<?xml version="1.0" encoding="utf-8"?>
<!--XBRL document created with Toppan Merrill Bridge  11.4.0.1 -->
<!--Based on XBRL 2.1-->
<!--Created on: 8/10/2026 11:25:33 PM-->
<!--Modified on: 8/10/2026 11:25:33 PM-->
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    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_RedeemableWarrantsMember" xlink:label="biii_RedeemableWarrantsMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_RedeemableWarrantsMember" xlink:to="biii_RedeemableWarrantsMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <loc xlink:type="locator" xlink:href="https://xbrl.sec.gov/dei/2026/dei-2026.xsd#dei_EntityAddressCountry" xlink:label="dei_EntityAddressCountry" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="dei_EntityAddressCountry" xlink:to="dei_EntityAddressCountry_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <loc xlink:type="locator" xlink:href="https://xbrl.sec.gov/dei/2026/dei-2026.xsd#dei_EntityAddressPostalZipCode" xlink:label="dei_EntityAddressPostalZipCode" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="dei_EntityAddressPostalZipCode" xlink:to="dei_EntityAddressPostalZipCode_lbl" 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="dei_EntitySmallBusiness" xlink:to="dei_EntitySmallBusiness_lbl" 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="dei_EntityExTransitionPeriod" xlink:to="dei_EntityExTransitionPeriod_lbl" 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="dei_EntityShellCompany" xlink:to="dei_EntityShellCompany_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <loc xlink:type="locator" xlink:href="https://xbrl.sec.gov/dei/2026/dei-2026.xsd#dei_EntityCommonStockSharesOutstanding" xlink:label="dei_EntityCommonStockSharesOutstanding" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="dei_EntityCommonStockSharesOutstanding" xlink:to="dei_EntityCommonStockSharesOutstanding_lbl" 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="dei_EntityCentralIndexKey" xlink:to="dei_EntityCentralIndexKey_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <loc xlink:type="locator" xlink:href="https://xbrl.sec.gov/dei/2026/dei-2026.xsd#dei_CurrentFiscalYearEndDate" xlink:label="dei_CurrentFiscalYearEndDate" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="dei_CurrentFiscalYearEndDate" xlink:to="dei_CurrentFiscalYearEndDate_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
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    <loc xlink:type="locator" xlink:href="https://xbrl.sec.gov/dei/2026/dei-2026.xsd#dei_DocumentFiscalYearFocus" xlink:label="dei_DocumentFiscalYearFocus" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="dei_DocumentFiscalYearFocus" xlink:to="dei_DocumentFiscalYearFocus_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="dei_DocumentFiscalYearFocus_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Document Fiscal Year Focus</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.sec.gov/dei/2026/dei-2026.xsd#dei_DocumentFiscalPeriodFocus" xlink:label="dei_DocumentFiscalPeriodFocus" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="dei_DocumentFiscalPeriodFocus" xlink:to="dei_DocumentFiscalPeriodFocus_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="dei_DocumentFiscalPeriodFocus_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Document Fiscal Period Focus</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.sec.gov/dei/2026/dei-2026.xsd#dei_AmendmentFlag" xlink:label="dei_AmendmentFlag" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="dei_AmendmentFlag" xlink:to="dei_AmendmentFlag_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="dei_AmendmentFlag_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amendment Flag</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_StatementOfFinancialPositionAbstract" xlink:label="us-gaap_StatementOfFinancialPositionAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_StatementOfFinancialPositionAbstract" xlink:to="us-gaap_StatementOfFinancialPositionAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_StatementOfFinancialPositionAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">CONDENSED BALANCE SHEETS</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_StatementTable" xlink:label="us-gaap_StatementTable" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_StatementTable" xlink:to="us-gaap_StatementTable_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_StatementTable_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Statement [Table]</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_CommonClassaSubjectToPossibleRedemptionMember" xlink:label="biii_CommonClassaSubjectToPossibleRedemptionMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_CommonClassaSubjectToPossibleRedemptionMember" xlink:to="biii_CommonClassaSubjectToPossibleRedemptionMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_CommonClassaSubjectToPossibleRedemptionMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents information pertaining to Common Class A shares subject to possible redemption.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_CommonClassaSubjectToPossibleRedemptionMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Common Class A Subject To Possible Redemption [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_CommonClassaSubjectToPossibleRedemptionMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class A ordinary shares subject to possible redemption</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_CommonClassaNotSubjectToPossibleRedemptionMember" xlink:label="biii_CommonClassaNotSubjectToPossibleRedemptionMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_CommonClassaNotSubjectToPossibleRedemptionMember" xlink:to="biii_CommonClassaNotSubjectToPossibleRedemptionMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_CommonClassaNotSubjectToPossibleRedemptionMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents information pertaining to Common Class A shares not subject to possible redemption.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_CommonClassaNotSubjectToPossibleRedemptionMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Common ClassA Not Subject To Possible Redemption [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_CommonClassaNotSubjectToPossibleRedemptionMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class A ordinary shares not subject to possible redemption</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_StatementLineItems" xlink:label="us-gaap_StatementLineItems" xmlns="http://www.xbrl.org/2003/linkbase" />
    <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" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_StatementLineItems_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Statement [Line Items]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_StatementLineItems_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Statement</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_AssetsAbstract" xlink:label="us-gaap_AssetsAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <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>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_AssetsAbstract_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/2026/elts/us-gaap-2026.xsd#us-gaap_AssetsCurrentAbstract" xlink:label="us-gaap_AssetsCurrentAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_AssetsCurrentAbstract" xlink:to="us-gaap_AssetsCurrentAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_AssetsCurrentAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Assets, Current [Abstract]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_AssetsCurrentAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Current assets</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_Cash" xlink:label="us-gaap_Cash" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_Cash" xlink:to="us-gaap_Cash_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_Cash_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cash</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_Cash_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cash</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_PrepaidExpenseExcludingInsuranceCurrent" xlink:label="biii_PrepaidExpenseExcludingInsuranceCurrent" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_PrepaidExpenseExcludingInsuranceCurrent" xlink:to="biii_PrepaidExpenseExcludingInsuranceCurrent_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_PrepaidExpenseExcludingInsuranceCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amount of asset related to consideration paid in advance for costs that provide economic benefits within a future period of one year or the normal operating cycle, if longer excluding prepaid insurance.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_PrepaidExpenseExcludingInsuranceCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Prepaid Expense Excluding Insurance, Current</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_PrepaidExpenseExcludingInsuranceCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Prepaid expenses</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_PrepaidInsurance" xlink:label="us-gaap_PrepaidInsurance" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_PrepaidInsurance" xlink:to="us-gaap_PrepaidInsurance_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_PrepaidInsurance_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Prepaid Insurance</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_PrepaidInsurance_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Prepaid insurance</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_AssetsCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Assets, Current</label>
    <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/2026/elts/us-gaap-2026.xsd#us-gaap_DeferredCosts" xlink:label="us-gaap_DeferredCosts" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_DeferredCosts" xlink:to="us-gaap_DeferredCosts_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_DeferredCosts_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Deferred Costs, Noncurrent</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_DeferredCosts_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Deferred offering costs</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_AssetsHeldInTrustNoncurrent" xlink:label="us-gaap_AssetsHeldInTrustNoncurrent" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_AssetsHeldInTrustNoncurrent" xlink:to="us-gaap_AssetsHeldInTrustNoncurrent_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_AssetsHeldInTrustNoncurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Asset, Held-in-Trust, Noncurrent</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_AssetsHeldInTrustNoncurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cash held in Trust Account</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_LiabilitiesAndStockholdersEquityAbstract" xlink:label="us-gaap_LiabilitiesAndStockholdersEquityAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <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, Class A Ordinary Shares Subject to Possible Redemption, and Shareholders' Deficit</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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="biii-20260630.xsd#biii_AccruedOfferingCostsCurrent" xlink:label="biii_AccruedOfferingCostsCurrent" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_AccruedOfferingCostsCurrent" xlink:to="biii_AccruedOfferingCostsCurrent_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_AccruedOfferingCostsCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Carrying value as of the balance sheet date of obligations incurred through that date and payable for offering costs. 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="biii_AccruedOfferingCostsCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Accrued Offering Costs, Current</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_AccruedOfferingCostsCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Accrued offering costs</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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 expenses</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_OtherLiabilitiesCurrent" xlink:label="us-gaap_OtherLiabilitiesCurrent" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_OtherLiabilitiesCurrent" xlink:to="us-gaap_OtherLiabilitiesCurrent_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_OtherLiabilitiesCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Advances from related party</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_OtherLiabilitiesCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Advances from related party</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_OtherLiabilityCurrentRelatedPartyTypeExtensibleEnumeration" xlink:label="us-gaap_OtherLiabilityCurrentRelatedPartyTypeExtensibleEnumeration" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_OtherLiabilityCurrentRelatedPartyTypeExtensibleEnumeration" xlink:to="us-gaap_OtherLiabilityCurrentRelatedPartyTypeExtensibleEnumeration_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_OtherLiabilityCurrentRelatedPartyTypeExtensibleEnumeration_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Other Liability, Current, Related Party [Extensible Enumeration]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_NotesPayableCurrent" xlink:label="us-gaap_NotesPayableCurrent" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_NotesPayableCurrent" xlink:to="us-gaap_NotesPayableCurrent_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_NotesPayableCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Notes Payable, Current</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_NotesPayableCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Promissory note - related party</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_NotesPayableCurrentRelatedPartyTypeExtensibleEnumeration" xlink:label="us-gaap_NotesPayableCurrentRelatedPartyTypeExtensibleEnumeration" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_NotesPayableCurrentRelatedPartyTypeExtensibleEnumeration" xlink:to="us-gaap_NotesPayableCurrentRelatedPartyTypeExtensibleEnumeration_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_NotesPayableCurrentRelatedPartyTypeExtensibleEnumeration_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Notes Payable, Current, Related Party [Extensible Enumeration]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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="biii-20260630.xsd#biii_DeferredUnderwritingFeeNonCurrent" xlink:label="biii_DeferredUnderwritingFeeNonCurrent" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_DeferredUnderwritingFeeNonCurrent" xlink:to="biii_DeferredUnderwritingFeeNonCurrent_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_DeferredUnderwritingFeeNonCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amount of underwriting fees payable classified as non-current.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_DeferredUnderwritingFeeNonCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Deferred Underwriting Fee, Non Current</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_DeferredUnderwritingFeeNonCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Deferred underwriting fee</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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/2026/elts/us-gaap-2026.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 (Note 6)</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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/terseLabel" xlink:label="us-gaap_TemporaryEquityCarryingAmountAttributableToParent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class A ordinary shares subject to possible redemption, $0.0001 par value; 17,250,000 and 0 shares at redemption value of $10.17 and $0 per share as of June 30, 2026 and December 31, 2025, respectively</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">Class A ordinary shares subject to possible redemption</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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">Shareholders' Deficit</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_PreferredStockValue" xlink:label="us-gaap_PreferredStockValue" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_PreferredStockValue" xlink:to="us-gaap_PreferredStockValue_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_PreferredStockValue_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Preferred Stock, Value, Issued</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_PreferredStockValue_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Preference shares, $0.0001 par value; 1,500,000 shares authorized; none issued or outstanding</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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/2026/elts/us-gaap-2026.xsd#us-gaap_RetainedEarningsAccumulatedDeficit" xlink:label="us-gaap_RetainedEarningsAccumulatedDeficit" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_RetainedEarningsAccumulatedDeficit" xlink:to="us-gaap_RetainedEarningsAccumulatedDeficit_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_RetainedEarningsAccumulatedDeficit_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Retained Earnings (Accumulated Deficit)</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_RetainedEarningsAccumulatedDeficit_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/2026/elts/us-gaap-2026.xsd#us-gaap_StockholdersEquity" xlink:label="us-gaap_StockholdersEquity" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_StockholdersEquity" xlink:to="us-gaap_StockholdersEquity_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <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>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/periodEndLabel" xlink:label="us-gaap_StockholdersEquity_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Balance at the end</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/periodStartLabel" xlink:label="us-gaap_StockholdersEquity_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Balance at the beginning</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:label="us-gaap_StockholdersEquity_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Total Shareholders' Deficit</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_LiabilitiesAndStockholdersEquity" xlink:label="us-gaap_LiabilitiesAndStockholdersEquity" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_LiabilitiesAndStockholdersEquity" xlink:to="us-gaap_LiabilitiesAndStockholdersEquity_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <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>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:label="us-gaap_LiabilitiesAndStockholdersEquity_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Total Liabilities, Class A Ordinary Shares Subject to Possible Redemption, and Shareholders' Deficit</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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/2026/elts/us-gaap-2026.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="biii-20260630.xsd#biii_FounderShareMember" xlink:label="biii_FounderShareMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_FounderShareMember" xlink:to="biii_FounderShareMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_FounderShareMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents information pertaining to founder share.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_FounderShareMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Founder Share [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_FounderShareMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Founder Shares</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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/2026/elts/us-gaap-2026.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/2026/elts/us-gaap-2026.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="biii-20260630.xsd#biii_SponsorMember" xlink:label="biii_SponsorMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_SponsorMember" xlink:to="biii_SponsorMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_SponsorMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents information pertaining to sponsor, a related party.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_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="biii_SponsorMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Sponsor</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_TemporaryEquityParOrStatedValuePerShare" xlink:label="us-gaap_TemporaryEquityParOrStatedValuePerShare" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_TemporaryEquityParOrStatedValuePerShare" xlink:to="us-gaap_TemporaryEquityParOrStatedValuePerShare_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_TemporaryEquityParOrStatedValuePerShare_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Temporary Equity, Par or Stated Value Per Share</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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">Preference shares, par value (in dollars, per share)</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_PreferredStockParOrStatedValuePerShare_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Preferred stock, par value per share</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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">Preference shares, shares authorized</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" 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="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_PreferredStockSharesIssued" xlink:label="us-gaap_PreferredStockSharesIssued" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_PreferredStockSharesIssued" xlink:to="us-gaap_PreferredStockSharesIssued_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_PreferredStockSharesIssued_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Preferred Stock, Shares Issued</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_PreferredStockSharesIssued_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Preference shares, shares issued</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_PreferredStockSharesIssued_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Preferred stock, shares issued</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_PreferredStockSharesOutstanding" xlink:label="us-gaap_PreferredStockSharesOutstanding" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_PreferredStockSharesOutstanding" xlink:to="us-gaap_PreferredStockSharesOutstanding_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_PreferredStockSharesOutstanding_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Preferred Stock, Shares Outstanding</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_PreferredStockSharesOutstanding_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Preference shares, shares outstanding</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_PreferredStockSharesOutstanding_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Preferred stock, shares outstanding</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_CommonStockParOrStatedValuePerShare" xlink:label="us-gaap_CommonStockParOrStatedValuePerShare" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_CommonStockParOrStatedValuePerShare" xlink:to="us-gaap_CommonStockParOrStatedValuePerShare_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_CommonStockParOrStatedValuePerShare_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Common 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_CommonStockParOrStatedValuePerShare_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Ordinary shares, par value (in dollars per share)</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_CommonStockSharesAuthorized" xlink:label="us-gaap_CommonStockSharesAuthorized" xmlns="http://www.xbrl.org/2003/linkbase" />
    <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">Ordinary shares, shares authorized</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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 at the end (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 at the beginning (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">Ordinary shares, shares issued</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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/terseLabel" xlink:label="us-gaap_CommonStockSharesOutstanding_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Ordinary shares, shares outstanding</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_NumberOfSharesSubjectToForfeiture" xlink:label="biii_NumberOfSharesSubjectToForfeiture" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_NumberOfSharesSubjectToForfeiture" xlink:to="biii_NumberOfSharesSubjectToForfeiture_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_NumberOfSharesSubjectToForfeiture_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of shares that may be surrendered to the extent to which the underwriters' over-allotment option is exercised.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_NumberOfSharesSubjectToForfeiture_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number Of Shares Subject To Forfeiture</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_NumberOfSharesSubjectToForfeiture_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of shares subject to forfeiture</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_NumberOfSharesNoLongerSubjectToForfeiture" xlink:label="biii_NumberOfSharesNoLongerSubjectToForfeiture" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_NumberOfSharesNoLongerSubjectToForfeiture" xlink:to="biii_NumberOfSharesNoLongerSubjectToForfeiture_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_NumberOfSharesNoLongerSubjectToForfeiture_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of shares that were no longer subject to forfeiture as a result of exercise of over-allotment option by the underwriters.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_NumberOfSharesNoLongerSubjectToForfeiture_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of Shares No Longer Subject To Forfeiture</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_NumberOfSharesNoLongerSubjectToForfeiture_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of shares no longer subject to forfeiture</label>
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    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_IncomeStatementAbstract" xlink:to="us-gaap_IncomeStatementAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_IncomeStatementAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">CONDENSED STATEMENTS OF OPERATIONS</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_GeneralAndAdministrativeExpense" xlink:label="us-gaap_GeneralAndAdministrativeExpense" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_GeneralAndAdministrativeExpense" xlink:to="us-gaap_GeneralAndAdministrativeExpense_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_GeneralAndAdministrativeExpense_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">General and Administrative Expense</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_GeneralAndAdministrativeExpense_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">General and administrative costs</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_OperatingIncomeLoss" xlink:label="us-gaap_OperatingIncomeLoss" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_OperatingIncomeLoss" xlink:to="us-gaap_OperatingIncomeLoss_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_OperatingIncomeLoss_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Operating Income (Loss)</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:label="us-gaap_OperatingIncomeLoss_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Loss from operations</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_OtherNonoperatingIncomeExpenseAbstract" xlink:label="us-gaap_OtherNonoperatingIncomeExpenseAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_OtherNonoperatingIncomeExpenseAbstract" xlink:to="us-gaap_OtherNonoperatingIncomeExpenseAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_OtherNonoperatingIncomeExpenseAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Other Nonoperating Income (Expense) [Abstract]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_OtherNonoperatingIncomeExpenseAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Other income:</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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/2009/role/negatedLabel" xlink:label="us-gaap_InvestmentIncomeInterest_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Interest earned on cash held in Trust Account</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 earned on cash held in Trust Account</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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/totalLabel" xlink:label="us-gaap_NonoperatingIncomeExpense_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Total other income</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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 income</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 income</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_WeightedAverageNumberOfSharesOutstandingBasic" xlink:label="us-gaap_WeightedAverageNumberOfSharesOutstandingBasic" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_WeightedAverageNumberOfSharesOutstandingBasic" xlink:to="us-gaap_WeightedAverageNumberOfSharesOutstandingBasic_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <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">Basic weighted average shares outstanding (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">Basic weighted average shares outstanding</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_WeightedAverageNumberOfDilutedSharesOutstanding" xlink:label="us-gaap_WeightedAverageNumberOfDilutedSharesOutstanding" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_WeightedAverageNumberOfDilutedSharesOutstanding" xlink:to="us-gaap_WeightedAverageNumberOfDilutedSharesOutstanding_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <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">Diluted weighted average shares outstanding (in shares)</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">Diluted weighted average shares outstanding</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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 income per ordinary share (in dollars per share)</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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 income per ordinary shares (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">Diluted net income per ordinary share</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_StatementOfStockholdersEquityAbstract" xlink:label="us-gaap_StatementOfStockholdersEquityAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_StatementOfStockholdersEquityAbstract" xlink:to="us-gaap_StatementOfStockholdersEquityAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <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">CONDENSED STATEMENTS OF CHANGES IN SHAREHOLDERS' DEFICIT</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_StatementEquityComponentsAxis" xlink:label="us-gaap_StatementEquityComponentsAxis" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_StatementEquityComponentsAxis" xlink:to="us-gaap_StatementEquityComponentsAxis_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_StatementEquityComponentsAxis_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Equity Components [Axis]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_EquityComponentDomain" xlink:label="us-gaap_EquityComponentDomain" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_EquityComponentDomain" xlink:to="us-gaap_EquityComponentDomain_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_EquityComponentDomain_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Equity Component [Domain]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_CommonStockMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Common Stock [Member]</label>
    <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">Ordinary Shares</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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/2026/elts/us-gaap-2026.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/2026/elts/us-gaap-2026.xsd#us-gaap_ClassOfWarrantOrRightAxis" xlink:label="us-gaap_ClassOfWarrantOrRightAxis" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ClassOfWarrantOrRightAxis" xlink:to="us-gaap_ClassOfWarrantOrRightAxis_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <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/2026/elts/us-gaap-2026.xsd#us-gaap_ClassOfWarrantOrRightDomain" xlink:label="us-gaap_ClassOfWarrantOrRightDomain" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ClassOfWarrantOrRightDomain" xlink:to="us-gaap_ClassOfWarrantOrRightDomain_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <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>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_PrivatePlacementWarrantsMember" xlink:label="biii_PrivatePlacementWarrantsMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_PrivatePlacementWarrantsMember" xlink:to="biii_PrivatePlacementWarrantsMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_PrivatePlacementWarrantsMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents information pertaining to private placement warrants.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_PrivatePlacementWarrantsMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Private Placement Warrants [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_PrivatePlacementWarrantsMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Private placement warrants</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_PublicWarrantsMember" xlink:label="biii_PublicWarrantsMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_PublicWarrantsMember" xlink:to="biii_PublicWarrantsMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_PublicWarrantsMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents information pertaining to public warrants.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_PublicWarrantsMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Public Warrants [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_PublicWarrantsMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Public warrants</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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">Warrants</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_AdjustmentsToAdditionalPaidInCapitalStockIssuedIssuanceCosts" xlink:label="us-gaap_AdjustmentsToAdditionalPaidInCapitalStockIssuedIssuanceCosts" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_AdjustmentsToAdditionalPaidInCapitalStockIssuedIssuanceCosts" xlink:to="us-gaap_AdjustmentsToAdditionalPaidInCapitalStockIssuedIssuanceCosts_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_AdjustmentsToAdditionalPaidInCapitalStockIssuedIssuanceCosts_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Adjustments to Additional Paid in Capital, Stock Issued, Issuance Costs</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2009/role/negatedLabel" xlink:label="us-gaap_AdjustmentsToAdditionalPaidInCapitalStockIssuedIssuanceCosts_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Allocated value of transaction costs to Class A shares</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_AdjustmentsToAdditionalPaidInCapitalIncreaseInCarryingAmountOfRedeemablePreferredStock" xlink:label="us-gaap_AdjustmentsToAdditionalPaidInCapitalIncreaseInCarryingAmountOfRedeemablePreferredStock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_AdjustmentsToAdditionalPaidInCapitalIncreaseInCarryingAmountOfRedeemablePreferredStock" xlink:to="us-gaap_AdjustmentsToAdditionalPaidInCapitalIncreaseInCarryingAmountOfRedeemablePreferredStock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_AdjustmentsToAdditionalPaidInCapitalIncreaseInCarryingAmountOfRedeemablePreferredStock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Adjustments to Additional Paid in Capital, Increase in Carrying Amount of Redeemable Preferred Stock</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2009/role/negatedLabel" xlink:label="us-gaap_AdjustmentsToAdditionalPaidInCapitalIncreaseInCarryingAmountOfRedeemablePreferredStock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Accretion for Class A ordinary shares to redemption amount</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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">CONDENSED STATEMENT OF CASH FLOWS</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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">Cash Provided by (Used in) Operating Activity, Including Discontinued Operation [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/2026/elts/us-gaap-2026.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 income</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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">Adjustment to Reconcile Net Income to Cash Provided by (Used in) Operating Activity [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 income to net cash used in operating activities:</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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">Adjustment to Reconcile Net Income to Cash Provided by (Used in) Operating Activity, 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">Changes in operating assets and liabilities:</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_IncreaseDecreaseInPrepaidExpenseExcludingPrepaidInsurance" xlink:label="biii_IncreaseDecreaseInPrepaidExpenseExcludingPrepaidInsurance" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_IncreaseDecreaseInPrepaidExpenseExcludingPrepaidInsurance" xlink:to="biii_IncreaseDecreaseInPrepaidExpenseExcludingPrepaidInsurance_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_IncreaseDecreaseInPrepaidExpenseExcludingPrepaidInsurance_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The increase (decrease) during the reporting period in the amount of outstanding money paid in advance for goods or services that bring economic benefits for future periods excluding insurance.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_IncreaseDecreaseInPrepaidExpenseExcludingPrepaidInsurance_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Increase (Decrease) in Prepaid Expense Excluding Prepaid Insurance</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2009/role/negatedLabel" xlink:label="biii_IncreaseDecreaseInPrepaidExpenseExcludingPrepaidInsurance_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Prepaid expenses</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_IncreaseDecreaseInPrepaidInsurance" xlink:label="us-gaap_IncreaseDecreaseInPrepaidInsurance" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_IncreaseDecreaseInPrepaidInsurance" xlink:to="us-gaap_IncreaseDecreaseInPrepaidInsurance_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_IncreaseDecreaseInPrepaidInsurance_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Increase (Decrease) in Prepaid Insurance</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel" xlink:label="us-gaap_IncreaseDecreaseInPrepaidInsurance_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Prepaid insurance</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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 expenses</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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">Cash Provided by (Used in) Operating Activity, Including Discontinued Operation</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/2026/elts/us-gaap-2026.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">Cash Flows from Investing Activities:</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_PaymentsForCashDepositedInTrustAccount" xlink:label="biii_PaymentsForCashDepositedInTrustAccount" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_PaymentsForCashDepositedInTrustAccount" xlink:to="biii_PaymentsForCashDepositedInTrustAccount_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_PaymentsForCashDepositedInTrustAccount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The amount of cash outflow associated with cash deposited in Trust Account.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_PaymentsForCashDepositedInTrustAccount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Payments for Cash Deposited in Trust Account</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2009/role/negatedLabel" xlink:label="biii_PaymentsForCashDepositedInTrustAccount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Investment of cash in Trust Account</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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">Cash Provided by (Used in) Investing Activity, Including Discontinued Operation</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/2026/elts/us-gaap-2026.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">Cash Provided by (Used in) Financing Activity, Including Discontinued Operation [Abstract]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" 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/2026/elts/us-gaap-2026.xsd#us-gaap_SaleOfStockConsiderationReceivedOnTransaction" xlink:label="us-gaap_SaleOfStockConsiderationReceivedOnTransaction" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_SaleOfStockConsiderationReceivedOnTransaction" xlink:to="us-gaap_SaleOfStockConsiderationReceivedOnTransaction_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_SaleOfStockConsiderationReceivedOnTransaction_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Sale of Stock, Consideration Received on Transaction</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_SaleOfStockConsiderationReceivedOnTransaction_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Proceeds from sale of Units, net of underwriting discounts paid</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_ProceedsFromIssuanceOfPrivatePlacement" xlink:label="us-gaap_ProceedsFromIssuanceOfPrivatePlacement" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ProceedsFromIssuanceOfPrivatePlacement" xlink:to="us-gaap_ProceedsFromIssuanceOfPrivatePlacement_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ProceedsFromIssuanceOfPrivatePlacement_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Proceeds from Issuance of Private Placement</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ProceedsFromIssuanceOfPrivatePlacement_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Gross proceeds from issuance of warrants</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_ProceedsFromIssuanceOfPrivatePlacement_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Proceeds from sale of Private Placement Warrants</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_RepaymentsOfRelatedPartyDebt" xlink:label="us-gaap_RepaymentsOfRelatedPartyDebt" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_RepaymentsOfRelatedPartyDebt" xlink:to="us-gaap_RepaymentsOfRelatedPartyDebt_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_RepaymentsOfRelatedPartyDebt_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Repayments of Related Party Debt</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2009/role/negatedLabel" xlink:label="us-gaap_RepaymentsOfRelatedPartyDebt_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Repayment of advances from related party</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_RepaymentsOfNotesPayable" xlink:label="us-gaap_RepaymentsOfNotesPayable" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_RepaymentsOfNotesPayable" xlink:to="us-gaap_RepaymentsOfNotesPayable_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_RepaymentsOfNotesPayable_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Repayments of Notes Payable</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2009/role/negatedLabel" xlink:label="us-gaap_RepaymentsOfNotesPayable_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Repayment of promissory note - related party</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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/2009/role/negatedLabel" xlink:label="us-gaap_PaymentsOfStockIssuanceCosts_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Payment of offering costs</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_NetCashProvidedByUsedInFinancingActivities" xlink:label="us-gaap_NetCashProvidedByUsedInFinancingActivities" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_NetCashProvidedByUsedInFinancingActivities" xlink:to="us-gaap_NetCashProvidedByUsedInFinancingActivities_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <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">Cash Provided by (Used in) Financing Activity, Including Discontinued Operation</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/2026/elts/us-gaap-2026.xsd#us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalentsPeriodIncreaseDecreaseExcludingExchangeRateEffect" xlink:label="us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalentsPeriodIncreaseDecreaseExcludingExchangeRateEffect" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalentsPeriodIncreaseDecreaseExcludingExchangeRateEffect" xlink:to="us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalentsPeriodIncreaseDecreaseExcludingExchangeRateEffect_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalentsPeriodIncreaseDecreaseExcludingExchangeRateEffect_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cash, Cash Equivalent, Restricted Cash, and Restricted Cash Equivalent, Period Increase (Decrease), Excluding Exchange Rate Effect, Including Discontinued Operation</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:label="us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalentsPeriodIncreaseDecreaseExcludingExchangeRateEffect_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Net Change in Cash</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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 Equivalent, Restricted Cash, and Restricted Cash Equivalent, Continuing Operation</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 - 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 - Beginning of period</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_SupplementalCashFlowInformationAbstract" xlink:label="us-gaap_SupplementalCashFlowInformationAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_SupplementalCashFlowInformationAbstract" xlink:to="us-gaap_SupplementalCashFlowInformationAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <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>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_SupplementalCashFlowInformationAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Non-cash Investing and Financing Activities</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_DeferredUnderwritingFeePayable" xlink:label="biii_DeferredUnderwritingFeePayable" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_DeferredUnderwritingFeePayable" xlink:to="biii_DeferredUnderwritingFeePayable_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_DeferredUnderwritingFeePayable_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amount of deferred underwriting fee payable.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_DeferredUnderwritingFeePayable_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Deferred underwriting fee payable</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_DeferredUnderwritingFeePayable_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Deferred underwriting fee payable</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_OfferingCostsIncludedInAccruedOfferingCosts" xlink:label="biii_OfferingCostsIncludedInAccruedOfferingCosts" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_OfferingCostsIncludedInAccruedOfferingCosts" xlink:to="biii_OfferingCostsIncludedInAccruedOfferingCosts_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_OfferingCostsIncludedInAccruedOfferingCosts_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amount of offering costs included in accrued offering costs.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_OfferingCostsIncludedInAccruedOfferingCosts_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Offering costs included in accrued offering costs</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_OfferingCostsIncludedInAccruedOfferingCosts_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Offering costs included in accrued offering costs</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_OrganizationConsolidationAndPresentationOfFinancialStatementsAbstract" xlink:label="us-gaap_OrganizationConsolidationAndPresentationOfFinancialStatementsAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_OrganizationConsolidationAndPresentationOfFinancialStatementsAbstract" xlink:to="us-gaap_OrganizationConsolidationAndPresentationOfFinancialStatementsAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_OrganizationConsolidationAndPresentationOfFinancialStatementsAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">DESCRIPTION OF ORGANIZATION AND BUSINESS OPERATIONS</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_OrganizationConsolidationAndPresentationOfFinancialStatementsDisclosureTextBlock" xlink:label="us-gaap_OrganizationConsolidationAndPresentationOfFinancialStatementsDisclosureTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_OrganizationConsolidationAndPresentationOfFinancialStatementsDisclosureTextBlock" xlink:to="us-gaap_OrganizationConsolidationAndPresentationOfFinancialStatementsDisclosureTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_OrganizationConsolidationAndPresentationOfFinancialStatementsDisclosureTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Organization, Consolidation and Presentation of Financial Statements Disclosure [Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_OrganizationConsolidationAndPresentationOfFinancialStatementsDisclosureTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">DESCRIPTION OF ORGANIZATION AND BUSINESS OPERATIONS</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_AccountingPoliciesAbstract" xlink:label="us-gaap_AccountingPoliciesAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_AccountingPoliciesAbstract" xlink:to="us-gaap_AccountingPoliciesAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <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/2026/elts/us-gaap-2026.xsd#us-gaap_SignificantAccountingPoliciesTextBlock" xlink:label="us-gaap_SignificantAccountingPoliciesTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_SignificantAccountingPoliciesTextBlock" xlink:to="us-gaap_SignificantAccountingPoliciesTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_SignificantAccountingPoliciesTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Significant Accounting Policies [Text Block]</label>
    <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="biii-20260630.xsd#biii_InitialPublicOfferingAbstract" xlink:label="biii_InitialPublicOfferingAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_InitialPublicOfferingAbstract" xlink:to="biii_InitialPublicOfferingAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_InitialPublicOfferingAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">N/A.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_InitialPublicOfferingAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">INITIAL PUBLIC OFFERING</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_InitialPublicOfferingTextBlock" xlink:label="biii_InitialPublicOfferingTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_InitialPublicOfferingTextBlock" xlink:to="biii_InitialPublicOfferingTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_InitialPublicOfferingTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The entire disclosure of information about initial public offering.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_InitialPublicOfferingTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Initial Public Offering [Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_InitialPublicOfferingTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">INITIAL PUBLIC OFFERING</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_PrivatePlacementAbstract" xlink:label="biii_PrivatePlacementAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_PrivatePlacementAbstract" xlink:to="biii_PrivatePlacementAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_PrivatePlacementAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">N/A.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_PrivatePlacementAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">PRIVATE PLACEMENTS</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_PrivatePlacementTextBlock" xlink:label="biii_PrivatePlacementTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_PrivatePlacementTextBlock" xlink:to="biii_PrivatePlacementTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_PrivatePlacementTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The entire disclosure of information about private placement.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_PrivatePlacementTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Private Placement [Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_PrivatePlacementTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">PRIVATE PLACEMENTS</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_RelatedPartyTransactionsAbstract" xlink:label="us-gaap_RelatedPartyTransactionsAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_RelatedPartyTransactionsAbstract" xlink:to="us-gaap_RelatedPartyTransactionsAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <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 PARTIES</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_RelatedPartyTransactionsDisclosureTextBlock" xlink:label="us-gaap_RelatedPartyTransactionsDisclosureTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_RelatedPartyTransactionsDisclosureTextBlock" xlink:to="us-gaap_RelatedPartyTransactionsDisclosureTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_RelatedPartyTransactionsDisclosureTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Related Party Transactions Disclosure [Text Block]</label>
    <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 PARTIES</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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/2026/elts/us-gaap-2026.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="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_EquityAbstract" xlink:label="us-gaap_EquityAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_EquityAbstract" xlink:to="us-gaap_EquityAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_EquityAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">SHAREHOLDERS' DEFICIT</label>
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    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_StockholdersEquityNoteDisclosureTextBlock" xlink:to="us-gaap_StockholdersEquityNoteDisclosureTextBlock_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_StockholdersEquityNoteDisclosureTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">SHAREHOLDERS' DEFICIT</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_WarrantsAndRightsNoteDisclosureAbstract" xlink:label="us-gaap_WarrantsAndRightsNoteDisclosureAbstract" 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_WarrantsAndRightsNoteDisclosureAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">WARRANTS</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_WarrantsAndRightsNoteDisclosureTextBlock" xlink:label="biii_WarrantsAndRightsNoteDisclosureTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_WarrantsAndRightsNoteDisclosureTextBlock" xlink:to="biii_WarrantsAndRightsNoteDisclosureTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_WarrantsAndRightsNoteDisclosureTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The entire disclosure of warrants.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_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="biii_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/2026/elts/us-gaap-2026.xsd#us-gaap_SegmentReportingAbstract" xlink:label="us-gaap_SegmentReportingAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <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 INFORMATION</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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 INFORMATION</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_FairValueDisclosuresAbstract" xlink:label="us-gaap_FairValueDisclosuresAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_FairValueDisclosuresAbstract" xlink:to="us-gaap_FairValueDisclosuresAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_FairValueDisclosuresAbstract_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/2026/elts/us-gaap-2026.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/2026/elts/us-gaap-2026.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/2026/elts/us-gaap-2026.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/2026/elts/us-gaap-2026.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="biii-20260630.xsd#biii_GoingConcernPolicyPolicyTextBlock" xlink:label="biii_GoingConcernPolicyPolicyTextBlock" 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="biii_GoingConcernPolicyPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Disclosure of accounting policy for going concern.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_GoingConcernPolicyPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Going Concern Policy [Policy Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_GoingConcernPolicyPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Going Concern</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_EntityEmergingGrowthCompanyPolicyTextBlock" xlink:label="biii_EntityEmergingGrowthCompanyPolicyTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_EntityEmergingGrowthCompanyPolicyTextBlock" xlink:to="biii_EntityEmergingGrowthCompanyPolicyTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_EntityEmergingGrowthCompanyPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Disclosure of accounting policy about benefits available for emerging growth company.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_EntityEmergingGrowthCompanyPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Entity, Emerging Growth Company [Policy Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_EntityEmergingGrowthCompanyPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Emerging Growth Company</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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="biii-20260630.xsd#biii_CashHeldInTrustAccountPolicyPolicyTextBlock" xlink:label="biii_CashHeldInTrustAccountPolicyPolicyTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_CashHeldInTrustAccountPolicyPolicyTextBlock" xlink:to="biii_CashHeldInTrustAccountPolicyPolicyTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_CashHeldInTrustAccountPolicyPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Disclosure of accounting policy for cash held in trust account.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_CashHeldInTrustAccountPolicyPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cash Held in Trust Account Policy [Policy Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_CashHeldInTrustAccountPolicyPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cash Held in Trust Account</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_ConcentrationRiskCreditRisk" xlink:label="us-gaap_ConcentrationRiskCreditRisk" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ConcentrationRiskCreditRisk" xlink:to="us-gaap_ConcentrationRiskCreditRisk_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <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">Concentration of Credit Risk</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_UseOfEstimates" xlink:label="us-gaap_UseOfEstimates" 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_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</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_OfferingCostsPolicyPolicyTextBlock" xlink:label="biii_OfferingCostsPolicyPolicyTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_OfferingCostsPolicyPolicyTextBlock" xlink:to="biii_OfferingCostsPolicyPolicyTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_OfferingCostsPolicyPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Disclosure of accounting policy for offering costs.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_OfferingCostsPolicyPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Offering Costs Policy [Policy Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_OfferingCostsPolicyPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Offering Costs</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_CommonStockSubjectedToPossibleRedemptionPolicyPolicyTextBlock" xlink:label="biii_CommonStockSubjectedToPossibleRedemptionPolicyPolicyTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_CommonStockSubjectedToPossibleRedemptionPolicyPolicyTextBlock" xlink:to="biii_CommonStockSubjectedToPossibleRedemptionPolicyPolicyTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_CommonStockSubjectedToPossibleRedemptionPolicyPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Disclosure of accounting policy for common stock subject to possible redemption.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_CommonStockSubjectedToPossibleRedemptionPolicyPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Common Stock Subjected to Possible Redemption Policy [Policy Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_CommonStockSubjectedToPossibleRedemptionPolicyPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class A Ordinary Shares Subject to Possible Redemption</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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/2026/elts/us-gaap-2026.xsd#us-gaap_EarningsPerSharePolicyTextBlock" xlink:label="us-gaap_EarningsPerSharePolicyTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_EarningsPerSharePolicyTextBlock" xlink:to="us-gaap_EarningsPerSharePolicyTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_EarningsPerSharePolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Earnings Per Share, Policy [Policy Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_EarningsPerSharePolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Net Income per Ordinary Share</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_FairValueOfFinancialInstrumentsPolicy" xlink:label="us-gaap_FairValueOfFinancialInstrumentsPolicy" 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_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 of Financial Instruments</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_WarrantsInstrumentsPolicyTextBlock" xlink:label="biii_WarrantsInstrumentsPolicyTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_WarrantsInstrumentsPolicyTextBlock" xlink:to="biii_WarrantsInstrumentsPolicyTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_WarrantsInstrumentsPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Disclosure of accounting policy for warrant instruments.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_WarrantsInstrumentsPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Warrants Instruments [Policy Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_WarrantsInstrumentsPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Warrant Instruments</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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">Share-Based Compensation</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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 Standards</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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 temporary equity</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_ScheduleOfEarningsPerShareBasicAndDilutedTableTextBlock" xlink:label="us-gaap_ScheduleOfEarningsPerShareBasicAndDilutedTableTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ScheduleOfEarningsPerShareBasicAndDilutedTableTextBlock" xlink:to="us-gaap_ScheduleOfEarningsPerShareBasicAndDilutedTableTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ScheduleOfEarningsPerShareBasicAndDilutedTableTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Schedule of Earnings Per Share, Basic and Diluted [Table Text Block]</label>
    <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 calculation of basic and diluted net income per ordinary share</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_ScheduleOfSegmentReportingInformationBySegmentTextBlock" xlink:label="us-gaap_ScheduleOfSegmentReportingInformationBySegmentTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ScheduleOfSegmentReportingInformationBySegmentTextBlock" xlink:to="us-gaap_ScheduleOfSegmentReportingInformationBySegmentTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <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>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ScheduleOfSegmentReportingInformationBySegmentTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Schedule of key metrics included in net income or loss and total assets</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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 quantitative information regarding market assumptions used in the Level 3 valuation of the Public Warrants</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_SubsidiaryOrEquityMethodInvesteeSaleOfStockBySubsidiaryOrEquityInvesteeTable" xlink:label="us-gaap_SubsidiaryOrEquityMethodInvesteeSaleOfStockBySubsidiaryOrEquityInvesteeTable" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_SubsidiaryOrEquityMethodInvesteeSaleOfStockBySubsidiaryOrEquityInvesteeTable" xlink:to="us-gaap_SubsidiaryOrEquityMethodInvesteeSaleOfStockBySubsidiaryOrEquityInvesteeTable_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <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/2026/elts/us-gaap-2026.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/2026/elts/us-gaap-2026.xsd#us-gaap_SaleOfStockNameOfTransactionDomain" xlink:label="us-gaap_SaleOfStockNameOfTransactionDomain" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_SaleOfStockNameOfTransactionDomain" xlink:to="us-gaap_SaleOfStockNameOfTransactionDomain_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <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/2026/elts/us-gaap-2026.xsd#us-gaap_IPOMember" xlink:label="us-gaap_IPOMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_IPOMember" xlink:to="us-gaap_IPOMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_IPOMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">IPO [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_IPOMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">IPO</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_InitialPublicOfferingAndOverAllotmentOptionMember" xlink:label="biii_InitialPublicOfferingAndOverAllotmentOptionMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_InitialPublicOfferingAndOverAllotmentOptionMember" xlink:to="biii_InitialPublicOfferingAndOverAllotmentOptionMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_InitialPublicOfferingAndOverAllotmentOptionMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents information pertaining to initial public offering and over allotment option.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_InitialPublicOfferingAndOverAllotmentOptionMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Initial Public Offering And Over Allotment Option [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_InitialPublicOfferingAndOverAllotmentOptionMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">IPO and over allotment</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_OverAllotmentOptionMember" xlink:label="us-gaap_OverAllotmentOptionMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_OverAllotmentOptionMember" xlink:to="us-gaap_OverAllotmentOptionMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_OverAllotmentOptionMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Over-Allotment Option [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_OverAllotmentOptionMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Over allotment</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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/srt/2026/elts/srt-2026.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/2026/elts/srt-2026.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="biii-20260630.xsd#biii_UnderwritersMember" xlink:label="biii_UnderwritersMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_UnderwritersMember" xlink:to="biii_UnderwritersMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_UnderwritersMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents information pertaining to underwriter.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_UnderwritersMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Underwriters [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_UnderwritersMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Underwriter</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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">DESCRIPTION OF ORGANIZATION AND BUSINESS OPERATIONS</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_SubsidiarySaleOfStockLineItems_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">PRIVATE PLACEMENTS</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_StockIssuedDuringPeriodUnitsNewIssues" xlink:label="biii_StockIssuedDuringPeriodUnitsNewIssues" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_StockIssuedDuringPeriodUnitsNewIssues" xlink:to="biii_StockIssuedDuringPeriodUnitsNewIssues_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_StockIssuedDuringPeriodUnitsNewIssues_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of new units issued during the period.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_StockIssuedDuringPeriodUnitsNewIssues_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Stock Issued During Period, Units, New Issues</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_StockIssuedDuringPeriodUnitsNewIssues_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of units issued</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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">Price per unit</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">Offering price per unit</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_ProceedsFromIssuanceOrSaleOfEquity" xlink:label="us-gaap_ProceedsFromIssuanceOrSaleOfEquity" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ProceedsFromIssuanceOrSaleOfEquity" xlink:to="us-gaap_ProceedsFromIssuanceOrSaleOfEquity_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ProceedsFromIssuanceOrSaleOfEquity_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Proceeds from Issuance or Sale of Equity</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ProceedsFromIssuanceOrSaleOfEquity_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Gross proceeds from issuance of units</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_ClassOfWarrantOrRightNumberOfWarrantsOrRightsIssued" xlink:label="biii_ClassOfWarrantOrRightNumberOfWarrantsOrRightsIssued" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_ClassOfWarrantOrRightNumberOfWarrantsOrRightsIssued" xlink:to="biii_ClassOfWarrantOrRightNumberOfWarrantsOrRightsIssued_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_ClassOfWarrantOrRightNumberOfWarrantsOrRightsIssued_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of warrants or rights issued during the period.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_ClassOfWarrantOrRightNumberOfWarrantsOrRightsIssued_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class Of Warrant Or Right, Number Of Warrants or Rights Issued</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_ClassOfWarrantOrRightNumberOfWarrantsOrRightsIssued_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Warrants issued</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_ClassOfWarrantOrRightIssuePricePerWarrantsOrRights" xlink:label="biii_ClassOfWarrantOrRightIssuePricePerWarrantsOrRights" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_ClassOfWarrantOrRightIssuePricePerWarrantsOrRights" xlink:to="biii_ClassOfWarrantOrRightIssuePricePerWarrantsOrRights_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_ClassOfWarrantOrRightIssuePricePerWarrantsOrRights_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Price of a single warrant or right issued during the period.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_ClassOfWarrantOrRightIssuePricePerWarrantsOrRights_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class Of Warrant Or Right, Issue Price Per Warrants Or Rights</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_ClassOfWarrantOrRightIssuePricePerWarrantsOrRights_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Issue price per warrant</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_StockOfferingCosts" xlink:label="biii_StockOfferingCosts" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_StockOfferingCosts" xlink:to="biii_StockOfferingCosts_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_StockOfferingCosts_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amount of stock issuance costs incurred during  the period.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_StockOfferingCosts_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Stock Offering Costs</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_StockOfferingCosts_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Transaction costs</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_CashUnderwritingFee" xlink:label="biii_CashUnderwritingFee" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_CashUnderwritingFee" xlink:to="biii_CashUnderwritingFee_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_CashUnderwritingFee_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amount of cash underwriting fees incurred during the period.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_CashUnderwritingFee_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cash Underwriting Fee</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_CashUnderwritingFee_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cash underwriting fees</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_DeferredUnderwritingFee" xlink:label="biii_DeferredUnderwritingFee" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_DeferredUnderwritingFee" xlink:to="biii_DeferredUnderwritingFee_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_DeferredUnderwritingFee_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amount of underwriting fee payable as of the balance sheet date.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_DeferredUnderwritingFee_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Deferred Underwriting Fee</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_DeferredUnderwritingFee_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Deferred underwriting fees</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="biii_DeferredUnderwritingFee_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Deferred fee</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_OtherStockOfferingCosts" xlink:label="biii_OtherStockOfferingCosts" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_OtherStockOfferingCosts" xlink:to="biii_OtherStockOfferingCosts_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_OtherStockOfferingCosts_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amount of other stock offering costs incurred during the period.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_OtherStockOfferingCosts_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Other Stock Offering Costs</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_OtherStockOfferingCosts_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Other offering costs</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_ThresholdFairMarketValueOfTargetBusinessAsPercentOfNetBalanceInTrustAccount" xlink:label="biii_ThresholdFairMarketValueOfTargetBusinessAsPercentOfNetBalanceInTrustAccount" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_ThresholdFairMarketValueOfTargetBusinessAsPercentOfNetBalanceInTrustAccount" xlink:to="biii_ThresholdFairMarketValueOfTargetBusinessAsPercentOfNetBalanceInTrustAccount_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_ThresholdFairMarketValueOfTargetBusinessAsPercentOfNetBalanceInTrustAccount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold fair market value of target businesses as a percent of net balance in trust account.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_ThresholdFairMarketValueOfTargetBusinessAsPercentOfNetBalanceInTrustAccount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold Fair Market Value Of Target Business As Percent Of Net Balance In Trust Account</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_ThresholdFairMarketValueOfTargetBusinessAsPercentOfNetBalanceInTrustAccount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Percent of fair market value of target businesses</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_ThresholdOwnershipPercentageInTargetCompany" xlink:label="biii_ThresholdOwnershipPercentageInTargetCompany" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_ThresholdOwnershipPercentageInTargetCompany" xlink:to="biii_ThresholdOwnershipPercentageInTargetCompany_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_ThresholdOwnershipPercentageInTargetCompany_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold percentage of ownership interest to be acquired in target company.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_ThresholdOwnershipPercentageInTargetCompany_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold Ownership Percentage In Target Company</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_ThresholdOwnershipPercentageInTargetCompany_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Percentage of ownership interest to be acquired</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_AssetsHeldInTrust" xlink:label="us-gaap_AssetsHeldInTrust" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_AssetsHeldInTrust" xlink:to="us-gaap_AssetsHeldInTrust_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_AssetsHeldInTrust_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Asset, Held-in-Trust</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_AssetsHeldInTrust_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amount held in trust account</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_RedemptionOfSharesThresholdPercentageOfRedemption" xlink:label="biii_RedemptionOfSharesThresholdPercentageOfRedemption" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_RedemptionOfSharesThresholdPercentageOfRedemption" xlink:to="biii_RedemptionOfSharesThresholdPercentageOfRedemption_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_RedemptionOfSharesThresholdPercentageOfRedemption_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold percentage of outstanding shares that can be redeemed.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_RedemptionOfSharesThresholdPercentageOfRedemption_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Redemption Of Shares, Threshold Percentage Of Redemption</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_RedemptionOfSharesThresholdPercentageOfRedemption_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Redemption percentage</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_PercentOfPublicSharesToBeRedeemedIfBusinessCombinationNotCompleted" xlink:label="biii_PercentOfPublicSharesToBeRedeemedIfBusinessCombinationNotCompleted" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_PercentOfPublicSharesToBeRedeemedIfBusinessCombinationNotCompleted" xlink:to="biii_PercentOfPublicSharesToBeRedeemedIfBusinessCombinationNotCompleted_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_PercentOfPublicSharesToBeRedeemedIfBusinessCombinationNotCompleted_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Percentage of public shares to be redeemed if the entity has not completed the business combination within the given time frame.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_PercentOfPublicSharesToBeRedeemedIfBusinessCombinationNotCompleted_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Percent Of Public Shares To Be Redeemed If Business Combination Not Completed</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_PercentOfPublicSharesToBeRedeemedIfBusinessCombinationNotCompleted_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Redemption percentage of shares if business combination not completed</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_MaximumPeriodToCompleteInitialBusinessCombinationAfterInitialPublicOffering" xlink:label="biii_MaximumPeriodToCompleteInitialBusinessCombinationAfterInitialPublicOffering" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_MaximumPeriodToCompleteInitialBusinessCombinationAfterInitialPublicOffering" xlink:to="biii_MaximumPeriodToCompleteInitialBusinessCombinationAfterInitialPublicOffering_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_MaximumPeriodToCompleteInitialBusinessCombinationAfterInitialPublicOffering_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Maximum period to complete the initial business combination after initial public offering,  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="biii_MaximumPeriodToCompleteInitialBusinessCombinationAfterInitialPublicOffering_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Maximum Period To Complete The Initial Business Combination After Initial Public Offering</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_MaximumPeriodToCompleteInitialBusinessCombinationAfterInitialPublicOffering_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Maximum period to complete the business combination</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_ExtendedPeriodToCompleteInitialBusinessCombinationAfterInitialPublicOffering" xlink:label="biii_ExtendedPeriodToCompleteInitialBusinessCombinationAfterInitialPublicOffering" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_ExtendedPeriodToCompleteInitialBusinessCombinationAfterInitialPublicOffering" xlink:to="biii_ExtendedPeriodToCompleteInitialBusinessCombinationAfterInitialPublicOffering_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_ExtendedPeriodToCompleteInitialBusinessCombinationAfterInitialPublicOffering_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Extended period to complete the initial business combination after initial public offering,  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="biii_ExtendedPeriodToCompleteInitialBusinessCombinationAfterInitialPublicOffering_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Extended Period To Complete The Initial Business Combination After Initial Public Offering</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_ExtendedPeriodToCompleteInitialBusinessCombinationAfterInitialPublicOffering_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Extended period to complete the business combination</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_MaximumPeriodToRedeemPublicSharesIfInitialBusinessCombinationNotCompleted" xlink:label="biii_MaximumPeriodToRedeemPublicSharesIfInitialBusinessCombinationNotCompleted" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_MaximumPeriodToRedeemPublicSharesIfInitialBusinessCombinationNotCompleted" xlink:to="biii_MaximumPeriodToRedeemPublicSharesIfInitialBusinessCombinationNotCompleted_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_MaximumPeriodToRedeemPublicSharesIfInitialBusinessCombinationNotCompleted_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Maximum period after the end of the completion window during which the company has to redeem public shares if it fails to complete a business combination.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_MaximumPeriodToRedeemPublicSharesIfInitialBusinessCombinationNotCompleted_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Maximum Period To Redeem Public Shares If Initial Business Combination Not Completed</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_MaximumPeriodToRedeemPublicSharesIfInitialBusinessCombinationNotCompleted_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Maximum period to redeem the shares</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_MaximumInterestEarnedFromTrustAccountUsedForDissolution" xlink:label="biii_MaximumInterestEarnedFromTrustAccountUsedForDissolution" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_MaximumInterestEarnedFromTrustAccountUsedForDissolution" xlink:to="biii_MaximumInterestEarnedFromTrustAccountUsedForDissolution_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_MaximumInterestEarnedFromTrustAccountUsedForDissolution_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Maximum amount of interest earned from trust account used for dissolution expense.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_MaximumInterestEarnedFromTrustAccountUsedForDissolution_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Maximum Interest Earned From Trust Account Used For Dissolution</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_MaximumInterestEarnedFromTrustAccountUsedForDissolution_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Dissolution expense</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_ThirdPartyClaimsLiabilityPerShare" xlink:label="biii_ThirdPartyClaimsLiabilityPerShare" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_ThirdPartyClaimsLiabilityPerShare" xlink:to="biii_ThirdPartyClaimsLiabilityPerShare_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_ThirdPartyClaimsLiabilityPerShare_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The per-share liability amount if the Trust Account at liquidation is less than $10.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_ThirdPartyClaimsLiabilityPerShare_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Third Party Claims, Liability Per Share</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_ThirdPartyClaimsLiabilityPerShare_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Claims by third party, liability floor rate</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_SignificantAccountingPolicyTable" xlink:label="biii_SignificantAccountingPolicyTable" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_SignificantAccountingPolicyTable" xlink:to="biii_SignificantAccountingPolicyTable_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_SignificantAccountingPolicyTable_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Disclosure of information about significant accounting policy.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_SignificantAccountingPolicyTable_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Significant Accounting Policy [Table]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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/2026/elts/us-gaap-2026.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="biii-20260630.xsd#biii_NotesPayableToSponsorMember" xlink:label="biii_NotesPayableToSponsorMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_NotesPayableToSponsorMember" xlink:to="biii_NotesPayableToSponsorMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_NotesPayableToSponsorMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents information pertaining to notes payable to sponsor.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_NotesPayableToSponsorMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Notes Payable To Sponsor [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_NotesPayableToSponsorMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Promissory note from sponsor</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_WorkingCapitalLoansMember" xlink:label="biii_WorkingCapitalLoansMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_WorkingCapitalLoansMember" xlink:to="biii_WorkingCapitalLoansMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_WorkingCapitalLoansMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents information pertaining to working capital loans.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_WorkingCapitalLoansMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Working Capital Loans [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_WorkingCapitalLoansMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Working capital loans</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_AntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareByAntidilutiveSecuritiesAxis" xlink:label="us-gaap_AntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareByAntidilutiveSecuritiesAxis" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_AntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareByAntidilutiveSecuritiesAxis" xlink:to="us-gaap_AntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareByAntidilutiveSecuritiesAxis_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_AntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareByAntidilutiveSecuritiesAxis_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Antidilutive Securities [Axis]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_AntidilutiveSecuritiesNameDomain" xlink:label="us-gaap_AntidilutiveSecuritiesNameDomain" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_AntidilutiveSecuritiesNameDomain" xlink:to="us-gaap_AntidilutiveSecuritiesNameDomain_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_AntidilutiveSecuritiesNameDomain_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Antidilutive Securities, Name [Domain]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_SignificantAccountingPolicyLineItems" xlink:label="biii_SignificantAccountingPolicyLineItems" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_SignificantAccountingPolicyLineItems" xlink:to="biii_SignificantAccountingPolicyLineItems_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_SignificantAccountingPolicyLineItems_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Line items represent significant accounting policy.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_SignificantAccountingPolicyLineItems_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Significant Accounting Policy [Line Items]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_SignificantAccountingPolicyLineItems_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/2026/elts/us-gaap-2026.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">Face amount of debt</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, face amount</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_NetWorkingCapitalAmount" xlink:label="biii_NetWorkingCapitalAmount" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_NetWorkingCapitalAmount" xlink:to="biii_NetWorkingCapitalAmount_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_NetWorkingCapitalAmount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amount of current assets excluding current liabilities.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_NetWorkingCapitalAmount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Net Working Capital, Amount</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_NetWorkingCapitalAmount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Working capital</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_DebtInstrumentConvertibleIntoWarrantsMaximumAmount" xlink:label="biii_DebtInstrumentConvertibleIntoWarrantsMaximumAmount" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_DebtInstrumentConvertibleIntoWarrantsMaximumAmount" xlink:to="biii_DebtInstrumentConvertibleIntoWarrantsMaximumAmount_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_DebtInstrumentConvertibleIntoWarrantsMaximumAmount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Maximum amount of debt that can be convertible to warrants.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_DebtInstrumentConvertibleIntoWarrantsMaximumAmount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Debt Instrument, Convertible Into Warrants, Maximum Amount</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_DebtInstrumentConvertibleIntoWarrantsMaximumAmount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Maximum convertible amount</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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">Debt conversion price per share</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_ShortTermBorrowings" xlink:label="us-gaap_ShortTermBorrowings" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ShortTermBorrowings" xlink:to="us-gaap_ShortTermBorrowings_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ShortTermBorrowings_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Short-Term Debt</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_ShortTermBorrowings_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Short term debt outstanding amount</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_CashAndCashEquivalentsAtCarryingValue" xlink:label="us-gaap_CashAndCashEquivalentsAtCarryingValue" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_CashAndCashEquivalentsAtCarryingValue" xlink:to="us-gaap_CashAndCashEquivalentsAtCarryingValue_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <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 Equivalent</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" 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/2026/elts/us-gaap-2026.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/verboseLabel" 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/2026/elts/us-gaap-2026.xsd#us-gaap_UnrecognizedTaxBenefitsIncomeTaxPenaltiesAndInterestAccrued" xlink:label="us-gaap_UnrecognizedTaxBenefitsIncomeTaxPenaltiesAndInterestAccrued" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_UnrecognizedTaxBenefitsIncomeTaxPenaltiesAndInterestAccrued" xlink:to="us-gaap_UnrecognizedTaxBenefitsIncomeTaxPenaltiesAndInterestAccrued_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_UnrecognizedTaxBenefitsIncomeTaxPenaltiesAndInterestAccrued_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Unrecognized Tax Benefits, Income Tax Penalties and Interest Accrued</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_UnrecognizedTaxBenefitsIncomeTaxPenaltiesAndInterestAccrued_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Accrued interest and penalties</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_IncomeTaxExpenseBenefit" xlink:label="us-gaap_IncomeTaxExpenseBenefit" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_IncomeTaxExpenseBenefit" xlink:to="us-gaap_IncomeTaxExpenseBenefit_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_IncomeTaxExpenseBenefit_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Income Tax Expense (Benefit)</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_IncomeTaxExpenseBenefit_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Income tax provision</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_AntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareAmount" xlink:label="us-gaap_AntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareAmount" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_AntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareAmount" xlink:to="us-gaap_AntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareAmount_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_AntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareAmount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Antidilutive Securities Excluded from Computation of Earnings Per Share, Amount</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_AntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareAmount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Anti-dilutive shares</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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">Class A ordinary shares</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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">Number of warrants outstanding</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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" />
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    <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>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_ProceedsFromIssuanceInitialPublicOffering" xlink:label="us-gaap_ProceedsFromIssuanceInitialPublicOffering" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ProceedsFromIssuanceInitialPublicOffering" xlink:to="us-gaap_ProceedsFromIssuanceInitialPublicOffering_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ProceedsFromIssuanceInitialPublicOffering_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Proceeds from Issuance Initial Public Offering</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ProceedsFromIssuanceInitialPublicOffering_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Gross proceeds</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_ProceedsAllocatedToPublicWarrants" xlink:label="biii_ProceedsAllocatedToPublicWarrants" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_ProceedsAllocatedToPublicWarrants" xlink:to="biii_ProceedsAllocatedToPublicWarrants_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_ProceedsAllocatedToPublicWarrants_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">A portion of gross proceeds from initial public offering allocated to public warrants</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_ProceedsAllocatedToPublicWarrants_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Proceeds Allocated to Public Warrants</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_ProceedsAllocatedToPublicWarrants_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Proceeds allocated to Public Warrants</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_ShareIssuanceCosts" xlink:label="biii_ShareIssuanceCosts" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_ShareIssuanceCosts" xlink:to="biii_ShareIssuanceCosts_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_ShareIssuanceCosts_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amount of issuance costs associated with the initial public offering.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_ShareIssuanceCosts_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Share Issuance Costs</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_ShareIssuanceCosts_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class A ordinary shares issuance cost</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_TemporaryEquityAccretionToRedemptionValue" xlink:label="us-gaap_TemporaryEquityAccretionToRedemptionValue" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_TemporaryEquityAccretionToRedemptionValue" xlink:to="us-gaap_TemporaryEquityAccretionToRedemptionValue_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_TemporaryEquityAccretionToRedemptionValue_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Temporary Equity, Accretion to Redemption Value</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_TemporaryEquityAccretionToRedemptionValue_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Accretion for Class A ordinary shares to redemption amount</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_ScheduleOfEarningsPerShareBasicByCommonClassTable" xlink:label="us-gaap_ScheduleOfEarningsPerShareBasicByCommonClassTable" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ScheduleOfEarningsPerShareBasicByCommonClassTable" xlink:to="us-gaap_ScheduleOfEarningsPerShareBasicByCommonClassTable_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ScheduleOfEarningsPerShareBasicByCommonClassTable_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Earnings Per Share, Basic, by Common Class, Including Two-Class Method [Table]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_EarningsPerShareBasicLineItems" xlink:label="us-gaap_EarningsPerShareBasicLineItems" xmlns="http://www.xbrl.org/2003/linkbase" />
    <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">Net income per ordinary shar</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_EarningsPerShareBasicAbstract" xlink:label="us-gaap_EarningsPerShareBasicAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_EarningsPerShareBasicAbstract" xlink:to="us-gaap_EarningsPerShareBasicAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_EarningsPerShareBasicAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Earnings Per Share, Basic [Abstract]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_EarningsPerShareBasicAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Basic net income per ordinary share</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_NetIncomeLossAvailableToCommonStockholdersBasicAbstract" xlink:label="us-gaap_NetIncomeLossAvailableToCommonStockholdersBasicAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_NetIncomeLossAvailableToCommonStockholdersBasicAbstract" xlink:to="us-gaap_NetIncomeLossAvailableToCommonStockholdersBasicAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_NetIncomeLossAvailableToCommonStockholdersBasicAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Net Income (Loss) Available to Common Stockholders, Basic [Abstract]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_NetIncomeLossAvailableToCommonStockholdersBasicAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Numerator:</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_NetIncomeLossAvailableToCommonStockholdersBasic" xlink:label="us-gaap_NetIncomeLossAvailableToCommonStockholdersBasic" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_NetIncomeLossAvailableToCommonStockholdersBasic" xlink:to="us-gaap_NetIncomeLossAvailableToCommonStockholdersBasic_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_NetIncomeLossAvailableToCommonStockholdersBasic_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Net Income (Loss) Available to Common Stockholders, Basic</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_NetIncomeLossAvailableToCommonStockholdersBasic_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Allocation of net income</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_WeightedAverageNumberOfSharesOutstandingBasicAbstract" xlink:label="us-gaap_WeightedAverageNumberOfSharesOutstandingBasicAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_WeightedAverageNumberOfSharesOutstandingBasicAbstract" xlink:to="us-gaap_WeightedAverageNumberOfSharesOutstandingBasicAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_WeightedAverageNumberOfSharesOutstandingBasicAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Weighted Average Number of Shares Outstanding, Basic [Abstract]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_WeightedAverageNumberOfSharesOutstandingBasicAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Denominator:</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_EarningsPerShareAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Earnings Per Share [Abstract]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_EarningsPerShareAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Diluted net income per ordinary share</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_NetIncomeLossAvailableToCommonStockholdersDilutedAbstract" xlink:label="us-gaap_NetIncomeLossAvailableToCommonStockholdersDilutedAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_NetIncomeLossAvailableToCommonStockholdersDilutedAbstract" xlink:to="us-gaap_NetIncomeLossAvailableToCommonStockholdersDilutedAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_NetIncomeLossAvailableToCommonStockholdersDilutedAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Net Income (Loss) Available to Common Stockholders, Diluted [Abstract]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_NetIncomeLossAvailableToCommonStockholdersDilutedAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Numerator:</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_NetIncomeLossAvailableToCommonStockholdersDiluted" xlink:label="us-gaap_NetIncomeLossAvailableToCommonStockholdersDiluted" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_NetIncomeLossAvailableToCommonStockholdersDiluted" xlink:to="us-gaap_NetIncomeLossAvailableToCommonStockholdersDiluted_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_NetIncomeLossAvailableToCommonStockholdersDiluted_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Net Income (Loss) Available to Common Stockholders, Diluted</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_NetIncomeLossAvailableToCommonStockholdersDiluted_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Allocation of net income</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_WeightedAverageNumberOfSharesOutstandingAbstract" xlink:label="us-gaap_WeightedAverageNumberOfSharesOutstandingAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_WeightedAverageNumberOfSharesOutstandingAbstract" xlink:to="us-gaap_WeightedAverageNumberOfSharesOutstandingAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_WeightedAverageNumberOfSharesOutstandingAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Weighted Average Number of Shares Outstanding, Diluted [Abstract]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_WeightedAverageNumberOfSharesOutstandingAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Denominator:</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_NumberOfSharesPerUnit" xlink:label="biii_NumberOfSharesPerUnit" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_NumberOfSharesPerUnit" xlink:to="biii_NumberOfSharesPerUnit_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_NumberOfSharesPerUnit_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of shares in a unit.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_NumberOfSharesPerUnit_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number Of Shares Per Unit</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_NumberOfSharesPerUnit_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of shares in unit</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_NumberOfWarrantsPerUnit" xlink:label="biii_NumberOfWarrantsPerUnit" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_NumberOfWarrantsPerUnit" xlink:to="biii_NumberOfWarrantsPerUnit_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_NumberOfWarrantsPerUnit_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of warrants in a unit.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_NumberOfWarrantsPerUnit_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number Of Warrants Per Unit</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_NumberOfWarrantsPerUnit_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of warrants in a unit</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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">Number of shares issued per warrant</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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 per warrant</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_ClassOfWarrantsOrRightsNumberOfFractionalWarrantsIssuedUponSeparationOfUnits" xlink:label="biii_ClassOfWarrantsOrRightsNumberOfFractionalWarrantsIssuedUponSeparationOfUnits" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_ClassOfWarrantsOrRightsNumberOfFractionalWarrantsIssuedUponSeparationOfUnits" xlink:to="biii_ClassOfWarrantsOrRightsNumberOfFractionalWarrantsIssuedUponSeparationOfUnits_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_ClassOfWarrantsOrRightsNumberOfFractionalWarrantsIssuedUponSeparationOfUnits_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of fractional warrants issued upon separation of units.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_ClassOfWarrantsOrRightsNumberOfFractionalWarrantsIssuedUponSeparationOfUnits_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class Of Warrants Or Rights, Number Of Fractional Warrants Issued Upon Separation Of Units</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_ClassOfWarrantsOrRightsNumberOfFractionalWarrantsIssuedUponSeparationOfUnits_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of fractional warrants issued upon separation of units</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_ThresholdNumberOfDaysToExerciseWarrantsFromCompletionOfInitialBusinessCombination" xlink:label="biii_ThresholdNumberOfDaysToExerciseWarrantsFromCompletionOfInitialBusinessCombination" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_ThresholdNumberOfDaysToExerciseWarrantsFromCompletionOfInitialBusinessCombination" xlink:to="biii_ThresholdNumberOfDaysToExerciseWarrantsFromCompletionOfInitialBusinessCombination_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_ThresholdNumberOfDaysToExerciseWarrantsFromCompletionOfInitialBusinessCombination_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold number of days to exercise warrants from completion of business combination.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_ThresholdNumberOfDaysToExerciseWarrantsFromCompletionOfInitialBusinessCombination_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold Number Of Days To Exercise Warrants From Completion Of Initial Business Combination</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_ThresholdNumberOfDaysToExerciseWarrantsFromCompletionOfInitialBusinessCombination_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold days to exercise warrants</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_WarrantsAndRightsOutstandingTerm_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Warrant term</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_ThresholdDaysFromBusinessCombinationForTransferOfWarrants" xlink:label="biii_ThresholdDaysFromBusinessCombinationForTransferOfWarrants" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_ThresholdDaysFromBusinessCombinationForTransferOfWarrants" xlink:to="biii_ThresholdDaysFromBusinessCombinationForTransferOfWarrants_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_ThresholdDaysFromBusinessCombinationForTransferOfWarrants_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold number of days from completion of business combination for warrants to be transferred, assigned or sold by the holders.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_ThresholdDaysFromBusinessCombinationForTransferOfWarrants_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold Days From Business Combination For Transfer Of Warrants</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_ThresholdDaysFromBusinessCombinationForTransferOfWarrants_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Restriction period to transfer warrants</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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 PARTIES</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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/verboseLabel" xlink:label="us-gaap_StockIssuedDuringPeriodSharesNewIssues_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of shares issued</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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/verboseLabel" xlink:label="us-gaap_StockIssuedDuringPeriodValueNewIssues_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Capital contribution</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_PercentageOfIssuedAndOutstandingSharesAfterInitialPublicOfferingHeldByStockholders" xlink:label="biii_PercentageOfIssuedAndOutstandingSharesAfterInitialPublicOfferingHeldByStockholders" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_PercentageOfIssuedAndOutstandingSharesAfterInitialPublicOfferingHeldByStockholders" xlink:to="biii_PercentageOfIssuedAndOutstandingSharesAfterInitialPublicOfferingHeldByStockholders_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_PercentageOfIssuedAndOutstandingSharesAfterInitialPublicOfferingHeldByStockholders_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Percentage of issued and outstanding shares after initial public offering held by the stockholders.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_PercentageOfIssuedAndOutstandingSharesAfterInitialPublicOfferingHeldByStockholders_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Percentage Of Issued And Outstanding Shares After Initial Public Offering Held By Stockholders</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_PercentageOfIssuedAndOutstandingSharesAfterInitialPublicOfferingHeldByStockholders_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Ownership percentage</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_NumberOfSharesTransferredToAffiliates" xlink:label="biii_NumberOfSharesTransferredToAffiliates" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_NumberOfSharesTransferredToAffiliates" xlink:to="biii_NumberOfSharesTransferredToAffiliates_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_NumberOfSharesTransferredToAffiliates_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of shares transferred to affiliates.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_NumberOfSharesTransferredToAffiliates_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number Of Shares Transferred To Affiliates</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_NumberOfSharesTransferredToAffiliates_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of shares transferred</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_SharesTransferredToAffiliatesNominalPricePerShare" xlink:label="biii_SharesTransferredToAffiliatesNominalPricePerShare" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_SharesTransferredToAffiliatesNominalPricePerShare" xlink:to="biii_SharesTransferredToAffiliatesNominalPricePerShare_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_SharesTransferredToAffiliatesNominalPricePerShare_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Nominal price per share of the shares transferred to affiliates.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_SharesTransferredToAffiliatesNominalPricePerShare_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Shares Transferred To Affiliates, Nominal Price Per Share</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_SharesTransferredToAffiliatesNominalPricePerShare_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Price per share</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_EquityClassifiedShareBasedPaymentGrantDateFairValue" xlink:label="biii_EquityClassifiedShareBasedPaymentGrantDateFairValue" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_EquityClassifiedShareBasedPaymentGrantDateFairValue" xlink:to="biii_EquityClassifiedShareBasedPaymentGrantDateFairValue_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_EquityClassifiedShareBasedPaymentGrantDateFairValue_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The grant date fair value of shares transferred under share based arrangement.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_EquityClassifiedShareBasedPaymentGrantDateFairValue_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Equity Classified Share Based Payment, Grant Date Fair Value</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_EquityClassifiedShareBasedPaymentGrantDateFairValue_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fair value of shares</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_EquityClassifiedShareBasedPaymentGrantDateFairValuePerShare" xlink:label="biii_EquityClassifiedShareBasedPaymentGrantDateFairValuePerShare" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_EquityClassifiedShareBasedPaymentGrantDateFairValuePerShare" xlink:to="biii_EquityClassifiedShareBasedPaymentGrantDateFairValuePerShare_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_EquityClassifiedShareBasedPaymentGrantDateFairValuePerShare_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The grant date fair value of per share transferred under share based arrangement.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_EquityClassifiedShareBasedPaymentGrantDateFairValuePerShare_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Equity Classified Share Based Payment, Grant Date Fair Value Per Share</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_EquityClassifiedShareBasedPaymentGrantDateFairValuePerShare_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fair value per share</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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">Implied share price</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_InstrumentSpecificMarketAdjustmentPercentage" xlink:label="biii_InstrumentSpecificMarketAdjustmentPercentage" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_InstrumentSpecificMarketAdjustmentPercentage" xlink:to="biii_InstrumentSpecificMarketAdjustmentPercentage_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_InstrumentSpecificMarketAdjustmentPercentage_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Percentage reduction applied to the implied market price of the underlying equity.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_InstrumentSpecificMarketAdjustmentPercentage_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Instrument Specific Market Adjustment, Percentage</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_InstrumentSpecificMarketAdjustmentPercentage_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Market adjustment</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_DiscountForLackOfMarketabilityPerShare" xlink:label="biii_DiscountForLackOfMarketabilityPerShare" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_DiscountForLackOfMarketabilityPerShare" xlink:to="biii_DiscountForLackOfMarketabilityPerShare_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_DiscountForLackOfMarketabilityPerShare_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Reduction applied to the value of equity interest to reflect that the shares cannot be sold freely in the market for a certain period.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_DiscountForLackOfMarketabilityPerShare_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Discount For Lack OF Marketability, Per Share</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_DiscountForLackOfMarketabilityPerShare_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Discount for lack of marketability applied to each share</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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">Compensation expense</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_ThresholdPeriodToTransferSharesAfterCompletionOfInitialBusinessCombination" xlink:label="biii_ThresholdPeriodToTransferSharesAfterCompletionOfInitialBusinessCombination" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_ThresholdPeriodToTransferSharesAfterCompletionOfInitialBusinessCombination" xlink:to="biii_ThresholdPeriodToTransferSharesAfterCompletionOfInitialBusinessCombination_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_ThresholdPeriodToTransferSharesAfterCompletionOfInitialBusinessCombination_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold period to transfer the shares after the initial business combination.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_ThresholdPeriodToTransferSharesAfterCompletionOfInitialBusinessCombination_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold Period To Transfer Shares After Completion Of Initial Business Combination</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_ThresholdPeriodToTransferSharesAfterCompletionOfInitialBusinessCombination_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold period to transfer shares</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_SaleOfSharesStockPriceTrigger" xlink:label="biii_SaleOfSharesStockPriceTrigger" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_SaleOfSharesStockPriceTrigger" xlink:to="biii_SaleOfSharesStockPriceTrigger_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_SaleOfSharesStockPriceTrigger_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Price per share required to sell the shares before the lock in period.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_SaleOfSharesStockPriceTrigger_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Sale Of Shares, Stock Price Trigger</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_SaleOfSharesStockPriceTrigger_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Target price per share</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_ThresholdTradingDaysForStockPriceTarget" xlink:label="biii_ThresholdTradingDaysForStockPriceTarget" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_ThresholdTradingDaysForStockPriceTarget" xlink:to="biii_ThresholdTradingDaysForStockPriceTarget_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_ThresholdTradingDaysForStockPriceTarget_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold trading days the stock price must be at or above the trigger price within a set time frame.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_ThresholdTradingDaysForStockPriceTarget_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold Trading Days For Stock Price Target</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_ThresholdTradingDaysForStockPriceTarget_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold trading days</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_ThresholdConsecutiveTradingDaysForStockPriceTarget" xlink:label="biii_ThresholdConsecutiveTradingDaysForStockPriceTarget" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_ThresholdConsecutiveTradingDaysForStockPriceTarget" xlink:to="biii_ThresholdConsecutiveTradingDaysForStockPriceTarget_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_ThresholdConsecutiveTradingDaysForStockPriceTarget_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold period of specified consecutive trading days the stock price must be at or above the trigger price within a set time frame.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_ThresholdConsecutiveTradingDaysForStockPriceTarget_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold Consecutive Trading Days For Stock Price Target</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_ThresholdConsecutiveTradingDaysForStockPriceTarget_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Consecutive trading days</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_ThresholdPeriodAfterInitialBusinessCombinationBeforeStockPriceConditionToTransferShares" xlink:label="biii_ThresholdPeriodAfterInitialBusinessCombinationBeforeStockPriceConditionToTransferShares" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_ThresholdPeriodAfterInitialBusinessCombinationBeforeStockPriceConditionToTransferShares" xlink:to="biii_ThresholdPeriodAfterInitialBusinessCombinationBeforeStockPriceConditionToTransferShares_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_ThresholdPeriodAfterInitialBusinessCombinationBeforeStockPriceConditionToTransferShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold period after the completion of the initial business combination before the stock price condition to transfer the shares.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_ThresholdPeriodAfterInitialBusinessCombinationBeforeStockPriceConditionToTransferShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold Period After Initial Business Combination Before Stock Price Condition To Transfer Shares</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_ThresholdPeriodAfterInitialBusinessCombinationBeforeStockPriceConditionToTransferShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Waiting period for transfer of shares</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_RelatedPartyUnsecuredPromissoryNoteMember" xlink:label="biii_RelatedPartyUnsecuredPromissoryNoteMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_RelatedPartyUnsecuredPromissoryNoteMember" xlink:to="biii_RelatedPartyUnsecuredPromissoryNoteMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_RelatedPartyUnsecuredPromissoryNoteMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents information pertaining to related party unsecured promissory note.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_RelatedPartyUnsecuredPromissoryNoteMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Related Party, Unsecured Promissory Note [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_RelatedPartyUnsecuredPromissoryNoteMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Promissory note</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/srt/2026/elts/srt-2026.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/2026/elts/srt-2026.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/2026/elts/srt-2026.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="biii-20260630.xsd#biii_SponsorOrAffiliateOfSponsorMember" xlink:label="biii_SponsorOrAffiliateOfSponsorMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_SponsorOrAffiliateOfSponsorMember" xlink:to="biii_SponsorOrAffiliateOfSponsorMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_SponsorOrAffiliateOfSponsorMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents information pertaining to sponsor or an affiliate of sponsor.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_SponsorOrAffiliateOfSponsorMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Sponsor Or An Affiliate Of Sponsor [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_SponsorOrAffiliateOfSponsorMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Sponsor or an affiliate</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_AdministrativeServicesAgreementMember" xlink:label="biii_AdministrativeServicesAgreementMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_AdministrativeServicesAgreementMember" xlink:to="biii_AdministrativeServicesAgreementMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_AdministrativeServicesAgreementMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents information pertaining to administrative services agreement.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_AdministrativeServicesAgreementMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Administrative Services Agreement [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_AdministrativeServicesAgreementMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Administrative Services Agreement</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_RelatedPartyTransactionMonthlyAdministrativeExpenses" xlink:label="biii_RelatedPartyTransactionMonthlyAdministrativeExpenses" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_RelatedPartyTransactionMonthlyAdministrativeExpenses" xlink:to="biii_RelatedPartyTransactionMonthlyAdministrativeExpenses_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_RelatedPartyTransactionMonthlyAdministrativeExpenses_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amount of monthly administrative expenses agreed under the related party transaction.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_RelatedPartyTransactionMonthlyAdministrativeExpenses_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Related Party Transaction, Monthly Administrative Expenses</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_RelatedPartyTransactionMonthlyAdministrativeExpenses_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Monthly expenses</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_AdministrativeFeesExpense" xlink:label="us-gaap_AdministrativeFeesExpense" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_AdministrativeFeesExpense" xlink:to="us-gaap_AdministrativeFeesExpense_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_AdministrativeFeesExpense_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Administrative Fees Expense</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_AdministrativeFeesExpense_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Administrative support fees</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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 conversion into warrants</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_OtherLiabilities" xlink:label="us-gaap_OtherLiabilities" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_OtherLiabilities" xlink:to="us-gaap_OtherLiabilities_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_OtherLiabilities_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Other Liabilities</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_OtherLiabilities_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Outstanding loan amount</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_ScheduleOfSubsidiaryOrEquityMethodInvesteeTable" xlink:label="us-gaap_ScheduleOfSubsidiaryOrEquityMethodInvesteeTable" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ScheduleOfSubsidiaryOrEquityMethodInvesteeTable" xlink:to="us-gaap_ScheduleOfSubsidiaryOrEquityMethodInvesteeTable_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ScheduleOfSubsidiaryOrEquityMethodInvesteeTable_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Subsidiary or Equity Method Investee, Sale of Stock [Table]</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_MaximumNumberOfDemandsExcludingShortFormRegistrationDemandsEntitledBySecurityHolders" xlink:label="biii_MaximumNumberOfDemandsExcludingShortFormRegistrationDemandsEntitledBySecurityHolders" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_MaximumNumberOfDemandsExcludingShortFormRegistrationDemandsEntitledBySecurityHolders" xlink:to="biii_MaximumNumberOfDemandsExcludingShortFormRegistrationDemandsEntitledBySecurityHolders_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_MaximumNumberOfDemandsExcludingShortFormRegistrationDemandsEntitledBySecurityHolders_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The maximum number of demands, excluding short form registration demands to be entitled by security holders.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_MaximumNumberOfDemandsExcludingShortFormRegistrationDemandsEntitledBySecurityHolders_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Maximum Number Of Demands, Excluding Short Form Registration Demands Entitled by Security Holders</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_MaximumNumberOfDemandsExcludingShortFormRegistrationDemandsEntitledBySecurityHolders_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of demands</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_OverAllotmentPeriod" xlink:label="biii_OverAllotmentPeriod" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_OverAllotmentPeriod" xlink:to="biii_OverAllotmentPeriod_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_OverAllotmentPeriod_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Period to exercise the over-allotment option, 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="biii_OverAllotmentPeriod_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Over Allotment Period</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_OverAllotmentPeriod_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Over allotment period</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_MaximumNumberOfAdditionalUnitsOfferedForPurchase" xlink:label="biii_MaximumNumberOfAdditionalUnitsOfferedForPurchase" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_MaximumNumberOfAdditionalUnitsOfferedForPurchase" xlink:to="biii_MaximumNumberOfAdditionalUnitsOfferedForPurchase_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_MaximumNumberOfAdditionalUnitsOfferedForPurchase_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Maximum number of additional units offered to purchase by the underwriters during the option period.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_MaximumNumberOfAdditionalUnitsOfferedForPurchase_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Maximum Number Of Additional Units Offered For Purchase</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_MaximumNumberOfAdditionalUnitsOfferedForPurchase_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Maximum number of additional units offered for purchase</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_NumberOfAdditionalUnitsPurchased" xlink:label="biii_NumberOfAdditionalUnitsPurchased" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_NumberOfAdditionalUnitsPurchased" xlink:to="biii_NumberOfAdditionalUnitsPurchased_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_NumberOfAdditionalUnitsPurchased_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The number of additional units purchased by the underwriter upon exercise the over allotment option.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_NumberOfAdditionalUnitsPurchased_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number Of Additional Units Purchased</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_NumberOfAdditionalUnitsPurchased_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of additional units exercised or purchased</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_NumberOfUnitsConsideredSold" xlink:label="biii_NumberOfUnitsConsideredSold" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_NumberOfUnitsConsideredSold" xlink:to="biii_NumberOfUnitsConsideredSold_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_NumberOfUnitsConsideredSold_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The number of units considered sold under the underwriting agreement.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_NumberOfUnitsConsideredSold_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number Of Units Considered Sold</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_NumberOfUnitsConsideredSold_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of units considered sold</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_PaymentsForUnderwritingDiscount" xlink:label="biii_PaymentsForUnderwritingDiscount" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_PaymentsForUnderwritingDiscount" xlink:to="biii_PaymentsForUnderwritingDiscount_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_PaymentsForUnderwritingDiscount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cash paid for expenses incurred during underwriting activities.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_PaymentsForUnderwritingDiscount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Payments For Underwriting Discount</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_PaymentsForUnderwritingDiscount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Underwriting discount paid</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_CashUnderwritingDiscountNotCharged" xlink:label="biii_CashUnderwritingDiscountNotCharged" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_CashUnderwritingDiscountNotCharged" xlink:to="biii_CashUnderwritingDiscountNotCharged_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_CashUnderwritingDiscountNotCharged_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amount of cash underwriting discount that was not charged on units sold to excluded investors.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_CashUnderwritingDiscountNotCharged_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cash Underwriting Discount Not Charged</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_CashUnderwritingDiscountNotCharged_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cash underwriting discount not charged</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_DeferredUnderwritingFeePerUnit" xlink:label="biii_DeferredUnderwritingFeePerUnit" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_DeferredUnderwritingFeePerUnit" xlink:to="biii_DeferredUnderwritingFeePerUnit_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_DeferredUnderwritingFeePerUnit_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The per unit value of deferred fee entitled to underwriters on units sold excluding the units sold under the option to purchase to underwriters.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_DeferredUnderwritingFeePerUnit_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Deferred Underwriting Fee Per Unit</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_DeferredUnderwritingFeePerUnit_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Deferred fee per unit</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_DeferredUnderwritingFeeNotCharged" xlink:label="biii_DeferredUnderwritingFeeNotCharged" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_DeferredUnderwritingFeeNotCharged" xlink:to="biii_DeferredUnderwritingFeeNotCharged_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_DeferredUnderwritingFeeNotCharged_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amount of deferred underwriting fee that was not charged on units sold to excluded investors.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_DeferredUnderwritingFeeNotCharged_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Deferred Underwriting Fee Not Charged</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_DeferredUnderwritingFeeNotCharged_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Deferred fee not charged</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_ClassOfWarrantOrRightLockUpPeriodAfterRegistrationStatementBecomesEffective" xlink:label="biii_ClassOfWarrantOrRightLockUpPeriodAfterRegistrationStatementBecomesEffective" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_ClassOfWarrantOrRightLockUpPeriodAfterRegistrationStatementBecomesEffective" xlink:to="biii_ClassOfWarrantOrRightLockUpPeriodAfterRegistrationStatementBecomesEffective_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_ClassOfWarrantOrRightLockUpPeriodAfterRegistrationStatementBecomesEffective_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Period of lock-up restrictions after registration statement becomes effective , 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="biii_ClassOfWarrantOrRightLockUpPeriodAfterRegistrationStatementBecomesEffective_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class Of Warrant Or Right, Lock Up Period After Registration Statement Becomes Effective</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_ClassOfWarrantOrRightLockUpPeriodAfterRegistrationStatementBecomesEffective_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Lock up period</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_DemandRegistrationRightsTerm" xlink:label="biii_DemandRegistrationRightsTerm" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_DemandRegistrationRightsTerm" xlink:to="biii_DemandRegistrationRightsTerm_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_DemandRegistrationRightsTerm_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Term of demand registration rights, 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="biii_DemandRegistrationRightsTerm_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Demand Registration Rights, Term</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_DemandRegistrationRightsTerm_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Term of demand registration rights</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_PiggyBackRegistrationRightsTerm" xlink:label="biii_PiggyBackRegistrationRightsTerm" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_PiggyBackRegistrationRightsTerm" xlink:to="biii_PiggyBackRegistrationRightsTerm_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_PiggyBackRegistrationRightsTerm_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Term of piggy-back registration rights, 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="biii_PiggyBackRegistrationRightsTerm_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Piggy Back Registration Rights, Term</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_PiggyBackRegistrationRightsTerm_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Term of piggy-back registration rights</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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/srt/2026/elts/srt-2026.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="biii-20260630.xsd#biii_FoundersSharesMember" xlink:label="biii_FoundersSharesMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_FoundersSharesMember" xlink:to="biii_FoundersSharesMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_FoundersSharesMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents information pertaining to founders shares.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_FoundersSharesMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Founders Shares [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_FoundersSharesMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Founder Shares</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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">SHAREHOLDERS' EQUITY</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_CommonStockNumberOfVotesPerShare" xlink:label="biii_CommonStockNumberOfVotesPerShare" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_CommonStockNumberOfVotesPerShare" xlink:to="biii_CommonStockNumberOfVotesPerShare_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_CommonStockNumberOfVotesPerShare_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of votes per share of common stock.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_CommonStockNumberOfVotesPerShare_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Common Stock, Number Of Votes Per Share</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_CommonStockNumberOfVotesPerShare_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of votes per share</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_ConversionOfStockConversionRatio" xlink:label="biii_ConversionOfStockConversionRatio" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_ConversionOfStockConversionRatio" xlink:to="biii_ConversionOfStockConversionRatio_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_ConversionOfStockConversionRatio_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of class A common shares issuable upon conversion for each share of other class of common stock.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_ConversionOfStockConversionRatio_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Conversion Of Stock, Conversion Ratio</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_ConversionOfStockConversionRatio_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Conversion ratio</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_ThresholdAggregatePercentageOfAdditionalSharesIssuableUponConversionInConnectionWithBusinessCombination" xlink:label="biii_ThresholdAggregatePercentageOfAdditionalSharesIssuableUponConversionInConnectionWithBusinessCombination" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_ThresholdAggregatePercentageOfAdditionalSharesIssuableUponConversionInConnectionWithBusinessCombination" xlink:to="biii_ThresholdAggregatePercentageOfAdditionalSharesIssuableUponConversionInConnectionWithBusinessCombination_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_ThresholdAggregatePercentageOfAdditionalSharesIssuableUponConversionInConnectionWithBusinessCombination_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold aggregate percentage of additional shares issuable upon conversion in connection with the business combination.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_ThresholdAggregatePercentageOfAdditionalSharesIssuableUponConversionInConnectionWithBusinessCombination_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold Aggregate Percentage Of Additional Shares Issuable Upon Conversion In Connection With Business Combination</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_ThresholdAggregatePercentageOfAdditionalSharesIssuableUponConversionInConnectionWithBusinessCombination_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold aggregate additional shares issuable, percentage</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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="biii-20260630.xsd#biii_ClassOfWarrantsOrRightsWarrantsExercisableTermPostBusinessCombination" xlink:label="biii_ClassOfWarrantsOrRightsWarrantsExercisableTermPostBusinessCombination" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_ClassOfWarrantsOrRightsWarrantsExercisableTermPostBusinessCombination" xlink:to="biii_ClassOfWarrantsOrRightsWarrantsExercisableTermPostBusinessCombination_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_ClassOfWarrantsOrRightsWarrantsExercisableTermPostBusinessCombination_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Period over which the warrants becomes exercisable post business combination expressed 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="biii_ClassOfWarrantsOrRightsWarrantsExercisableTermPostBusinessCombination_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class Of Warrants Or Rights, Warrants Exercisable Term Post Business Combination</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_ClassOfWarrantsOrRightsWarrantsExercisableTermPostBusinessCombination_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Exercisable term</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_ClassOfWarrantOrRightExercisableForCashOrCashlessBasis" xlink:label="biii_ClassOfWarrantOrRightExercisableForCashOrCashlessBasis" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_ClassOfWarrantOrRightExercisableForCashOrCashlessBasis" xlink:to="biii_ClassOfWarrantOrRightExercisableForCashOrCashlessBasis_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_ClassOfWarrantOrRightExercisableForCashOrCashlessBasis_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The number of warrants or rights exercisable for cash or cashless basis.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_ClassOfWarrantOrRightExercisableForCashOrCashlessBasis_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class of Warrant or Right, Exercisable For Cash Or Cashless Basis</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="biii_ClassOfWarrantOrRightExercisableForCashOrCashlessBasis_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of warrants or rights exercisable for cash</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_ThresholdNumberOfBusinessDaysForFilingOfRegistrationStatementPostInitialBusinessCombination" xlink:label="biii_ThresholdNumberOfBusinessDaysForFilingOfRegistrationStatementPostInitialBusinessCombination" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_ThresholdNumberOfBusinessDaysForFilingOfRegistrationStatementPostInitialBusinessCombination" xlink:to="biii_ThresholdNumberOfBusinessDaysForFilingOfRegistrationStatementPostInitialBusinessCombination_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_ThresholdNumberOfBusinessDaysForFilingOfRegistrationStatementPostInitialBusinessCombination_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold number of business days available for company to file the registration statement post initial business combination.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_ThresholdNumberOfBusinessDaysForFilingOfRegistrationStatementPostInitialBusinessCombination_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold Number Of Business Days For Filing Of Registration Statement Post Initial Business Combination</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_ThresholdNumberOfBusinessDaysForFilingOfRegistrationStatementPostInitialBusinessCombination_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold number of business days for filing of registration statement</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_ThresholdBusinessDaysForMakingRegistrationStatementEffective" xlink:label="biii_ThresholdBusinessDaysForMakingRegistrationStatementEffective" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_ThresholdBusinessDaysForMakingRegistrationStatementEffective" xlink:to="biii_ThresholdBusinessDaysForMakingRegistrationStatementEffective_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_ThresholdBusinessDaysForMakingRegistrationStatementEffective_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The threshold number of business days for making the registration statement effective within number of days of initial business combination.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_ThresholdBusinessDaysForMakingRegistrationStatementEffective_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold Business Days For Making Registration Statement Effective</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_ThresholdBusinessDaysForMakingRegistrationStatementEffective_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold business days for making the registration statement effective</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_ClassOfWarrantsOrRightsRedemptionsStockPriceTrigger" xlink:label="biii_ClassOfWarrantsOrRightsRedemptionsStockPriceTrigger" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_ClassOfWarrantsOrRightsRedemptionsStockPriceTrigger" xlink:to="biii_ClassOfWarrantsOrRightsRedemptionsStockPriceTrigger_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_ClassOfWarrantsOrRightsRedemptionsStockPriceTrigger_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Price of the entity's common stock which would be required to be attained for the redemption feature in the warrant to become effective.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_ClassOfWarrantsOrRightsRedemptionsStockPriceTrigger_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class Of Warrants Or Rights, Redemptions, Stock Price Trigger</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_ClassOfWarrantsOrRightsRedemptionsStockPriceTrigger_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Warrant redemption, stock price trigger</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_ClassOfWarrantsOrRightsRedemptionPricePerWarrant" xlink:label="biii_ClassOfWarrantsOrRightsRedemptionPricePerWarrant" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_ClassOfWarrantsOrRightsRedemptionPricePerWarrant" xlink:to="biii_ClassOfWarrantsOrRightsRedemptionPricePerWarrant_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_ClassOfWarrantsOrRightsRedemptionPricePerWarrant_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Redemption price per warrant.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_ClassOfWarrantsOrRightsRedemptionPricePerWarrant_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class Of Warrants OR Rights, Redemption Price Per Warrant</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_ClassOfWarrantsOrRightsRedemptionPricePerWarrant_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Redemption price per warrant</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_ClassOfWarrantsOrRightsPriorNoticeOfRedemptionToBeIssuedNumberOfDays" xlink:label="biii_ClassOfWarrantsOrRightsPriorNoticeOfRedemptionToBeIssuedNumberOfDays" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_ClassOfWarrantsOrRightsPriorNoticeOfRedemptionToBeIssuedNumberOfDays" xlink:to="biii_ClassOfWarrantsOrRightsPriorNoticeOfRedemptionToBeIssuedNumberOfDays_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_ClassOfWarrantsOrRightsPriorNoticeOfRedemptionToBeIssuedNumberOfDays_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of days of prior notice to be issued for redemption of warrants.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_ClassOfWarrantsOrRightsPriorNoticeOfRedemptionToBeIssuedNumberOfDays_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class Of Warrants Or Rights, Prior Notice Of Redemption To Be Issued, Number Of Days</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_ClassOfWarrantsOrRightsPriorNoticeOfRedemptionToBeIssuedNumberOfDays_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of days of prior notice of redemption to be issued</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_ClassOfWarrantsOrRightsRedemptionsThresholdTradingDays" xlink:label="biii_ClassOfWarrantsOrRightsRedemptionsThresholdTradingDays" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_ClassOfWarrantsOrRightsRedemptionsThresholdTradingDays" xlink:to="biii_ClassOfWarrantsOrRightsRedemptionsThresholdTradingDays_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_ClassOfWarrantsOrRightsRedemptionsThresholdTradingDays_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold number of specified trading days that common stock price to redemption price of warrant must exceed threshold percentage within a specified consecutive trading period to trigger redemption feature.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_ClassOfWarrantsOrRightsRedemptionsThresholdTradingDays_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class Of Warrants Or Rights, Redemptions, Threshold Trading Days</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_ClassOfWarrantsOrRightsRedemptionsThresholdTradingDays_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold trading days</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_ClassOfWarrantsOrRightsRedemptionsThresholdConsecutiveTradingDays" xlink:label="biii_ClassOfWarrantsOrRightsRedemptionsThresholdConsecutiveTradingDays" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_ClassOfWarrantsOrRightsRedemptionsThresholdConsecutiveTradingDays" xlink:to="biii_ClassOfWarrantsOrRightsRedemptionsThresholdConsecutiveTradingDays_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_ClassOfWarrantsOrRightsRedemptionsThresholdConsecutiveTradingDays_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold period of specified consecutive trading days within which common stock price to redemption price of warrant must exceed threshold percentage for specified number of trading days to trigger redemption feature.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_ClassOfWarrantsOrRightsRedemptionsThresholdConsecutiveTradingDays_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class Of Warrants Or Rights, Redemptions, Threshold Consecutive Trading Days</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_ClassOfWarrantsOrRightsRedemptionsThresholdConsecutiveTradingDays_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold consecutive trading days</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_ClassOfWarrantOrRightFairValueVolumeWeightedAveragePricePerShareTradingDaysPriorToWrittenNoticeForExercise" xlink:label="biii_ClassOfWarrantOrRightFairValueVolumeWeightedAveragePricePerShareTradingDaysPriorToWrittenNoticeForExercise" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_ClassOfWarrantOrRightFairValueVolumeWeightedAveragePricePerShareTradingDaysPriorToWrittenNoticeForExercise" xlink:to="biii_ClassOfWarrantOrRightFairValueVolumeWeightedAveragePricePerShareTradingDaysPriorToWrittenNoticeForExercise_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_ClassOfWarrantOrRightFairValueVolumeWeightedAveragePricePerShareTradingDaysPriorToWrittenNoticeForExercise_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The number of trading days prior to the date on which the written notice for exercise is received by the warrant agent considered for volume weighted average price of per share.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_ClassOfWarrantOrRightFairValueVolumeWeightedAveragePricePerShareTradingDaysPriorToWrittenNoticeForExercise_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class Of Warrant Or Right, Fair Value ,Volume Weighted Average Price Per Share, Trading Days Prior To Written Notice For Exercise</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_ClassOfWarrantOrRightFairValueVolumeWeightedAveragePricePerShareTradingDaysPriorToWrittenNoticeForExercise_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Trading day prior to the date on which the notice of exercise is received</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_ThresholdPeriodForNotToTransferAssignOrSellAnySharesOrWarrantsAfterCompletionOfInitialBusinessCombination" xlink:label="biii_ThresholdPeriodForNotToTransferAssignOrSellAnySharesOrWarrantsAfterCompletionOfInitialBusinessCombination" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_ThresholdPeriodForNotToTransferAssignOrSellAnySharesOrWarrantsAfterCompletionOfInitialBusinessCombination" xlink:to="biii_ThresholdPeriodForNotToTransferAssignOrSellAnySharesOrWarrantsAfterCompletionOfInitialBusinessCombination_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_ThresholdPeriodForNotToTransferAssignOrSellAnySharesOrWarrantsAfterCompletionOfInitialBusinessCombination_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The period of time which must elapse after completion of a business combination before the Sponsor can transfer, assign or sell any Founder Shares unless other specified conditions are met.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_ThresholdPeriodForNotToTransferAssignOrSellAnySharesOrWarrantsAfterCompletionOfInitialBusinessCombination_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold Period For Not To Transfer, Assign Or Sell Any Shares Or Warrants After Completion Of Initial Business Combination</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_ThresholdPeriodForNotToTransferAssignOrSellAnySharesOrWarrantsAfterCompletionOfInitialBusinessCombination_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold period for not to transfer, assign or sell any of their shares or warrants after the completion of the initial business combination</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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/2026/elts/us-gaap-2026.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/2026/elts/us-gaap-2026.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="biii-20260630.xsd#biii_SingleReportableSegmentMember" xlink:label="biii_SingleReportableSegmentMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_SingleReportableSegmentMember" xlink:to="biii_SingleReportableSegmentMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_SingleReportableSegmentMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents information pertaining to single reportable segment.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_SingleReportableSegmentMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Single reportable segment</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_SingleReportableSegmentMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Single reportable segment</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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 INFORMATION</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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 reportable segments</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.xsd#us-gaap_NumberOfOperatingSegments" xlink:label="us-gaap_NumberOfOperatingSegments" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_NumberOfOperatingSegments" xlink:to="us-gaap_NumberOfOperatingSegments_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_NumberOfOperatingSegments_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of Operating Segments</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_NumberOfOperatingSegments_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of operating segments</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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/2026/elts/us-gaap-2026.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/2026/elts/us-gaap-2026.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">Implied Class A share price</label>
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    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_MeasurementInputExercisePriceMember" xlink:to="us-gaap_MeasurementInputExercisePriceMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_MeasurementInputExercisePriceMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Measurement Input, Exercise Price [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_MeasurementInputExercisePriceMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Exercise price</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_MeasurementInputExpectedTermOfDeSpacMember" xlink:label="biii_MeasurementInputExpectedTermOfDeSpacMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_MeasurementInputExpectedTermOfDeSpacMember" xlink:to="biii_MeasurementInputExpectedTermOfDeSpacMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_MeasurementInputExpectedTermOfDeSpacMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents information pertaining to expected term for De-SPAC.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_MeasurementInputExpectedTermOfDeSpacMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Measurement Input, Expected Term of De-SPAC [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_MeasurementInputExpectedTermOfDeSpacMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Expected term to de-SPAC (years)</label>
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    <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">Warrant term</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_MeasurementInputProbabilityOfDeSpacAndMarketAdjustmentMember" xlink:label="biii_MeasurementInputProbabilityOfDeSpacAndMarketAdjustmentMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="biii_MeasurementInputProbabilityOfDeSpacAndMarketAdjustmentMember" xlink:to="biii_MeasurementInputProbabilityOfDeSpacAndMarketAdjustmentMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="biii_MeasurementInputProbabilityOfDeSpacAndMarketAdjustmentMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents information pertaining to probability of de-SPAC and market adjustment.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_MeasurementInputProbabilityOfDeSpacAndMarketAdjustmentMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Measurement Input, Probability of De-SPAC And Market Adjustment [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_MeasurementInputProbabilityOfDeSpacAndMarketAdjustmentMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Probability of de-SPAC and market adjustment</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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 rate (continuous)</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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">Selected volatility</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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/2026/elts/us-gaap-2026.xsd#us-gaap_WarrantsAndRightsOutstanding" xlink:label="us-gaap_WarrantsAndRightsOutstanding" 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_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">Fair value of warrants</label>
    <loc xlink:type="locator" xlink:href="biii-20260630.xsd#biii_WarrantsAndRightsOutstandingFairValuePerShare" xlink:label="biii_WarrantsAndRightsOutstandingFairValuePerShare" 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="biii_WarrantsAndRightsOutstandingFairValuePerShare_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fair value per outstanding derivative security that permit the holder the right to purchase securities (usually equity) from the issuer at a specified price.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="biii_WarrantsAndRightsOutstandingFairValuePerShare_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Warrants And Rights Outstanding, Fair Value Per Share</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="biii_WarrantsAndRightsOutstandingFairValuePerShare_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fair value per warrant</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2026/elts/us-gaap-2026.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 input</label>
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