Amount of cash underwriting discount that was not charged on units sold to excluded investors.

No definition available.

Name:	biii_CashUnderwritingDiscountNotCharged
Namespace Prefix:	biii_
Data Type:	xbrli:monetaryItemType
Balance Type:	debit
Period Type:	duration

- Definition

Price of a single warrant or right issued during the period.

+ References

No definition available.

+ Details

Name:	biii_ClassOfWarrantOrRightIssuePricePerWarrantsOrRights
Namespace Prefix:	biii_
Data Type:	dtr-types:perShareItemType
Balance Type:	na
Period Type:	duration

- Definition

Number of warrants or rights issued during the period.

+ References

No definition available.

+ Details

Name:	biii_ClassOfWarrantOrRightNumberOfWarrantsOrRightsIssued
Namespace Prefix:	biii_
Data Type:	xbrli:sharesItemType
Balance Type:	na
Period Type:	duration

- Definition

Amount of underwriting fee payable as of the balance sheet date.

+ References

No definition available.

+ Details

Name:	biii_DeferredUnderwritingFee
Namespace Prefix:	biii_
Data Type:	xbrli:monetaryItemType
Balance Type:	debit
Period Type:	instant

- Definition

Amount of deferred underwriting fee that was not charged on units sold to excluded investors.

+ References

No definition available.

+ Details

Name:	biii_DeferredUnderwritingFeeNotCharged
Namespace Prefix:	biii_
Data Type:	xbrli:monetaryItemType
Balance Type:	debit
Period Type:	instant

- Definition

The per unit value of deferred fee entitled to underwriters on units sold excluding the units sold under the option to purchase to underwriters.

+ References

No definition available.

+ Details

Name:	biii_DeferredUnderwritingFeePerUnit
Namespace Prefix:	biii_
Data Type:	dtr-types:perShareItemType
Balance Type:	na
Period Type:	instant

- Definition

Number of shares that were no longer subject to forfeiture as a result of exercise of over-allotment option by the underwriters.

+ References

No definition available.

+ Details

Name:	biii_NumberOfSharesNoLongerSubjectToForfeiture
Namespace Prefix:	biii_
Data Type:	xbrli:sharesItemType
Balance Type:	na
Period Type:	instant

- Definition

Cash paid for expenses incurred during underwriting activities.

+ References

No definition available.

+ Details

Name:	biii_PaymentsForUnderwritingDiscount
Namespace Prefix:	biii_
Data Type:	xbrli:monetaryItemType
Balance Type:	debit
Period Type:	duration

- Definition

Amount of monthly administrative expenses agreed under the related party transaction.

+ References

No definition available.

+ Details

Name:	biii_RelatedPartyTransactionMonthlyAdministrativeExpenses
Namespace Prefix:	biii_
Data Type:	xbrli:monetaryItemType
Balance Type:	debit
Period Type:	duration

- Definition

Number of new units issued during the period.

+ References

No definition available.

+ Details

Name:	biii_StockIssuedDuringPeriodUnitsNewIssues
Namespace Prefix:	biii_
Data Type:	xbrli:sharesItemType
Balance Type:	na
Period Type:	duration

- Definition

The total amount of cash and securities held by third party trustees pursuant to terms of debt instruments or other agreements as of the date of each statement of financial position presented, which can be used by the trustee only to pay the noncurrent portion of specified obligations.

+ References

Reference 1: http://www.xbrl.org/2009/role/commonPracticeRef
-Topic 860
-SubTopic 30
-Name Accounting Standards Codification
-Section S50
-Paragraph 1
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147480031/860-30-S50-1

Reference 2: http://www.xbrl.org/2009/role/commonPracticeRef
-Topic 235
-SubTopic 10
-Name Accounting Standards Codification
-Section S99
-Paragraph 1
-Subparagraph (SX 210.4-08(b))
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147480678/235-10-S99-1

Reference 3: http://www.xbrl.org/2009/role/commonPracticeRef
-Topic 235
-SubTopic 10
-Name Accounting Standards Codification
-Section S50
-Paragraph 1
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147480738/235-10-S50-1

+ Details

Name:	us-gaap_AssetsHeldInTrust
Namespace Prefix:	us-gaap_
Data Type:	xbrli:monetaryItemType
Balance Type:	debit
Period Type:	instant

- Definition

Sum of the carrying values as of the balance sheet date of the portions of long-term notes payable due within one year or the operating cycle if longer.

+ References

Reference 1: http://www.xbrl.org/2009/role/commonPracticeRef
-Topic 210
-SubTopic 10
-Name Accounting Standards Codification
-Section S99
-Paragraph 1
-Subparagraph (SX 210.5-02(20))
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147480566/210-10-S99-1

+ Details

Name:	us-gaap_NotesPayableCurrent
Namespace Prefix:	us-gaap_
Data Type:	xbrli:monetaryItemType
Balance Type:	credit
Period Type:	instant

- Definition

Indicates status and type of related party for notes payable classified as current.

+ References

Reference 1: http://www.xbrl.org/2009/role/commonPracticeRef
-Topic 850
-SubTopic 10
-Name Accounting Standards Codification
-Section 50
-Paragraph 3
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147483326/850-10-50-3

+ Details

Name:	us-gaap_NotesPayableCurrentRelatedPartyTypeExtensibleEnumeration
Namespace Prefix:	us-gaap_
Data Type:	enum2:enumerationSetItemType
Balance Type:	na
Period Type:	instant

- Definition

Amount of liabilities classified as other, due within one year or the normal operating cycle, if longer.

+ References

Reference 1: http://www.xbrl.org/2003/role/disclosureRef
-Topic 210
-SubTopic 10
-Name Accounting Standards Codification
-Section S50
-Paragraph 8
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147480657/210-10-S50-8

Reference 2: http://www.xbrl.org/2003/role/disclosureRef
-Topic 210
-SubTopic 10
-Name Accounting Standards Codification
-Section S99
-Paragraph 1
-Subparagraph (SX 210.5-02(19)(a)(5))
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147480566/210-10-S99-1

Reference 3: http://www.xbrl.org/2003/role/disclosureRef
-Topic 210
-SubTopic 10
-Name Accounting Standards Codification
-Section S99
-Paragraph 1
-Subparagraph (SX 210.5-02(20))
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147480566/210-10-S99-1

Reference 4: http://www.xbrl.org/2003/role/exampleRef
-Topic 852
-SubTopic 10
-Name Accounting Standards Codification
-Section 55
-Paragraph 10
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147481372/852-10-55-10

+ Details

Name:	us-gaap_OtherLiabilitiesCurrent
Namespace Prefix:	us-gaap_
Data Type:	xbrli:monetaryItemType
Balance Type:	credit
Period Type:	instant

- Definition

Indicates status and type of related party for liability classified as other and current.

+ References

Reference 1: http://www.xbrl.org/2003/role/disclosureRef
-Topic 210
-SubTopic 10
-Name Accounting Standards Codification
-Section S50
-Paragraph 8
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147480657/210-10-S50-8

Reference 2: http://www.xbrl.org/2003/role/disclosureRef
-Topic 210
-SubTopic 10
-Name Accounting Standards Codification
-Section S99
-Paragraph 1
-Subparagraph (SX 210.5-02(19)(a)(5))
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147480566/210-10-S99-1

Reference 3: http://www.xbrl.org/2009/role/commonPracticeRef
-Topic 850
-SubTopic 10
-Name Accounting Standards Codification
-Section 50
-Paragraph 3
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147483326/850-10-50-3

+ Details

Name:	us-gaap_OtherLiabilityCurrentRelatedPartyTypeExtensibleEnumeration
Namespace Prefix:	us-gaap_
Data Type:	enum2:enumerationSetItemType
Balance Type:	na
Period Type:	instant

- Definition

The cash inflow associated with the amount received from entity's raising of capital via private rather than public placement.

+ References

+ Details

Name:	us-gaap_ProceedsFromIssuanceOfPrivatePlacement
Namespace Prefix:	us-gaap_
Data Type:	xbrli:monetaryItemType
Balance Type:	debit
Period Type:	duration

- Definition

The cash inflow from the issuance of common stock, preferred stock, treasury stock, stock options, and other types of equity.

+ References

Reference 1: http://fasb.org/us-gaap/role/ref/legacyRef
-Name Accounting Standards Codification
-Section 45
-Paragraph 14
-Subparagraph (a)
-SubTopic 10
-Topic 230
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147482740/230-10-45-14

Reference 2: http://www.xbrl.org/2003/role/disclosureRef
-Topic 946
-SubTopic 10
-Name Accounting Standards Codification
-Section S99
-Paragraph 3
-Subparagraph (SX 210.6-03(i)(1))
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147479886/946-10-S99-3

+ Details

Name:	us-gaap_ProceedsFromIssuanceOrSaleOfEquity
Namespace Prefix:	us-gaap_
Data Type:	xbrli:monetaryItemType
Balance Type:	debit
Period Type:	duration

- Definition

Per share or per unit amount of equity securities issued.

+ References

No definition available.

+ Details

Name:	us-gaap_SharesIssuedPricePerShare
Namespace Prefix:	us-gaap_
Data Type:	dtr-types:perShareItemType
Balance Type:	na
Period Type:	instant

- Definition

Detail information of subsequent event by type. User is expected to use existing line items from elsewhere in the taxonomy as the primary line items for this disclosure, which is further associated with dimension and member elements pertaining to a subsequent event.

+ References

Reference 1: http://www.xbrl.org/2003/role/disclosureRef
-Topic 830
-SubTopic 30
-Name Accounting Standards Codification
-Section 50
-Paragraph 2
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147481674/830-30-50-2

Reference 2: http://www.xbrl.org/2003/role/disclosureRef
-Topic 855
-SubTopic 10
-Name Accounting Standards Codification
-Section 50
-Paragraph 2
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147483399/855-10-50-2

+ Details

Name:	us-gaap_SubsequentEventLineItems
Namespace Prefix:	us-gaap_
Data Type:	xbrli:stringItemType
Balance Type:	na
Period Type:	duration

- Details

Name:	us-gaap_RelatedPartyTransactionAxis=biii_RelatedPartyUnsecuredPromissoryNoteMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

- Details

Name:	us-gaap_RelatedPartyTransactionsByRelatedPartyAxis=us-gaap_RelatedPartyMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

- Details

Name:	us-gaap_RelatedPartyTransactionsByRelatedPartyAxis=biii_SponsorOrAffiliateOfSponsorMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

- Details

Name:	us-gaap_RelatedPartyTransactionAxis=biii_AdministrativeServicesAgreementMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

- Details

Name:	us-gaap_SubsidiarySaleOfStockAxis=biii_InitialPublicOfferingAndOverAllotmentOptionMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

- Details

Name:	us-gaap_SubsidiarySaleOfStockAxis=us-gaap_OverAllotmentOptionMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

- Details

Name:	us-gaap_ClassOfWarrantOrRightAxis=biii_PrivatePlacementWarrantsMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

- Details

Name:	us-gaap_SubsidiarySaleOfStockAxis=us-gaap_PrivatePlacementMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

- Details

Name:	srt_CounterpartyNameAxis=biii_UnderwritersMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

- Details

Name:	us-gaap_RelatedPartyTransactionsByRelatedPartyAxis=biii_SponsorMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

- Details

Name:	us-gaap_SubsequentEventTypeAxis=us-gaap_SubsequentEventMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

- Details

Name:	us-gaap_RelatedPartyTransactionAxis=biii_FoundersShareMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

- Details

Name:	us-gaap_StatementClassOfStockAxis=us-gaap_CommonClassBMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

SUBSEQUENT EVENTS (Details) - USD ($)			4 Months Ended	6 Months Ended	12 Months Ended
SUBSEQUENT EVENTS (Details) - USD ($)	Jan. 07, 2026	Jan. 05, 2026	Dec. 31, 2025	Jun. 30, 2026	Dec. 31, 2025
SUBSEQUENT EVENTS
Warrants issued
Gross proceeds from issuance of warrants				$ 4,075,000
Amount held in trust account	$ 172,500,000
Promissory note - related party			$ 123,988		$ 123,988
Due to related party			3,275,000		3,275,000
Other Liability, Current, Related Party [Extensible Enumeration]	Related party
Deferred underwriting fees	$ 6,876,000
Promissory note
SUBSEQUENT EVENTS
Promissory note - related party			123,988	123,988	123,988
Related party
SUBSEQUENT EVENTS
Promissory note - related party			123,988		123,988
Due to related party			3,275,000		$ 3,275,000
Sponsor or an affiliate \| Administrative Services Agreement
SUBSEQUENT EVENTS
Monthly expenses			$ 20,000	$ 20,000
IPO and over allotment
SUBSEQUENT EVENTS
Number of units issued	17,250,000
Price per unit	$ 10
Gross proceeds from issuance of units	$ 172,500,000
Over allotment
SUBSEQUENT EVENTS
Number of units issued	2,250,000
Price per unit		$ 10
Underwriting discount paid	$ 2,292,000
Cash underwriting discount not charged	$ 8,000
Deferred fee per unit	$ 0.4
Deferred underwriting fees	$ 6,876,000
Deferred fee not charged	$ 24,000
Over allotment \| Private placement warrants
SUBSEQUENT EVENTS
Warrants issued	1,150,000
Issue price per warrant	$ 0.5
Gross proceeds from issuance of warrants	$ 575,000
Private placement \| Private placement warrants
SUBSEQUENT EVENTS
Warrants issued	8,150,000
Issue price per warrant	$ 0.5
Gross proceeds from issuance of warrants	$ 4,075,000
Private placement \| Private placement warrants \| Underwriter
SUBSEQUENT EVENTS
Warrants issued	1,150,000
Private placement \| Private placement warrants \| Sponsor
SUBSEQUENT EVENTS
Warrants issued	7,000,000
Subsequent events
SUBSEQUENT EVENTS
Amount held in trust account	$ 172,500,000
Due to related party	72,804
Deferred underwriting fees	6,876,000
Subsequent events \| Promissory note
SUBSEQUENT EVENTS
Promissory note - related party	$ 123,988
Notes Payable, Current, Related Party [Extensible Enumeration]	Related party
Subsequent events \| Sponsor \| Founder share \| Class B ordinary shares
SUBSEQUENT EVENTS
Number of shares no longer subject to forfeiture	750,000
Subsequent events \| Sponsor or an affiliate \| Administrative Services Agreement
SUBSEQUENT EVENTS
Monthly expenses		$ 20,000
Subsequent events \| Private placement warrants \| Sponsor
SUBSEQUENT EVENTS
Warrants issued	7,000,000
Subsequent events \| IPO and over allotment
SUBSEQUENT EVENTS
Number of units issued	17,250,000
Price per unit	$ 10
Gross proceeds from issuance of units	$ 172,500,000
Amount held in trust account	$ 172,500,000
Subsequent events \| Over allotment
SUBSEQUENT EVENTS
Number of units issued	2,250,000
Price per unit		$ 10
Underwriting discount paid	$ 2,292,000
Cash underwriting discount not charged	$ 8,000
Deferred fee per unit	$ 0.4
Deferred underwriting fees	$ 6,876,000
Deferred fee not charged	$ 24,000
Subsequent events \| Over allotment \| Private placement warrants
SUBSEQUENT EVENTS
Warrants issued	1,150,000
Issue price per warrant	$ 0.5
Gross proceeds from issuance of warrants	$ 575,000
Subsequent events \| Private placement \| Private placement warrants
SUBSEQUENT EVENTS
Warrants issued	8,150,000
Issue price per warrant	$ 0.5
Gross proceeds from issuance of warrants	$ 4,075,000
Subsequent events \| Private placement \| Private placement warrants \| Underwriter
SUBSEQUENT EVENTS
Warrants issued	1,150,000
Subsequent events \| Private placement \| Private placement warrants \| Sponsor
SUBSEQUENT EVENTS
Warrants issued	7,000,000