockConvertibleFromConvertiblePreferredStockOutstandingForVoting_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The minimum number of shares of common stock that can be converted from preferred stock, subject to appropriate adjustment in the event of any stock dividends, stock split, combination or other similar recapitalization with respect to the common stock, which defines the voting rights.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_ThresholdMinimumNumberOfCommonStockConvertibleFromConvertiblePreferredStockOutstandingForVoting_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold Minimum Number of Common Stock Convertible From Convertible Preferred Stock Outstanding For Voting</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_ThresholdMinimumNumberOfCommonStockConvertibleFromConvertiblePreferredStockOutstandingForVoting_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Minimum shares of common stock converted from preferred stock</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_ConvertiblePreferredStockVotingRightsNumberOfDirectorsElected" xlink:label="adgm_ConvertiblePreferredStockVotingRightsNumberOfDirectorsElected" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_ConvertiblePreferredStockVotingRightsNumberOfDirectorsElected" xlink:to="adgm_ConvertiblePreferredStockVotingRightsNumberOfDirectorsElected_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_ConvertiblePreferredStockVotingRightsNumberOfDirectorsElected_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The number of members of the Board of Directors that can be elected by the holders of convertible preferred stock, voting as a separate class on an as-converted to common stock basis, voting as a separate class.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_ConvertiblePreferredStockVotingRightsNumberOfDirectorsElected_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Convertible Preferred Stock, Voting Rights, Number of Directors Elected</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_ConvertiblePreferredStockVotingRightsNumberOfDirectorsElected_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of directors elected by holders of preferred stock</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_CommonStockVotingRightsNumberOfDirectorsElected" xlink:label="adgm_CommonStockVotingRightsNumberOfDirectorsElected" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_CommonStockVotingRightsNumberOfDirectorsElected" xlink:to="adgm_CommonStockVotingRightsNumberOfDirectorsElected_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_CommonStockVotingRightsNumberOfDirectorsElected_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The number of members of the Board of Directors that can be elected by the holders of common stock, voting as a separate class.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_CommonStockVotingRightsNumberOfDirectorsElected_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Common Stock, Voting Rights, Number of Directors Elected</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_CommonStockVotingRightsNumberOfDirectorsElected_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of directors elected by holders of common stock</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_PreferredStockConvertibleConversionPriceAfterAdjustments" xlink:label="adgm_PreferredStockConvertibleConversionPriceAfterAdjustments" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_PreferredStockConvertibleConversionPriceAfterAdjustments" xlink:to="adgm_PreferredStockConvertibleConversionPriceAfterAdjustments_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_PreferredStockConvertibleConversionPriceAfterAdjustments_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Per share conversion value of preferred stock after adjustments.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_PreferredStockConvertibleConversionPriceAfterAdjustments_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Preferred Stock, Convertible, Conversion Price After Adjustments</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_PreferredStockConvertibleConversionPriceAfterAdjustments_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Conversion price after adjustment</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_ProtectiveProvisionsThresholdPreferredStockOutstanding" xlink:label="adgm_ProtectiveProvisionsThresholdPreferredStockOutstanding" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_ProtectiveProvisionsThresholdPreferredStockOutstanding" xlink:to="adgm_ProtectiveProvisionsThresholdPreferredStockOutstanding_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_ProtectiveProvisionsThresholdPreferredStockOutstanding_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Minimum number of preferred shares outstanding, in which the company cannot (by merger, reclassification, amendment or otherwise), (a) amend the certificate of incorporation or bylaws; (b) adversely alter or change the rights, preferences or privileges of the preferred stock; (c) increase or decrease the aggregate number of authorized shares of any class of the capital stock of the Company.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_ProtectiveProvisionsThresholdPreferredStockOutstanding_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Protective Provisions, Threshold Preferred Stock Outstanding</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_ProtectiveProvisionsThresholdPreferredStockOutstanding_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold number of preferred stock outstanding</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_ProtectiveProvisionsThresholdVotingPercentageOfPreferredStockholders" xlink:label="adgm_ProtectiveProvisionsThresholdVotingPercentageOfPreferredStockholders" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_ProtectiveProvisionsThresholdVotingPercentageOfPreferredStockholders" xlink:to="adgm_ProtectiveProvisionsThresholdVotingPercentageOfPreferredStockholders_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_ProtectiveProvisionsThresholdVotingPercentageOfPreferredStockholders_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Minimum percentage of approval of the holders of the then outstanding shares of preferred stock, voting separately as a class, to, among other things: (a) amend the certificate of incorporation or bylaws; (b) adversely alter or change the rights, preferences or privileges of the preferred stock; (c) increase or decrease the aggregate number of authorized shares of any class of the capital stock of the Company.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_ProtectiveProvisionsThresholdVotingPercentageOfPreferredStockholders_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Protective Provisions, Threshold Voting Percentage of Preferred Stockholders</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_ProtectiveProvisionsThresholdVotingPercentageOfPreferredStockholders_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold percentage of voting by preferred stockholders</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_ProtectiveProvisionsThresholdMinimumNumberOfCommonStockOutOfOutstandingConvertibleFromConvertiblePreferredStock" xlink:label="adgm_ProtectiveProvisionsThresholdMinimumNumberOfCommonStockOutOfOutstandingConvertibleFromConvertiblePreferredStock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_ProtectiveProvisionsThresholdMinimumNumberOfCommonStockOutOfOutstandingConvertibleFromConvertiblePreferredStock" xlink:to="adgm_ProtectiveProvisionsThresholdMinimumNumberOfCommonStockOutOfOutstandingConvertibleFromConvertiblePreferredStock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_ProtectiveProvisionsThresholdMinimumNumberOfCommonStockOutOfOutstandingConvertibleFromConvertiblePreferredStock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Minimum number of common stock convertible from the preferred stock, at which the company shall: (a) amend, alter, repeal or waive any provision of the certificate of incorporation or bylaws of the Company in a manner that adversely affects the holders of the Series E preferred stock in a manner different from any other series of preferred stock; (b) create or authorize the creation of or issue any other security convertible into to exercisable for any equity security having rights, preferences or privileges senior to the Series E preferred stock; (c) increase or decrease the authorized number of shares of Series E preferred stock</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_ProtectiveProvisionsThresholdMinimumNumberOfCommonStockOutOfOutstandingConvertibleFromConvertiblePreferredStock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Protective Provisions, Threshold Minimum Number of Common Stock Out of Outstanding Convertible From Convertible Preferred Stock</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_ProtectiveProvisionsThresholdMinimumNumberOfCommonStockOutOfOutstandingConvertibleFromConvertiblePreferredStock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold common stock convertible by Series E convertible preferred stock</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ScheduleOfStockByClassTable" xlink:label="us-gaap_ScheduleOfStockByClassTable" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ScheduleOfStockByClassTable" xlink:to="us-gaap_ScheduleOfStockByClassTable_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ScheduleOfStockByClassTable_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Stock, Class of Stock [Table]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_RelatedPartyTransactionAxis" xlink:label="us-gaap_RelatedPartyTransactionAxis" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_RelatedPartyTransactionAxis" xlink:to="us-gaap_RelatedPartyTransactionAxis_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_RelatedPartyTransactionAxis_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Related Party Transaction [Axis]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_RelatedPartyTransactionDomain" xlink:label="us-gaap_RelatedPartyTransactionDomain" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_RelatedPartyTransactionDomain" xlink:to="us-gaap_RelatedPartyTransactionDomain_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_RelatedPartyTransactionDomain_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Related Party Transaction [Domain]</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_SponsorEarnoutMember" xlink:label="adgm_SponsorEarnoutMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_SponsorEarnoutMember" xlink:to="adgm_SponsorEarnoutMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_SponsorEarnoutMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The represents information related to sponsor earnout</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_SponsorEarnoutMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Sponsor Earnout [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_SponsorEarnoutMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Sponsor Earnout</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ClassOfStockLineItems" xlink:label="us-gaap_ClassOfStockLineItems" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ClassOfStockLineItems" xlink:to="us-gaap_ClassOfStockLineItems_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ClassOfStockLineItems_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class of Stock [Line Items]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ClassOfStockLineItems_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Common Stock</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_CommonStockVotingRightsVotesPerShare" xlink:label="adgm_CommonStockVotingRightsVotesPerShare" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_CommonStockVotingRightsVotesPerShare" xlink:to="adgm_CommonStockVotingRightsVotesPerShare_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_CommonStockVotingRightsVotesPerShare_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The number of votes entitled to each share of common stock.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_CommonStockVotingRightsVotesPerShare_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Common Stock, Voting Rights, Votes Per Share</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_CommonStockVotingRightsVotesPerShare_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Votes per share of common stock</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardNumberOfSharesAvailableForGrant" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardNumberOfSharesAvailableForGrant" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardNumberOfSharesAvailableForGrant" xlink:to="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardNumberOfSharesAvailableForGrant_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardNumberOfSharesAvailableForGrant_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Share-Based Compensation Arrangement by Share-Based Payment Award, Number of Shares Available for Grant</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardNumberOfSharesAvailableForGrant_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of shares available for issuance</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardNumberOfSharesAvailableForGrant_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Common shares available for future grant under the 2012 and 2022 Plan</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_PreferredStockSharesAuthorized" xlink:label="us-gaap_PreferredStockSharesAuthorized" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_PreferredStockSharesAuthorized" xlink:to="us-gaap_PreferredStockSharesAuthorized_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_PreferredStockSharesAuthorized_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Preferred Stock, Shares Authorized</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_PreferredStockSharesAuthorized_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Preferred stock, shares authorized</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_CommonStockTriggerPrice" xlink:label="adgm_CommonStockTriggerPrice" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_CommonStockTriggerPrice" xlink:to="adgm_CommonStockTriggerPrice_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_CommonStockTriggerPrice_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">the information related to common stock trigger price</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_CommonStockTriggerPrice_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Common Stock Trigger Price</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_CommonStockTriggerPrice_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Common stock trigger price</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_SponsorAgreementShareTriggerPriceTradingDays" xlink:label="adgm_SponsorAgreementShareTriggerPriceTradingDays" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_SponsorAgreementShareTriggerPriceTradingDays" xlink:to="adgm_SponsorAgreementShareTriggerPriceTradingDays_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_SponsorAgreementShareTriggerPriceTradingDays_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The number of trading days of the entity's shares at a certain price (Trigger Price) and during the earn out period for the shares issued to sponsor to vest.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_SponsorAgreementShareTriggerPriceTradingDays_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Sponsor Agreement, Share Trigger Price, Trading Days</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_SponsorAgreementShareTriggerPriceTradingDays_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Trading days of trigger price</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_SponsorAgreementShareTriggerPriceTradingDaysPeriod" xlink:label="adgm_SponsorAgreementShareTriggerPriceTradingDaysPeriod" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_SponsorAgreementShareTriggerPriceTradingDaysPeriod" xlink:to="adgm_SponsorAgreementShareTriggerPriceTradingDaysPeriod_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_SponsorAgreementShareTriggerPriceTradingDaysPeriod_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The period over which the number of trading days of the entity's shares at a certain price (Trigger Price) and during the earn-out period for the shares issued to sponsor to vest.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_SponsorAgreementShareTriggerPriceTradingDaysPeriod_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Sponsor Agreement, Share Trigger Price, Trading Days Period</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_SponsorAgreementShareTriggerPriceTradingDaysPeriod_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Trading day period</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_CashDividendToDate" xlink:label="adgm_CashDividendToDate" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_CashDividendToDate" xlink:to="adgm_CashDividendToDate_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_CashDividendToDate_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The number of vote the entity's common stock is entitled to.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_CashDividendToDate_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cash Dividend To Date</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_CashDividendToDate_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cash dividend to date</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/srt/2024/elts/srt-2024.xsd#srt_CounterpartyNameAxis" xlink:label="srt_CounterpartyNameAxis" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="srt_CounterpartyNameAxis" xlink:to="srt_CounterpartyNameAxis_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="srt_CounterpartyNameAxis_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Counterparty Name [Axis]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/srt/2024/elts/srt-2024.xsd#srt_RepurchaseAgreementCounterpartyNameDomain" xlink:label="srt_RepurchaseAgreementCounterpartyNameDomain" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="srt_RepurchaseAgreementCounterpartyNameDomain" xlink:to="srt_RepurchaseAgreementCounterpartyNameDomain_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="srt_RepurchaseAgreementCounterpartyNameDomain_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Counterparty Name [Domain]</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_PipeInvestorMember" xlink:label="adgm_PipeInvestorMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_PipeInvestorMember" xlink:to="adgm_PipeInvestorMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_PipeInvestorMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents information pertaining to certain PIPE investor.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_PipeInvestorMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">PIPE Investor [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_PipeInvestorMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">PIPE Investor</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_StockIssuedDuringPeriodSharesConversionOfTemporaryEquityShares" xlink:label="adgm_StockIssuedDuringPeriodSharesConversionOfTemporaryEquityShares" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_StockIssuedDuringPeriodSharesConversionOfTemporaryEquityShares" xlink:to="adgm_StockIssuedDuringPeriodSharesConversionOfTemporaryEquityShares_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_StockIssuedDuringPeriodSharesConversionOfTemporaryEquityShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of shares issued by conversion of temporary equity shares of acquiree entity during the period pursuant to merger process.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_StockIssuedDuringPeriodSharesConversionOfTemporaryEquityShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Stock Issued During Period, Shares, Conversion of Temporary Equity Shares</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_StockIssuedDuringPeriodSharesConversionOfTemporaryEquityShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Conversion of Class A ordinary shares subject to redemption</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_PlanNameAxis" xlink:label="us-gaap_PlanNameAxis" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_PlanNameAxis" xlink:to="us-gaap_PlanNameAxis_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_PlanNameAxis_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Plan Name [Axis]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_PlanNameDomain" xlink:label="us-gaap_PlanNameDomain" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_PlanNameDomain" xlink:to="us-gaap_PlanNameDomain_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_PlanNameDomain_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Plan Name [Domain]</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_EquityIncentivePlan2024Member" xlink:label="adgm_EquityIncentivePlan2024Member" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_EquityIncentivePlan2024Member" xlink:to="adgm_EquityIncentivePlan2024Member_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_EquityIncentivePlan2024Member_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents information pertaining to 2024 Equity Incentive Plan.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_EquityIncentivePlan2024Member_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Equity Incentive Plan, 2024 [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_EquityIncentivePlan2024Member_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">2024 Equity incentive plan</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_KeyEmployeeEquityIncentivePlan2024Member" xlink:label="adgm_KeyEmployeeEquityIncentivePlan2024Member" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_KeyEmployeeEquityIncentivePlan2024Member" xlink:to="adgm_KeyEmployeeEquityIncentivePlan2024Member_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_KeyEmployeeEquityIncentivePlan2024Member_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents information pertaining to 2024 Key Employee Equity Incentive Plan.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_KeyEmployeeEquityIncentivePlan2024Member_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Key Employee Equity Incentive Plan, 2024 [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_KeyEmployeeEquityIncentivePlan2024Member_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">2024 Key employee equity incentive plan</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_EmployeeStockPurchasePlan2024Member" xlink:label="adgm_EmployeeStockPurchasePlan2024Member" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_EmployeeStockPurchasePlan2024Member" xlink:to="adgm_EmployeeStockPurchasePlan2024Member_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_EmployeeStockPurchasePlan2024Member_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents information pertaining to 2024 Employee Stock Purchase Plan.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_EmployeeStockPurchasePlan2024Member_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Employee Stock Purchase Plan, 2024 [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_EmployeeStockPurchasePlan2024Member_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">2024 Employee Stock Purchase Plan</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_PipeMilestoneWarrantsWarrantsMember" xlink:label="adgm_PipeMilestoneWarrantsWarrantsMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_PipeMilestoneWarrantsWarrantsMember" xlink:to="adgm_PipeMilestoneWarrantsWarrantsMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_PipeMilestoneWarrantsWarrantsMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents information pertaining to PIPE milestone warrants.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_PipeMilestoneWarrantsWarrantsMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">PIPE Milestone Warrants Warrants [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_PipeMilestoneWarrantsWarrantsMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">2025 PIPE Milestone Warrants</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="adgm_PipeMilestoneWarrantsWarrantsMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">2025 Milestone Warrants</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingNumber" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingNumber" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingNumber" xlink:to="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingNumber_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingNumber_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Share-Based Compensation Arrangement by Share-Based Payment Award, Options, Outstanding, Number</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/periodEndLabel" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingNumber_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Outstanding (shares)</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/periodStartLabel" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingNumber_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Outstanding (shares)</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingNumber_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Stock options issued and outstanding under the 2012 and 2022 Plan</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_ConvertibleDebtSharesIssuableShares" xlink:label="adgm_ConvertibleDebtSharesIssuableShares" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_ConvertibleDebtSharesIssuableShares" xlink:to="adgm_ConvertibleDebtSharesIssuableShares_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_ConvertibleDebtSharesIssuableShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of common shares issuable upon conversion of debt.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_ConvertibleDebtSharesIssuableShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Convertible Debt, Shares Issuable Shares</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_ConvertibleDebtSharesIssuableShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Convertible Securities Notes</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_StockIssuableUponExerciseOfOutstandingOptionsEquityPlansAssumedInBusinessCombination" xlink:label="adgm_StockIssuableUponExerciseOfOutstandingOptionsEquityPlansAssumedInBusinessCombination" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_StockIssuableUponExerciseOfOutstandingOptionsEquityPlansAssumedInBusinessCombination" xlink:to="adgm_StockIssuableUponExerciseOfOutstandingOptionsEquityPlansAssumedInBusinessCombination_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_StockIssuableUponExerciseOfOutstandingOptionsEquityPlansAssumedInBusinessCombination_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents the number of shares issuable upon the exercise of outstanding options that were assumed in the Business Combination.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_StockIssuableUponExerciseOfOutstandingOptionsEquityPlansAssumedInBusinessCombination_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Stock Issuable upon Exercise of Outstanding Options Equity Plans Assumed in Business Combination</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_StockIssuableUponExerciseOfOutstandingOptionsEquityPlansAssumedInBusinessCombination_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Company's common stock issuable upon the exercise of outstanding options Legacy Adagio's equity plans that were assumed in the Business Combination</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ScheduleOfShareBasedCompensationArrangementsByShareBasedPaymentAwardTable" xlink:label="us-gaap_ScheduleOfShareBasedCompensationArrangementsByShareBasedPaymentAwardTable" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ScheduleOfShareBasedCompensationArrangementsByShareBasedPaymentAwardTable" xlink:to="us-gaap_ScheduleOfShareBasedCompensationArrangementsByShareBasedPaymentAwardTable_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ScheduleOfShareBasedCompensationArrangementsByShareBasedPaymentAwardTable_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Schedule of Share-Based Compensation Arrangements by Share-Based Payment Award [Table]</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_StockIncentivePlan2012Member" xlink:label="adgm_StockIncentivePlan2012Member" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_StockIncentivePlan2012Member" xlink:to="adgm_StockIncentivePlan2012Member_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_StockIncentivePlan2012Member_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents information related to Stock Incentive Plan 2012.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_StockIncentivePlan2012Member_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Stock Incentive Plan 2012 [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_StockIncentivePlan2012Member_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">2012 Stock Incentive Plan</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardLineItems" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardLineItems" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardLineItems" xlink:to="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardLineItems_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardLineItems_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Share-Based Compensation Arrangement by Share-Based Payment Award [Line Items]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardLineItems_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Stock-Based Compensation</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardNumberOfSharesAuthorized" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardNumberOfSharesAuthorized" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardNumberOfSharesAuthorized" xlink:to="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardNumberOfSharesAuthorized_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardNumberOfSharesAuthorized_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Share-Based Compensation Arrangement by Share-Based Payment Award, Number of Shares Authorized</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardNumberOfSharesAuthorized_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Maximum number of shares that can be granted</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardExpirationPeriod" xlink:label="us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardExpirationPeriod" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardExpirationPeriod" xlink:to="us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardExpirationPeriod_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardExpirationPeriod_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Share-Based Compensation Arrangement by Share-Based Payment Award, Expiration Period</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardExpirationPeriod_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Maximum term</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_StockIncentivePlan2022Member" xlink:label="adgm_StockIncentivePlan2022Member" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_StockIncentivePlan2022Member" xlink:to="adgm_StockIncentivePlan2022Member_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_StockIncentivePlan2022Member_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents information related to Stock Incentive Plan 2022.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_StockIncentivePlan2022Member_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Stock Incentive Plan 2022 [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_StockIncentivePlan2022Member_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">2022 Stock Incentive Plan</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_ShareBasedCompensationArrangementByShareBasedPaymentAwardNumberOfNonEmployeesWhosePerformanceObligationWereDeemedToBeImmaterial" xlink:label="adgm_ShareBasedCompensationArrangementByShareBasedPaymentAwardNumberOfNonEmployeesWhosePerformanceObligationWereDeemedToBeImmaterial" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_ShareBasedCompensationArrangementByShareBasedPaymentAwardNumberOfNonEmployeesWhosePerformanceObligationWereDeemedToBeImmaterial" xlink:to="adgm_ShareBasedCompensationArrangementByShareBasedPaymentAwardNumberOfNonEmployeesWhosePerformanceObligationWereDeemedToBeImmaterial_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_ShareBasedCompensationArrangementByShareBasedPaymentAwardNumberOfNonEmployeesWhosePerformanceObligationWereDeemedToBeImmaterial_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The number of non employees whose performance obligation were deemed to be immaterial.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_ShareBasedCompensationArrangementByShareBasedPaymentAwardNumberOfNonEmployeesWhosePerformanceObligationWereDeemedToBeImmaterial_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Share-Based Compensation Arrangement by Share-Based Payment Award, Number of Non Employees Whose Performance Obligation were Deemed to be Immaterial</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_ShareBasedCompensationArrangementByShareBasedPaymentAwardNumberOfNonEmployeesWhosePerformanceObligationWereDeemedToBeImmaterial_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of non employees whose performance obligation were deemed to be immaterial</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardAwardVestingPeriod1" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardAwardVestingPeriod1" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardAwardVestingPeriod1" xlink:to="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardAwardVestingPeriod1_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardAwardVestingPeriod1_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Share-Based Compensation Arrangement by Share-Based Payment Award, Award Vesting Period</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardAwardVestingPeriod1_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Vesting period</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardFairValueAssumptionsExpectedDividendRate" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardFairValueAssumptionsExpectedDividendRate" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardFairValueAssumptionsExpectedDividendRate" xlink:to="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardFairValueAssumptionsExpectedDividendRate_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardFairValueAssumptionsExpectedDividendRate_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Share-Based Compensation Arrangement by Share-Based Payment Award, Fair Value Assumptions, Expected Dividend Rate</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardFairValueAssumptionsExpectedDividendRate_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Expected dividend yield</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingRollForward" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingRollForward" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingRollForward" xlink:to="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingRollForward_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingRollForward_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Share-Based Compensation Arrangement by Share-Based Payment Award, Options, Outstanding [Roll Forward]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingRollForward_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of shares</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsGrantsInPeriodGross" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsGrantsInPeriodGross" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsGrantsInPeriodGross" xlink:to="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsGrantsInPeriodGross_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsGrantsInPeriodGross_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Share-Based Compensation Arrangement by Share-Based Payment Award, Options, Grants in Period, Gross</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsGrantsInPeriodGross_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Options issued as part of the Business Combination (shares)</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsForfeituresInPeriod" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsForfeituresInPeriod" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsForfeituresInPeriod_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Share-Based Compensation Arrangement by Share-Based Payment Award, Options, Forfeitures in Period</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2009/role/negatedLabel" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsForfeituresInPeriod_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Forfeited (shares)</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsVestedAndExpectedToVestOutstandingNumber" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsVestedAndExpectedToVestOutstandingNumber" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsVestedAndExpectedToVestOutstandingNumber_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Share-Based Compensation Arrangement by Share-Based Payment Award, Options, Vested and Expected to Vest, Outstanding, Number</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsVestedAndExpectedToVestOutstandingNumber_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Vested and expected to vest (shares)</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsExercisableNumber" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsExercisableNumber" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsExercisableNumber_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Share-Based Compensation Arrangement by Share-Based Payment Award, Options, Exercisable, Number</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsExercisableNumber_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Vested and exercisable (shares)</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingWeightedAverageExercisePriceRollforward" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingWeightedAverageExercisePriceRollforward" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingWeightedAverageExercisePriceRollforward_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Share-Based Compensation Arrangement by Share-Based Payment Award, Options, Outstanding, Weighted Average Exercise Price [Abstract]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingWeightedAverageExercisePriceRollforward_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Weighted Average Exercise Price</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingWeightedAverageExercisePrice" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingWeightedAverageExercisePrice" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingWeightedAverageExercisePrice_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Share-Based Compensation Arrangement by Share-Based Payment Award, Options, Outstanding, Weighted Average Exercise Price</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/periodEndLabel" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingWeightedAverageExercisePrice_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Outstanding (in dollars per share)</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/periodStartLabel" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingWeightedAverageExercisePrice_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Outstanding (in dollars per share)</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ShareBasedCompensationArrangementsByShareBasedPaymentAwardOptionsGrantsInPeriodWeightedAverageExercisePrice" xlink:label="us-gaap_ShareBasedCompensationArrangementsByShareBasedPaymentAwardOptionsGrantsInPeriodWeightedAverageExercisePrice" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ShareBasedCompensationArrangementsByShareBasedPaymentAwardOptionsGrantsInPeriodWeightedAverageExercisePrice_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Share-Based Compensation Arrangements by Share-Based Payment Award, Options, Grants in Period, Weighted Average Exercise Price</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ShareBasedCompensationArrangementsByShareBasedPaymentAwardOptionsGrantsInPeriodWeightedAverageExercisePrice_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Options issued as part of the Business Combination (in dollars per share)</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ShareBasedCompensationArrangementsByShareBasedPaymentAwardOptionsForfeituresInPeriodWeightedAverageExercisePrice" xlink:label="us-gaap_ShareBasedCompensationArrangementsByShareBasedPaymentAwardOptionsForfeituresInPeriodWeightedAverageExercisePrice" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ShareBasedCompensationArrangementsByShareBasedPaymentAwardOptionsForfeituresInPeriodWeightedAverageExercisePrice_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Share-Based Compensation Arrangements by Share-Based Payment Award, Options, Forfeitures in Period, Weighted Average Exercise Price</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ShareBasedCompensationArrangementsByShareBasedPaymentAwardOptionsForfeituresInPeriodWeightedAverageExercisePrice_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Forfeited (in dollars per share)</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsVestedAndExpectedToVestOutstandingWeightedAverageExercisePrice" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsVestedAndExpectedToVestOutstandingWeightedAverageExercisePrice" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsVestedAndExpectedToVestOutstandingWeightedAverageExercisePrice" xlink:to="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsVestedAndExpectedToVestOutstandingWeightedAverageExercisePrice_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsVestedAndExpectedToVestOutstandingWeightedAverageExercisePrice_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Share-Based Compensation Arrangement by Share-Based Payment Award, Options, Vested and Expected to Vest, Outstanding, Weighted Average Exercise Price</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsVestedAndExpectedToVestOutstandingWeightedAverageExercisePrice_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Vested and expected to vest (in dollars per share)</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsExercisableWeightedAverageExercisePrice" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsExercisableWeightedAverageExercisePrice" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsExercisableWeightedAverageExercisePrice_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Share-Based Compensation Arrangement by Share-Based Payment Award, Options, Exercisable, Weighted Average Exercise Price</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsExercisableWeightedAverageExercisePrice_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Vested and exercisable (in dollars per share)</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_WeightedAverageRemainingContractualLifeAbstract" xlink:label="adgm_WeightedAverageRemainingContractualLifeAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_WeightedAverageRemainingContractualLifeAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Weighted Average Remaining Contractual Life (in years)</label>
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardOptionsOutstandingWeightedAverageRemainingContractualTerm2_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Weighted Average Remaining Contractual Life (in years)</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardOptionsVestedAndExpectedToVestOutstandingWeightedAverageRemainingContractualTerm1" xlink:label="us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardOptionsVestedAndExpectedToVestOutstandingWeightedAverageRemainingContractualTerm1" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardOptionsVestedAndExpectedToVestOutstandingWeightedAverageRemainingContractualTerm1_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Vested and expected to vest</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardOptionsExercisableWeightedAverageRemainingContractualTerm1" xlink:label="us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardOptionsExercisableWeightedAverageRemainingContractualTerm1" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardOptionsExercisableWeightedAverageRemainingContractualTerm1_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Share-Based Compensation Arrangement by Share-Based Payment Award, Options, Exercisable, Weighted Average Remaining Contractual Term</label>
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    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_AggregateIntrinsicValueAbstract" xlink:label="adgm_AggregateIntrinsicValueAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingIntrinsicValue" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingIntrinsicValue" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsVestedAndExpectedToVestOutstandingAggregateIntrinsicValue" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsVestedAndExpectedToVestOutstandingAggregateIntrinsicValue" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardOptionsExercisableIntrinsicValue1" xlink:to="us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardOptionsExercisableIntrinsicValue1_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardOptionsExercisableIntrinsicValue1_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Share-Based Compensation Arrangement by Share-Based Payment Award, Options, Exercisable, Intrinsic Value</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardOptionsExercisableIntrinsicValue1_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Vested and exercisable</label>
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    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardOptionsVestedInPeriodFairValue1" xlink:label="us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardOptionsVestedInPeriodFairValue1" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardOptionsVestedInPeriodFairValue1_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Share-Based Compensation Arrangement by Share-Based Payment Award, Options, Vested in Period, Fair Value</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardOptionsVestedInPeriodFairValue1_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fair value of awards vested</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_EmployeeServiceShareBasedCompensationNonvestedAwardsTotalCompensationCostNotYetRecognizedStockOptions" xlink:label="us-gaap_EmployeeServiceShareBasedCompensationNonvestedAwardsTotalCompensationCostNotYetRecognizedStockOptions" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_EmployeeServiceShareBasedCompensationNonvestedAwardsTotalCompensationCostNotYetRecognizedStockOptions" xlink:to="us-gaap_EmployeeServiceShareBasedCompensationNonvestedAwardsTotalCompensationCostNotYetRecognizedStockOptions_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_EmployeeServiceShareBasedCompensationNonvestedAwardsTotalCompensationCostNotYetRecognizedStockOptions_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Share-Based Payment Arrangement, Nonvested Award, Option, Cost Not yet Recognized, Amount</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_EmployeeServiceShareBasedCompensationNonvestedAwardsTotalCompensationCostNotYetRecognizedStockOptions_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Unrecognized compensation cost</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ScheduleOfEmployeeServiceShareBasedCompensationAllocationOfRecognizedPeriodCostsTable" xlink:label="us-gaap_ScheduleOfEmployeeServiceShareBasedCompensationAllocationOfRecognizedPeriodCostsTable" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ScheduleOfEmployeeServiceShareBasedCompensationAllocationOfRecognizedPeriodCostsTable" xlink:to="us-gaap_ScheduleOfEmployeeServiceShareBasedCompensationAllocationOfRecognizedPeriodCostsTable_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ScheduleOfEmployeeServiceShareBasedCompensationAllocationOfRecognizedPeriodCostsTable_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Share-Based Payment Arrangement, Expensed and Capitalized, Amount [Table]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_IncomeStatementLocationAxis" xlink:label="us-gaap_IncomeStatementLocationAxis" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_IncomeStatementLocationAxis" xlink:to="us-gaap_IncomeStatementLocationAxis_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_IncomeStatementLocationAxis_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Statement of Income Location, Balance [Axis]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_IncomeStatementLocationDomain" xlink:label="us-gaap_IncomeStatementLocationDomain" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_IncomeStatementLocationDomain" xlink:to="us-gaap_IncomeStatementLocationDomain_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_IncomeStatementLocationDomain_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Statement of Income Location, Balance [Domain]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_SellingGeneralAndAdministrativeExpensesMember" xlink:label="us-gaap_SellingGeneralAndAdministrativeExpensesMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_SellingGeneralAndAdministrativeExpensesMember" xlink:to="us-gaap_SellingGeneralAndAdministrativeExpensesMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_SellingGeneralAndAdministrativeExpensesMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Selling, General and Administrative Expenses [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_SellingGeneralAndAdministrativeExpensesMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Selling, general and administrative</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ResearchAndDevelopmentExpenseMember" xlink:label="us-gaap_ResearchAndDevelopmentExpenseMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ResearchAndDevelopmentExpenseMember" xlink:to="us-gaap_ResearchAndDevelopmentExpenseMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ResearchAndDevelopmentExpenseMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Research and Development Expense [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ResearchAndDevelopmentExpenseMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Research and development</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_EmployeeServiceShareBasedCompensationAllocationOfRecognizedPeriodCostsLineItems" xlink:label="us-gaap_EmployeeServiceShareBasedCompensationAllocationOfRecognizedPeriodCostsLineItems" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_EmployeeServiceShareBasedCompensationAllocationOfRecognizedPeriodCostsLineItems" xlink:to="us-gaap_EmployeeServiceShareBasedCompensationAllocationOfRecognizedPeriodCostsLineItems_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_EmployeeServiceShareBasedCompensationAllocationOfRecognizedPeriodCostsLineItems_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Share-Based Payment Arrangement, Expensed and Capitalized, Amount [Line Items]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_EmployeeServiceShareBasedCompensationAllocationOfRecognizedPeriodCostsLineItems_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Stock-Based Compensation</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_AllocatedShareBasedCompensationExpense" xlink:label="us-gaap_AllocatedShareBasedCompensationExpense" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_AllocatedShareBasedCompensationExpense" xlink:to="us-gaap_AllocatedShareBasedCompensationExpense_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_AllocatedShareBasedCompensationExpense_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Share-Based Payment Arrangement, Expense</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_AllocatedShareBasedCompensationExpense_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Total stock-based compensation</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_ShareBasedCompensationArrangementByShareBasedPaymentAwardAnnualPercentageIncreaseToNumberOfSharesAuthorized" xlink:label="adgm_ShareBasedCompensationArrangementByShareBasedPaymentAwardAnnualPercentageIncreaseToNumberOfSharesAuthorized" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_ShareBasedCompensationArrangementByShareBasedPaymentAwardAnnualPercentageIncreaseToNumberOfSharesAuthorized" xlink:to="adgm_ShareBasedCompensationArrangementByShareBasedPaymentAwardAnnualPercentageIncreaseToNumberOfSharesAuthorized_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_ShareBasedCompensationArrangementByShareBasedPaymentAwardAnnualPercentageIncreaseToNumberOfSharesAuthorized_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Annual percentage increase to the number of shares authorized for issuance under share-based payment arrangement.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_ShareBasedCompensationArrangementByShareBasedPaymentAwardAnnualPercentageIncreaseToNumberOfSharesAuthorized_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Share-Based Compensation Arrangement by Share-Based Payment Award, Annual Percentage Increase To The Number of Shares Authorized</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_ShareBasedCompensationArrangementByShareBasedPaymentAwardAnnualPercentageIncreaseToNumberOfSharesAuthorized_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Percentage increase to the number of shares authorized</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardNonOptionEquityInstrumentsGranted" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardNonOptionEquityInstrumentsGranted" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardNonOptionEquityInstrumentsGranted" xlink:to="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardNonOptionEquityInstrumentsGranted_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardNonOptionEquityInstrumentsGranted_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Share-Based Compensation Arrangement by Share-Based Payment Award, Non-Option Equity Instruments, Granted</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardNonOptionEquityInstrumentsGranted_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Awards granted</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_ShareBasedCompensationAutomaticIncreasePeriod" xlink:label="adgm_ShareBasedCompensationAutomaticIncreasePeriod" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_ShareBasedCompensationAutomaticIncreasePeriod_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The period from the beginning of share based compensation plan to the end of automatic increase in authorized awards that can be granted under share-based payment arrangement.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_ShareBasedCompensationAutomaticIncreasePeriod_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Share Based Compensation, Automatic Increase Period</label>
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    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_ShareBasedCompensationArrangementByShareBasedPaymentAwardGrantsToDateNetOfForfeitures" xlink:label="adgm_ShareBasedCompensationArrangementByShareBasedPaymentAwardGrantsToDateNetOfForfeitures" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_ShareBasedCompensationArrangementByShareBasedPaymentAwardGrantsToDateNetOfForfeitures_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of share share based awards granted as of reporting date.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_ShareBasedCompensationArrangementByShareBasedPaymentAwardGrantsToDateNetOfForfeitures_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Share-Based Compensation Arrangement by Share-Based Payment Award, Grants To Date, Net of Forfeitures</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_ShareBasedCompensationArrangementByShareBasedPaymentAwardGrantsToDateNetOfForfeitures_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number purchase rights granted as of reporting December 31, 2025</label>
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    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_EarnOutSharesMember" xlink:label="adgm_EarnOutSharesMember" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_EarnOutSharesMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Sponsor Earnout</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_NetIncomeLossAvailableToCommonStockholdersBasic" xlink:label="us-gaap_NetIncomeLossAvailableToCommonStockholdersBasic" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_IncomeTaxRateReconciliationFairValueAdjustmentsOfWarrantLiabilityAmount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amount of the difference between reported income tax expense (benefit) and expected income tax expense (benefit) computed by applying the domestic federal statutory income tax rates to pretax income (loss) from continuing operations attributable to fair value adjustments of warrant liability.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_IncomeTaxRateReconciliationFairValueAdjustmentsOfWarrantLiabilityAmount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Income Tax Rate Reconciliation, Fair Value Adjustments of Warrant Liability, Amount</label>
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_IncomeTaxReconciliationChangeInDeferredTaxAssetsValuationAllowance_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Change in federal valuation allowance</label>
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_EffectiveIncomeTaxRateReconciliationNonTaxableIncomeNondeductibleExpenseAmount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Effective Income Tax Rate Reconciliation, Non Taxable Income, Nondeductible Expense, Amount</label>
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    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_EffectiveIncomeTaxRateReconciliationCrossBorderTaxEffectPercent" xlink:label="us-gaap_EffectiveIncomeTaxRateReconciliationCrossBorderTaxEffectPercent" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_EffectiveIncomeTaxRateReconciliationCrossBorderTaxEffectPercent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Foreign tax effects (Germany)</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_EffectiveIncomeTaxRateReconciliationNondeductibleExpense" xlink:label="us-gaap_EffectiveIncomeTaxRateReconciliationNondeductibleExpense" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_EffectiveIncomeTaxRateReconciliationNondeductibleExpenseImpairmentLosses" xlink:label="us-gaap_EffectiveIncomeTaxRateReconciliationNondeductibleExpenseImpairmentLosses" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_EffectiveIncomeTaxRateReconciliationNondeductibleExpenseImpairmentLosses_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Goodwill impairment</label>
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    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_IncomeTaxRateReconciliationFairValueAdjustmentsOfConvertibleNotesPercent" xlink:to="adgm_IncomeTaxRateReconciliationFairValueAdjustmentsOfConvertibleNotesPercent_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_IncomeTaxRateReconciliationFairValueAdjustmentsOfConvertibleNotesPercent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Percentage of the difference between reported income tax expense (benefit) and expected income tax expense (benefit) computed by applying the domestic federal statutory income tax rates to pretax income (loss) from continuing operations attributable to fair value adjustments of convertible notes.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_IncomeTaxRateReconciliationFairValueAdjustmentsOfConvertibleNotesPercent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Income Tax Rate Reconciliation, Fair Value Adjustments of Convertible Notes, Percent</label>
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_IncomeTaxRateReconciliationFairValueAdjustmentsOfWarrantLiabilityPercent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Income Tax Rate Reconciliation, Fair Value Adjustments of Warrant Liability, Percent</label>
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    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_DeferredTaxAssetsStateTaxes" xlink:label="us-gaap_DeferredTaxAssetsStateTaxes" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_DeferredTaxAssetsStateTaxes_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">State taxes</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_DeferredTaxAssetsOther" xlink:label="us-gaap_DeferredTaxAssetsOther" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_DeferredTaxAssetsOther_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Other</label>
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    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_DeferredTaxAssetsValuationAllowance" xlink:label="us-gaap_DeferredTaxAssetsValuationAllowance" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2009/role/negatedLabel" xlink:label="us-gaap_DeferredTaxAssetsValuationAllowance_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Less: Valuation allowance</label>
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    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_DeferredTaxAssetsNet" xlink:label="us-gaap_DeferredTaxAssetsNet" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_DeferredTaxAssetsNet" xlink:to="us-gaap_DeferredTaxAssetsNet_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:label="us-gaap_DeferredTaxAssetsNet_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Total deferred tax assets, net of valuation allowance</label>
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2009/role/negatedLabel" xlink:label="us-gaap_DeferredTaxLiabilitiesLeasingArrangements_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Right-of-use assets</label>
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2009/role/negatedLabel" xlink:label="us-gaap_DeferredTaxLiabilitiesGoodwillAndIntangibleAssetsIntangibleAssets_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Intangible assets</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_DeferredTaxLiabilitiesUnrecognizedTaxBenefit" xlink:label="adgm_DeferredTaxLiabilitiesUnrecognizedTaxBenefit" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_DeferredTaxLiabilitiesUnrecognizedTaxBenefit_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Deferred Tax Liabilities, Unrecognized Tax Benefit</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2009/role/negatedLabel" xlink:label="adgm_DeferredTaxLiabilitiesUnrecognizedTaxBenefit_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Unrecognized tax benefit</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_DeferredTaxLiabilitiesOther" xlink:label="us-gaap_DeferredTaxLiabilitiesOther" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_DeferredTaxLiabilitiesOther" xlink:to="us-gaap_DeferredTaxLiabilitiesOther_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_DeferredTaxLiabilitiesOther_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Deferred Tax Liabilities, Other</label>
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    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_DeferredIncomeTaxLiabilities" xlink:label="us-gaap_DeferredIncomeTaxLiabilities" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_DeferredIncomeTaxLiabilities" xlink:to="us-gaap_DeferredIncomeTaxLiabilities_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_DeferredIncomeTaxLiabilities_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Deferred Tax Liabilities, Gross</label>
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    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_AdagioMedicalGmbhMember" xlink:label="adgm_AdagioMedicalGmbhMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_AdagioMedicalGmbhMember" xlink:to="adgm_AdagioMedicalGmbhMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_AdagioMedicalGmbhMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Adagio Medical GmbH [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_AdagioMedicalGmbhMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Adagio Medical GmbH</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_IncomeTaxAuthorityAxis" xlink:label="us-gaap_IncomeTaxAuthorityAxis" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_IncomeTaxAuthorityAxis" xlink:to="us-gaap_IncomeTaxAuthorityAxis_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_IncomeTaxAuthorityAxis_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Income Tax Jurisdiction [Axis]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_IncomeTaxAuthorityDomain" xlink:label="us-gaap_IncomeTaxAuthorityDomain" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_IncomeTaxAuthorityDomain" xlink:to="us-gaap_IncomeTaxAuthorityDomain_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_IncomeTaxAuthorityDomain_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Income Tax Jurisdiction [Domain]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_DomesticCountryMember" xlink:label="us-gaap_DomesticCountryMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_DomesticCountryMember" xlink:to="us-gaap_DomesticCountryMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_DomesticCountryMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Domestic Tax Jurisdiction [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_DomesticCountryMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Federal</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_StateAndLocalJurisdictionMember" xlink:label="us-gaap_StateAndLocalJurisdictionMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_StateAndLocalJurisdictionMember" xlink:to="us-gaap_StateAndLocalJurisdictionMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_StateAndLocalJurisdictionMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">State and Local Jurisdiction [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_StateAndLocalJurisdictionMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">State</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_OperatingLossCarryforwardsLineItems" xlink:label="us-gaap_OperatingLossCarryforwardsLineItems" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_OperatingLossCarryforwardsLineItems" xlink:to="us-gaap_OperatingLossCarryforwardsLineItems_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_OperatingLossCarryforwardsLineItems_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Operating Loss Carryforwards [Line Items]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_OperatingLossCarryforwardsLineItems_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Income Taxes</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_OperatingLossCarryforwards" xlink:label="us-gaap_OperatingLossCarryforwards" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_OperatingLossCarryforwards" xlink:to="us-gaap_OperatingLossCarryforwards_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_OperatingLossCarryforwards_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Operating Loss Carryforwards</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_OperatingLossCarryforwards_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Net operating loss carryforwards</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_OperatingLossCarryforwardsSubjectToExpiration" xlink:label="adgm_OperatingLossCarryforwardsSubjectToExpiration" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_OperatingLossCarryforwardsSubjectToExpiration" xlink:to="adgm_OperatingLossCarryforwardsSubjectToExpiration_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_OperatingLossCarryforwardsSubjectToExpiration_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amount of operating loss carryforwards that are subject to expiration dates.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_OperatingLossCarryforwardsSubjectToExpiration_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Operating Loss Carryforwards Subject To Expiration</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_OperatingLossCarryforwardsSubjectToExpiration_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Net operating loss subject to expiration</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_OperatingLossCarryforwardsNotSubjectToExpiration" xlink:label="adgm_OperatingLossCarryforwardsNotSubjectToExpiration" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_OperatingLossCarryforwardsNotSubjectToExpiration" xlink:to="adgm_OperatingLossCarryforwardsNotSubjectToExpiration_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_OperatingLossCarryforwardsNotSubjectToExpiration_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amount of operating loss carryforwards that are not subject to expiration dates.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_OperatingLossCarryforwardsNotSubjectToExpiration_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Operating Loss Carryforwards, Not Subject to Expiration</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_OperatingLossCarryforwardsNotSubjectToExpiration_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Net operating loss not subject to expiration</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ReconciliationOfUnrecognizedTaxBenefitsExcludingAmountsPertainingToExaminedTaxReturnsRollForward" xlink:label="us-gaap_ReconciliationOfUnrecognizedTaxBenefitsExcludingAmountsPertainingToExaminedTaxReturnsRollForward" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ReconciliationOfUnrecognizedTaxBenefitsExcludingAmountsPertainingToExaminedTaxReturnsRollForward" xlink:to="us-gaap_ReconciliationOfUnrecognizedTaxBenefitsExcludingAmountsPertainingToExaminedTaxReturnsRollForward_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ReconciliationOfUnrecognizedTaxBenefitsExcludingAmountsPertainingToExaminedTaxReturnsRollForward_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Unrecognized Tax Benefits [Roll Forward]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ReconciliationOfUnrecognizedTaxBenefitsExcludingAmountsPertainingToExaminedTaxReturnsRollForward_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Unrecognized tax benefits</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_UnrecognizedTaxBenefits" xlink:label="us-gaap_UnrecognizedTaxBenefits" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_UnrecognizedTaxBenefits" xlink:to="us-gaap_UnrecognizedTaxBenefits_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_UnrecognizedTaxBenefits_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Unrecognized Tax Benefits</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/periodEndLabel" xlink:label="us-gaap_UnrecognizedTaxBenefits_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Balance at end of year</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/periodStartLabel" xlink:label="us-gaap_UnrecognizedTaxBenefits_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Balance at beginning of year</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_UnrecognizedTaxBenefits_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Unrecognized tax benefits</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_UnrecognizedTaxBenefitsThatWouldImpactEffectiveTaxRate" xlink:label="us-gaap_UnrecognizedTaxBenefitsThatWouldImpactEffectiveTaxRate" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_UnrecognizedTaxBenefitsThatWouldImpactEffectiveTaxRate" xlink:to="us-gaap_UnrecognizedTaxBenefitsThatWouldImpactEffectiveTaxRate_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_UnrecognizedTaxBenefitsThatWouldImpactEffectiveTaxRate_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Unrecognized Tax Benefits that Would Impact Effective Tax Rate</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_UnrecognizedTaxBenefitsThatWouldImpactEffectiveTaxRate_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Unrecognized tax benefits that will affect the effective tax rate</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_CapitalizationOfResearchAndExperimentalCosts" xlink:label="adgm_CapitalizationOfResearchAndExperimentalCosts" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_CapitalizationOfResearchAndExperimentalCosts" xlink:to="adgm_CapitalizationOfResearchAndExperimentalCosts_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_CapitalizationOfResearchAndExperimentalCosts_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amount of research and experimental costs required to be capitalized as per Tax Cuts and Jobs Act.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_CapitalizationOfResearchAndExperimentalCosts_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Capitalization of Research and Experimental Costs</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_CapitalizationOfResearchAndExperimentalCosts_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Capitalization of R &amp; E</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ScheduleOfRelatedPartyTransactionsByRelatedPartyTable" xlink:label="us-gaap_ScheduleOfRelatedPartyTransactionsByRelatedPartyTable" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ScheduleOfRelatedPartyTransactionsByRelatedPartyTable" xlink:to="us-gaap_ScheduleOfRelatedPartyTransactionsByRelatedPartyTable_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ScheduleOfRelatedPartyTransactionsByRelatedPartyTable_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Related Party Transaction [Table]</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_FjordVenturesMember" xlink:label="adgm_FjordVenturesMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_FjordVenturesMember" xlink:to="adgm_FjordVenturesMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_FjordVenturesMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents information pertaining to Fjord Ventures.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_FjordVenturesMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fjord Ventures [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_FjordVenturesMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fjord</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_SharedServicesAgreementMember" xlink:label="adgm_SharedServicesAgreementMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_SharedServicesAgreementMember" xlink:to="adgm_SharedServicesAgreementMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_SharedServicesAgreementMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents information pertaining to shared services agreement.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_SharedServicesAgreementMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Shared Services Agreement [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_SharedServicesAgreementMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Shared Services Agreement</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_RelatedPartyTransactionLineItems" xlink:label="us-gaap_RelatedPartyTransactionLineItems" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_RelatedPartyTransactionLineItems" xlink:to="us-gaap_RelatedPartyTransactionLineItems_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_RelatedPartyTransactionLineItems_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Related Party Transaction [Line Items]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_RelatedPartyTransactionLineItems_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Related Party Transactions</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_RelatedPartyTransactionAmountsOfTransaction" xlink:label="us-gaap_RelatedPartyTransactionAmountsOfTransaction" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_RelatedPartyTransactionAmountsOfTransaction" xlink:to="us-gaap_RelatedPartyTransactionAmountsOfTransaction_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_RelatedPartyTransactionAmountsOfTransaction_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Related Party Transaction, Amounts of Transaction</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_RelatedPartyTransactionAmountsOfTransaction_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amount of transaction</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_SubLeaseOfOfficeAndManufacturingSpaceMember" xlink:label="adgm_SubLeaseOfOfficeAndManufacturingSpaceMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_SubLeaseOfOfficeAndManufacturingSpaceMember" xlink:to="adgm_SubLeaseOfOfficeAndManufacturingSpaceMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_SubLeaseOfOfficeAndManufacturingSpaceMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents information pertaining to sub-lease of office and manufacturing space.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_SubLeaseOfOfficeAndManufacturingSpaceMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Sub-lease of Office and Manufacturing Space [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_SubLeaseOfOfficeAndManufacturingSpaceMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Sub-lease of office and manufacturing space</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_SubleaseArea" xlink:label="adgm_SubleaseArea" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_SubleaseArea" xlink:to="adgm_SubleaseArea_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_SubleaseArea_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Area represented by square feet of office and manufacturing space subleased by the company.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_SubleaseArea_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Sublease Area</label>
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    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_FjordinvestLlcMember" xlink:label="adgm_FjordinvestLlcMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_FjordinvestLlcMember" xlink:to="adgm_FjordinvestLlcMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_FjordinvestLlcMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents information pertaining to Fjordinvest, LLC.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_FjordinvestLlcMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fjordinvest, LLC [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_FjordinvestLlcMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fjordinvest</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_PIPEPreFundedWarrants2025AndMilestoneWarrantsMember" xlink:label="adgm_PIPEPreFundedWarrants2025AndMilestoneWarrantsMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_PIPEPreFundedWarrants2025AndMilestoneWarrantsMember" xlink:to="adgm_PIPEPreFundedWarrants2025AndMilestoneWarrantsMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_PIPEPreFundedWarrants2025AndMilestoneWarrantsMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents Pre-Funded warrants issued in 2025 and milestone warrants.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_PIPEPreFundedWarrants2025AndMilestoneWarrantsMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">P I P E Pre-Funded Warrants, 2025 and Milestone Warrants [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_PIPEPreFundedWarrants2025AndMilestoneWarrantsMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">2025 PIPE Pre-Funded Warrants and Milestone Warrants</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_StockIssuedDuringPeriodSharesIssuedForAdditionalCashInvestment" xlink:label="adgm_StockIssuedDuringPeriodSharesIssuedForAdditionalCashInvestment" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_StockIssuedDuringPeriodSharesIssuedForAdditionalCashInvestment" xlink:to="adgm_StockIssuedDuringPeriodSharesIssuedForAdditionalCashInvestment_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_StockIssuedDuringPeriodSharesIssuedForAdditionalCashInvestment_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of shares issued during the period for additional cash investment.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_StockIssuedDuringPeriodSharesIssuedForAdditionalCashInvestment_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Stock issued During Period, Shares Issued For Additional Cash Investment</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_StockIssuedDuringPeriodSharesIssuedForAdditionalCashInvestment_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Issuance of common stock</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_ClassOfWarrantOrRightNumberOfWarrantsIssuedForAdditionalCashInvestment" xlink:label="adgm_ClassOfWarrantOrRightNumberOfWarrantsIssuedForAdditionalCashInvestment" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_ClassOfWarrantOrRightNumberOfWarrantsIssuedForAdditionalCashInvestment" xlink:to="adgm_ClassOfWarrantOrRightNumberOfWarrantsIssuedForAdditionalCashInvestment_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_ClassOfWarrantOrRightNumberOfWarrantsIssuedForAdditionalCashInvestment_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of warrants issued to purchase stock  for additional cash investment.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_ClassOfWarrantOrRightNumberOfWarrantsIssuedForAdditionalCashInvestment_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class Of Warrant Or Right Number Of Warrants Issued For Additional Cash Investment</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="adgm_ClassOfWarrantOrRightNumberOfWarrantsIssuedForAdditionalCashInvestment_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of warrants issued</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ProceedsFromIssuanceOfWarrants" xlink:label="us-gaap_ProceedsFromIssuanceOfWarrants" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ProceedsFromIssuanceOfWarrants" xlink:to="us-gaap_ProceedsFromIssuanceOfWarrants_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ProceedsFromIssuanceOfWarrants_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Proceeds from Issuance of Warrants</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ProceedsFromIssuanceOfWarrants_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Purchase price of issuance of warrants</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_ScheduleOfSegmentReportingInformationBySegmentTable" xlink:label="us-gaap_ScheduleOfSegmentReportingInformationBySegmentTable" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ScheduleOfSegmentReportingInformationBySegmentTable" xlink:to="us-gaap_ScheduleOfSegmentReportingInformationBySegmentTable_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ScheduleOfSegmentReportingInformationBySegmentTable_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Schedule of Segment Reporting Information, by Segment [Table]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_StatementBusinessSegmentsAxis" xlink:label="us-gaap_StatementBusinessSegmentsAxis" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_StatementBusinessSegmentsAxis" xlink:to="us-gaap_StatementBusinessSegmentsAxis_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_StatementBusinessSegmentsAxis_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Segments [Axis]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_SegmentDomain" xlink:label="us-gaap_SegmentDomain" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_SegmentDomain" xlink:to="us-gaap_SegmentDomain_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_SegmentDomain_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Segments [Domain]</label>
    <loc xlink:type="locator" xlink:href="adgm-20251231.xsd#adgm_SingleReportableSegmentMember" xlink:label="adgm_SingleReportableSegmentMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="adgm_SingleReportableSegmentMember" xlink:to="adgm_SingleReportableSegmentMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="adgm_SingleReportableSegmentMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents single reportable segment, not identified in the disclosure.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="adgm_SingleReportableSegmentMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Single Reportable Segment [Member]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_SegmentReportingInformationLineItems" xlink:label="us-gaap_SegmentReportingInformationLineItems" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_SegmentReportingInformationLineItems" xlink:to="us-gaap_SegmentReportingInformationLineItems_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_SegmentReportingInformationLineItems_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Segment Reporting Information [Line Items]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_SegmentReportingInformationLineItems_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Segment Reporting</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2024/elts/us-gaap-2024.xsd#us-gaap_CostOfGoodsAndServicesSoldDepreciationAndAmortizationAbstract" xlink:label="us-gaap_CostOfGoodsAndServicesSoldDepreciationAndAmortizationAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_CostOfGoodsAndServicesSoldDepreciationAndAmortizationAbstract" xlink:to="us-gaap_CostOfGoodsAndServicesSoldDepreciationAndAmortizationAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_CostOfGoodsAndServicesSoldDepreciationAndAmortizationAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cost, Depreciation and Amortization [Abstract]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_CostOfGoodsAndServicesSoldDepreciationAndAmortizationAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Less:</label>
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