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<!--XBRL document created with Toppan Merrill Bridge  10.6.2.5 -->
<!--Based on XBRL 2.1-->
<!--Created on: 6/5/2024 9:15:06 PM-->
<!--Modified on: 6/5/2024 9:15:06 PM-->
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_UnitsEachConsistingOfOneShareOfClassaCommonStockAndOneRedeemableWarrantToPurchaseOneShareOfClassCommonStockFor11.50PerShareMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">This member represents Units, each consisting of one share of class A common stock and one redeemable warrant to purchase one share of class A common stock for $11.50 per share.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_UnitsEachConsistingOfOneShareOfClassaCommonStockAndOneRedeemableWarrantToPurchaseOneShareOfClassCommonStockFor11.50PerShareMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Units Each Consisting Of One Share Of Class A Common Stock And One Redeemable Warrant To Purchase One Share Of Class A Common Stock For $11.50 Per Share [Member]</label>
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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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    <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>
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    <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" />
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    <loc xlink:type="locator" xlink:href="https://xbrl.sec.gov/dei/2023/dei-2023.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/2023/elts/us-gaap-2023.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 CONSOLIDATED BALANCE SHEETS</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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="brkh-20240331.xsd#brkh_CommonClassaSubjectToRedemptionMember" xlink:label="brkh_CommonClassaSubjectToRedemptionMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_CommonClassaSubjectToRedemptionMember" xlink:to="brkh_CommonClassaSubjectToRedemptionMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_CommonClassaSubjectToRedemptionMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">This member stands for classification of common stock representing ownership interest in a corporation that is subject to redemption.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_CommonClassaSubjectToRedemptionMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Common Class a Subject To Redemption [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_CommonClassaSubjectToRedemptionMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class A common stock subject to possible redemption</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="brkh_CommonClassaSubjectToRedemptionMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class A common stock subject to redemption</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_CommonClassaNotSubjectToRedemptionMember" xlink:label="brkh_CommonClassaNotSubjectToRedemptionMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_CommonClassaNotSubjectToRedemptionMember" xlink:to="brkh_CommonClassaNotSubjectToRedemptionMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_CommonClassaNotSubjectToRedemptionMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">This member stands for classification of common stock representing ownership interest in a corporation that is not subject to redemption.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_CommonClassaNotSubjectToRedemptionMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Common Class a Not Subject To Redemption [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_CommonClassaNotSubjectToRedemptionMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Non-redeemable Class A common stock</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="brkh_CommonClassaNotSubjectToRedemptionMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class A common stock not subject to possible redemption</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_CommonClassBMember" xlink:label="us-gaap_CommonClassBMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_CommonClassBMember" xlink:to="us-gaap_CommonClassBMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_CommonClassBMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Common Class B [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_CommonClassBMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class B Common Stock</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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/verboseLabel" 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/2023/elts/us-gaap-2023.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/2023/elts/us-gaap-2023.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/2023/elts/us-gaap-2023.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 held outside the Trust Account</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/periodEndLabel" xlink:label="us-gaap_CashAndCashEquivalentsAtCarryingValue_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_CashAndCashEquivalentsAtCarryingValue_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cash - Beginning of period</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_CashAndCashEquivalentsAtCarryingValue_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cash</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_RestrictedCashCurrent" xlink:label="us-gaap_RestrictedCashCurrent" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_RestrictedCashCurrent" xlink:to="us-gaap_RestrictedCashCurrent_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_RestrictedCashCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Restricted Cash, Current</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/periodEndLabel" xlink:label="us-gaap_RestrictedCashCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Restricted Cash - End of Period</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/periodStartLabel" xlink:label="us-gaap_RestrictedCashCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Restricted Cash - Beginning of period</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_RestrictedCashCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Restricted Cash</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_DueFromSponsorCurrent" xlink:label="brkh_DueFromSponsorCurrent" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_DueFromSponsorCurrent" xlink:to="brkh_DueFromSponsorCurrent_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_DueFromSponsorCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents the amount of receivables to be collected from an entity that is controlling, under the control of, or within the same control group as the reporting entity, within one year.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_DueFromSponsorCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Due From Sponsor, Current</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_DueFromSponsorCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Due from sponsor</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_AssetsHeldInTrustCurrent" xlink:label="us-gaap_AssetsHeldInTrustCurrent" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_AssetsHeldInTrustCurrent" xlink:to="us-gaap_AssetsHeldInTrustCurrent_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_AssetsHeldInTrustCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Asset, Held-in-Trust, Current</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_AssetsHeldInTrustCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Trust Account - Restricted for Redeeming Shareholders</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_PrepaidExpenseAndOtherAssetsCurrent" xlink:label="us-gaap_PrepaidExpenseAndOtherAssetsCurrent" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_PrepaidExpenseAndOtherAssetsCurrent" xlink:to="us-gaap_PrepaidExpenseAndOtherAssetsCurrent_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_PrepaidExpenseAndOtherAssetsCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Prepaid Expense and Other Assets, Current</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_PrepaidExpenseAndOtherAssetsCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Prepaid expenses and other Assets</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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/2023/elts/us-gaap-2023.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">Assets Held-in-trust, Noncurrent</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_AssetsHeldInTrustNoncurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Investments held in Trust Account</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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/2023/elts/us-gaap-2023.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/verboseLabel" xlink:label="us-gaap_LiabilitiesAndStockholdersEquityAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Liabilities and Stockholders' Deficit</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_LiabilitiesCurrentAbstract" xlink:label="us-gaap_LiabilitiesCurrentAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_LiabilitiesCurrentAbstract" xlink:to="us-gaap_LiabilitiesCurrentAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_LiabilitiesCurrentAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Liabilities, Current [Abstract]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_LiabilitiesCurrentAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Current liabilities:</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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="brkh-20240331.xsd#brkh_FranchiseTaxPayable" xlink:label="brkh_FranchiseTaxPayable" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_FranchiseTaxPayable" xlink:to="brkh_FranchiseTaxPayable_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_FranchiseTaxPayable_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Carrying value as of the balance sheet date of obligations incurred and payable, pertaining to franchise tax.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_FranchiseTaxPayable_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Franchise Tax Payable</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_FranchiseTaxPayable_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Franchise tax payable</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_DueToTrustAccountCurrent" xlink:label="brkh_DueToTrustAccountCurrent" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_DueToTrustAccountCurrent" xlink:to="brkh_DueToTrustAccountCurrent_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_DueToTrustAccountCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents the amount of payables to be paid from an entity that is controlling, under the control of, or within the same control group as the reporting entity, within one year.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_DueToTrustAccountCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Due To Trust Account, Current</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_DueToTrustAccountCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Due to Trust Account</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_PayableCommonStockRedeemed" xlink:label="us-gaap_PayableCommonStockRedeemed" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_PayableCommonStockRedeemed" xlink:to="us-gaap_PayableCommonStockRedeemed_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_PayableCommonStockRedeemed_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Payable, Common Stock Redeemed</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_PayableCommonStockRedeemed_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Redeemed stock payable to public stockholders</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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, Related Parties, 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">Convertible promissory note - related party</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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">Other Liabilities, Current</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_OtherLiabilitiesCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Advances from sponsor</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 owed to the Sponsor</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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, Type [Extensible Enumeration]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_AccruedIncomeTaxesCurrent" xlink:label="us-gaap_AccruedIncomeTaxesCurrent" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_AccruedIncomeTaxesCurrent" xlink:to="us-gaap_AccruedIncomeTaxesCurrent_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_AccruedIncomeTaxesCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Accrued Income Taxes, Current</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_AccruedIncomeTaxesCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Income tax payable</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_SalesAndExciseTaxPayableCurrent" xlink:label="us-gaap_SalesAndExciseTaxPayableCurrent" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_SalesAndExciseTaxPayableCurrent" xlink:to="us-gaap_SalesAndExciseTaxPayableCurrent_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_SalesAndExciseTaxPayableCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Sales and Excise Tax Payable, Current</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_SalesAndExciseTaxPayableCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Excise Tax Payable</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_SalesAndExciseTaxPayableCurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Excise tax liability recorded</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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="brkh-20240331.xsd#brkh_DeferredOfferingCostsNoncurrent" xlink:label="brkh_DeferredOfferingCostsNoncurrent" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_DeferredOfferingCostsNoncurrent" xlink:to="brkh_DeferredOfferingCostsNoncurrent_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_DeferredOfferingCostsNoncurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The carrying value as of balance sheet date of underwriting fees payable or deferred, classified as noncurrent.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_DeferredOfferingCostsNoncurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Deferred Offering Costs Non-Current</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_DeferredOfferingCostsNoncurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Deferred underwriting commissions</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="brkh_DeferredOfferingCostsNoncurrent_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Deferred underwriting fee payable</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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/2023/elts/us-gaap-2023.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/verboseLabel" 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/2023/elts/us-gaap-2023.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</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 common stock subject to possible redemption, 4,345,663 shares at redemption value of approximately $11.00 and $10.81 as of March 31, 2024 and December 31, 2023, respectively</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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">Stockholders' Equity Attributable to Parent [Abstract]</label>
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    <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>
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    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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">Stockholders' 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>
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    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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 and Stockholders' Deficit</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_TemporaryEquitySharesOutstanding" xlink:label="us-gaap_TemporaryEquitySharesOutstanding" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_TemporaryEquitySharesOutstanding" xlink:to="us-gaap_TemporaryEquitySharesOutstanding_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_TemporaryEquitySharesOutstanding_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Temporary Equity, Shares Outstanding</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_TemporaryEquitySharesOutstanding_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class A common stock subject to possible redemption, outstanding (in shares)</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_TemporaryEquitySharesOutstanding_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of shares outstanding</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_TemporaryEquityRedemptionPricePerShare" xlink:label="us-gaap_TemporaryEquityRedemptionPricePerShare" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_TemporaryEquityRedemptionPricePerShare" xlink:to="us-gaap_TemporaryEquityRedemptionPricePerShare_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_TemporaryEquityRedemptionPricePerShare_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Temporary Equity, Redemption Price Per Share</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_TemporaryEquityRedemptionPricePerShare_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Redemption price (per share)</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_TemporaryEquityRedemptionPricePerShare_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class A common stock subject to possible redemption, redemption value (per share)</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_PreferredStockParOrStatedValuePerShare" xlink:label="us-gaap_PreferredStockParOrStatedValuePerShare" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_PreferredStockParOrStatedValuePerShare" xlink:to="us-gaap_PreferredStockParOrStatedValuePerShare_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_PreferredStockParOrStatedValuePerShare_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Preferred Stock, Par or Stated Value Per Share</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_PreferredStockParOrStatedValuePerShare_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Preferred stock, par value, (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/2023/elts/us-gaap-2023.xsd#us-gaap_PreferredStockSharesAuthorized" xlink:label="us-gaap_PreferredStockSharesAuthorized" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_PreferredStockSharesAuthorized" xlink:to="us-gaap_PreferredStockSharesAuthorized_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_PreferredStockSharesAuthorized_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Preferred Stock, Shares Authorized</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_PreferredStockSharesAuthorized_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Preferred stock, shares authorized</label>
    <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 shares, shares authorized</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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">Preferred stock, 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 shares, shares outstanding</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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">Preferred stock, 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 shares, shares issued</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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">Common stock, par value (per share)</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_CommonStockParOrStatedValuePerShare_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Common shares, par value (in dollars per share)</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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">Common stock, shares authorized</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_CommonStockSharesAuthorized_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Common shares, shares authorized (in shares)</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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">Common stock, shares outstanding</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_CommonStockSharesOutstanding_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Common shares, shares outstanding (in shares)</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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/terseLabel" 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/verboseLabel" xlink:label="us-gaap_CommonStockSharesIssued_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Common shares, shares issued (in shares)</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_IncomeStatementAbstract" xlink:label="us-gaap_IncomeStatementAbstract" 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_IncomeStatementAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">CONDENSED CONSOLIDATED STATEMENTS OF OPERATIONS</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_OperatingCostsAndExpenses" xlink:label="us-gaap_OperatingCostsAndExpenses" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_OperatingCostsAndExpenses" xlink:to="us-gaap_OperatingCostsAndExpenses_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_OperatingCostsAndExpenses_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Operating Costs and Expenses</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_OperatingCostsAndExpenses_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Operating costs</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_FranchiseTaxExpense" xlink:label="brkh_FranchiseTaxExpense" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_FranchiseTaxExpense" xlink:to="brkh_FranchiseTaxExpense_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_FranchiseTaxExpense_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amount of franchise tax expense incurred during the period.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_FranchiseTaxExpense_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Franchise Tax Expense</label>
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    <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/2023/elts/us-gaap-2023.xsd#us-gaap_NonoperatingIncomeExpenseAbstract" xlink:label="us-gaap_NonoperatingIncomeExpenseAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_NonoperatingIncomeExpenseAbstract" xlink:to="us-gaap_NonoperatingIncomeExpenseAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_NonoperatingIncomeExpenseAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Nonoperating Income (Expense) [Abstract]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_NonoperatingIncomeExpenseAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Other income:</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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 and investments 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 income earned on Trust</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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/2023/elts/us-gaap-2023.xsd#us-gaap_IncomeLossFromContinuingOperationsBeforeIncomeTaxesExtraordinaryItemsNoncontrollingInterest" xlink:label="us-gaap_IncomeLossFromContinuingOperationsBeforeIncomeTaxesExtraordinaryItemsNoncontrollingInterest" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_IncomeLossFromContinuingOperationsBeforeIncomeTaxesExtraordinaryItemsNoncontrollingInterest" xlink:to="us-gaap_IncomeLossFromContinuingOperationsBeforeIncomeTaxesExtraordinaryItemsNoncontrollingInterest_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_IncomeLossFromContinuingOperationsBeforeIncomeTaxesExtraordinaryItemsNoncontrollingInterest_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Income (Loss) from Continuing Operations before Income Taxes, Noncontrolling Interest</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:label="us-gaap_IncomeLossFromContinuingOperationsBeforeIncomeTaxesExtraordinaryItemsNoncontrollingInterest_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Income before provision for income taxes</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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/2009/role/negatedLabel" xlink:label="us-gaap_IncomeTaxExpenseBenefit_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Provision for income taxes</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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 (loss)</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_NetIncomeLoss_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Net income (loss)</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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">Weighted average shares outstanding, basic (in shares)</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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">Weighted average shares outstanding, diluted (in shares)</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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 common stock</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_EarningsPerShareBasic_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Basic net income (loss) per share (in dollars per share)</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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 common stock</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 (loss) per share (in dollars per share)</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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 CONSOLIDATED STATEMENTS OF CHANGES IN STOCKHOLDERS' DEFICIT</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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/2023/elts/us-gaap-2023.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/2023/elts/us-gaap-2023.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">Common Stock</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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/2023/elts/us-gaap-2023.xsd#us-gaap_IncreaseDecreaseInStockholdersEquityRollForward" xlink:label="us-gaap_IncreaseDecreaseInStockholdersEquityRollForward" 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_IncreaseDecreaseInStockholdersEquityRollForward_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Increase (Decrease) in Stockholders' Equity [Roll Forward]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_IncreaseDecreaseInStockholdersEquityRollForward_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Increase (decrease) in Stockholders' deficit</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_SharesIssued" xlink:label="us-gaap_SharesIssued" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_SharesIssued" xlink:to="us-gaap_SharesIssued_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_SharesIssued_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Shares, Issued</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/periodEndLabel" xlink:label="us-gaap_SharesIssued_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Balance at the end (in shares)</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/periodStartLabel" xlink:label="us-gaap_SharesIssued_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Balance at the beginning (in shares)</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_EquityFairValueStockholderNonRedemptionAgreement" xlink:label="brkh_EquityFairValueStockholderNonRedemptionAgreement" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_EquityFairValueStockholderNonRedemptionAgreement" xlink:to="brkh_EquityFairValueStockholderNonRedemptionAgreement_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_EquityFairValueStockholderNonRedemptionAgreement_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The amount of fair value of equity transferrable to the Non-Redeeming Stockholders pursuant to the Non-Redemption Agreement.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_EquityFairValueStockholderNonRedemptionAgreement_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Equity Fair Value, Stockholder Non-Redemption Agreement</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_EquityFairValueStockholderNonRedemptionAgreement_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Aggregate fair value</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="brkh_EquityFairValueStockholderNonRedemptionAgreement_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Stockholder Non-Redemption Agreement</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_EquityStockholderNonRedemptionAgreement" xlink:label="brkh_EquityStockholderNonRedemptionAgreement" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_EquityStockholderNonRedemptionAgreement" xlink:to="brkh_EquityStockholderNonRedemptionAgreement_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_EquityStockholderNonRedemptionAgreement_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The amount of fair value of equity transferrable to the Non-Redeeming Stockholders pursuant to the Non-Redemption Agreement.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_EquityStockholderNonRedemptionAgreement_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Equity, Stockholder Non-Redemption Agreement</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2009/role/negatedLabel" xlink:label="brkh_EquityStockholderNonRedemptionAgreement_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Stockholder Non-Redemption Agreement</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_ExciseTaxLiabilityArisingFromRedemptionOfOrdinaryShares" xlink:label="brkh_ExciseTaxLiabilityArisingFromRedemptionOfOrdinaryShares" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_ExciseTaxLiabilityArisingFromRedemptionOfOrdinaryShares" xlink:to="brkh_ExciseTaxLiabilityArisingFromRedemptionOfOrdinaryShares_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_ExciseTaxLiabilityArisingFromRedemptionOfOrdinaryShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The amount of excise tax liability arising from redemption of Class A shares.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_ExciseTaxLiabilityArisingFromRedemptionOfOrdinaryShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Excise Tax Liability Arising from Redemption of Ordinary Shares</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2009/role/negatedLabel" xlink:label="brkh_ExciseTaxLiabilityArisingFromRedemptionOfOrdinaryShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Excise tax liability arising from redemption of Class A shares</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_TemporaryEquityAccretionToRedemptionValueAdjustment" xlink:label="us-gaap_TemporaryEquityAccretionToRedemptionValueAdjustment" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_TemporaryEquityAccretionToRedemptionValueAdjustment" xlink:to="us-gaap_TemporaryEquityAccretionToRedemptionValueAdjustment_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_TemporaryEquityAccretionToRedemptionValueAdjustment_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Temporary Equity, Accretion to Redemption Value, Adjustment</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2009/role/negatedLabel" xlink:label="us-gaap_TemporaryEquityAccretionToRedemptionValueAdjustment_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Accretion for common stock subject to redemption amount</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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 CONSOLIDATED STATEMENTS OF CASH FLOWS</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_NetCashProvidedByUsedInOperatingActivitiesAbstract" xlink:label="us-gaap_NetCashProvidedByUsedInOperatingActivitiesAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_NetCashProvidedByUsedInOperatingActivitiesAbstract" xlink:to="us-gaap_NetCashProvidedByUsedInOperatingActivitiesAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_NetCashProvidedByUsedInOperatingActivitiesAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Net Cash Provided by (Used in) Operating Activities [Abstract]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_NetCashProvidedByUsedInOperatingActivitiesAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cash flows from operating activities:</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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/terseLabel" xlink:label="us-gaap_ProfitLoss_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Net income (loss)</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_AdjustmentsToReconcileNetIncomeLossToCashProvidedByUsedInOperatingActivitiesAbstract" xlink:label="us-gaap_AdjustmentsToReconcileNetIncomeLossToCashProvidedByUsedInOperatingActivitiesAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_AdjustmentsToReconcileNetIncomeLossToCashProvidedByUsedInOperatingActivitiesAbstract" xlink:to="us-gaap_AdjustmentsToReconcileNetIncomeLossToCashProvidedByUsedInOperatingActivitiesAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_AdjustmentsToReconcileNetIncomeLossToCashProvidedByUsedInOperatingActivitiesAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Adjustments to Reconcile Net Income (Loss) to Cash Provided by (Used in) Operating Activities [Abstract]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_AdjustmentsToReconcileNetIncomeLossToCashProvidedByUsedInOperatingActivitiesAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Adjustments to reconcile net income (loss) to net cash used in operating activities:</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_IncreaseDecreaseInOperatingCapitalAbstract" xlink:label="us-gaap_IncreaseDecreaseInOperatingCapitalAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_IncreaseDecreaseInOperatingCapitalAbstract" xlink:to="us-gaap_IncreaseDecreaseInOperatingCapitalAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_IncreaseDecreaseInOperatingCapitalAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Increase (Decrease) in Operating Capital [Abstract]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_IncreaseDecreaseInOperatingCapitalAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Changes in current assets and liabilities:</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_IncreaseDecreaseInOtherAssets" xlink:label="brkh_IncreaseDecreaseInOtherAssets" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_IncreaseDecreaseInOtherAssets" xlink:to="brkh_IncreaseDecreaseInOtherAssets_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_IncreaseDecreaseInOtherAssets_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The increase (decrease) during the reporting period of other assets.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_IncreaseDecreaseInOtherAssets_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Increase Decrease In Other Assets</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_IncreaseDecreaseInOtherAssets_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Other Assets</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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="brkh-20240331.xsd#brkh_IncreaseDecreaseInFranchiseTaxPayable" xlink:label="brkh_IncreaseDecreaseInFranchiseTaxPayable" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_IncreaseDecreaseInFranchiseTaxPayable" xlink:to="brkh_IncreaseDecreaseInFranchiseTaxPayable_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_IncreaseDecreaseInFranchiseTaxPayable_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The increase (decrease) during the reporting period of franchise tax payable.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_IncreaseDecreaseInFranchiseTaxPayable_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Increase (Decrease) in Franchise Tax Payable</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_IncreaseDecreaseInFranchiseTaxPayable_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Franchise tax payable</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_IncreaseDecreaseInAccruedIncomeTaxesPayable" xlink:label="us-gaap_IncreaseDecreaseInAccruedIncomeTaxesPayable" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_IncreaseDecreaseInAccruedIncomeTaxesPayable" xlink:to="us-gaap_IncreaseDecreaseInAccruedIncomeTaxesPayable_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_IncreaseDecreaseInAccruedIncomeTaxesPayable_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Increase (Decrease) in Income Taxes Payable</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_IncreaseDecreaseInAccruedIncomeTaxesPayable_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Income tax payable</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_NetCashProvidedByUsedInOperatingActivities" xlink:label="us-gaap_NetCashProvidedByUsedInOperatingActivities" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_NetCashProvidedByUsedInOperatingActivities" xlink:to="us-gaap_NetCashProvidedByUsedInOperatingActivities_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_NetCashProvidedByUsedInOperatingActivities_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Net Cash Provided by (Used in) Operating Activities</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/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/2023/elts/us-gaap-2023.xsd#us-gaap_NetCashProvidedByUsedInInvestingActivitiesAbstract" xlink:label="us-gaap_NetCashProvidedByUsedInInvestingActivitiesAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_NetCashProvidedByUsedInInvestingActivitiesAbstract" xlink:to="us-gaap_NetCashProvidedByUsedInInvestingActivitiesAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_NetCashProvidedByUsedInInvestingActivitiesAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Net Cash Provided by (Used in) Investing Activities [Abstract]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_NetCashProvidedByUsedInInvestingActivitiesAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cash Flows from Investing Activities:</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_ProceedsFromInvestmentIntoTrustAccountToPayFranchiseAndIncomeTaxes" xlink:label="brkh_ProceedsFromInvestmentIntoTrustAccountToPayFranchiseAndIncomeTaxes" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_ProceedsFromInvestmentIntoTrustAccountToPayFranchiseAndIncomeTaxes" xlink:to="brkh_ProceedsFromInvestmentIntoTrustAccountToPayFranchiseAndIncomeTaxes_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_ProceedsFromInvestmentIntoTrustAccountToPayFranchiseAndIncomeTaxes_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The amount of cash inflow associated with proceeds from investments into Trust Account to pay franchise and income taxes.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_ProceedsFromInvestmentIntoTrustAccountToPayFranchiseAndIncomeTaxes_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Proceeds from Investment into Trust Account to Pay Franchise and Income Taxes</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_ProceedsFromInvestmentIntoTrustAccountToPayFranchiseAndIncomeTaxes_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cash withdrawn from Trust Account to pay franchise and income taxes</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_PaymentsForDepositsIntoTrustAccountForExtension" xlink:label="brkh_PaymentsForDepositsIntoTrustAccountForExtension" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_PaymentsForDepositsIntoTrustAccountForExtension" xlink:to="brkh_PaymentsForDepositsIntoTrustAccountForExtension_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_PaymentsForDepositsIntoTrustAccountForExtension_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The cash outflow for the deposits into trust account for extension.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_PaymentsForDepositsIntoTrustAccountForExtension_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Payments for Deposits into Trust Account for Extension</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2009/role/negatedLabel" xlink:label="brkh_PaymentsForDepositsIntoTrustAccountForExtension_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Principal deposits into Trust Account for Extension</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_ProceedsFromInvestmentIntoTrustAccountInConnectionWithRedemptions" xlink:label="brkh_ProceedsFromInvestmentIntoTrustAccountInConnectionWithRedemptions" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_ProceedsFromInvestmentIntoTrustAccountInConnectionWithRedemptions" xlink:to="brkh_ProceedsFromInvestmentIntoTrustAccountInConnectionWithRedemptions_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_ProceedsFromInvestmentIntoTrustAccountInConnectionWithRedemptions_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The amount of cash inflow associated with proceeds from investments into Trust Account in connection with redemptions.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_ProceedsFromInvestmentIntoTrustAccountInConnectionWithRedemptions_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Proceeds from Investment into Trust Account In Connection with Redemptions</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_ProceedsFromInvestmentIntoTrustAccountInConnectionWithRedemptions_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cash withdrawn from Trust Account in connection with redemptions</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_NetCashProvidedByUsedInInvestingActivities" xlink:label="us-gaap_NetCashProvidedByUsedInInvestingActivities" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_NetCashProvidedByUsedInInvestingActivities" xlink:to="us-gaap_NetCashProvidedByUsedInInvestingActivities_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_NetCashProvidedByUsedInInvestingActivities_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Net Cash Provided by (Used in) Investing Activities</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:label="us-gaap_NetCashProvidedByUsedInInvestingActivities_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Net cash provided by investing activities</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_NetCashProvidedByUsedInFinancingActivitiesAbstract" xlink:label="us-gaap_NetCashProvidedByUsedInFinancingActivitiesAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_NetCashProvidedByUsedInFinancingActivitiesAbstract" xlink:to="us-gaap_NetCashProvidedByUsedInFinancingActivitiesAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_NetCashProvidedByUsedInFinancingActivitiesAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Net Cash Provided by (Used in) Financing Activities [Abstract]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_NetCashProvidedByUsedInFinancingActivitiesAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cash flows from financing activities:</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_ProceedsFromRelatedPartyDebt" xlink:label="us-gaap_ProceedsFromRelatedPartyDebt" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ProceedsFromRelatedPartyDebt" xlink:to="us-gaap_ProceedsFromRelatedPartyDebt_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ProceedsFromRelatedPartyDebt_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Proceeds from Related Party Debt</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ProceedsFromRelatedPartyDebt_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Proceeds from Advances from sponsor</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_ProceedsFromConvertibleDebt" xlink:label="us-gaap_ProceedsFromConvertibleDebt" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ProceedsFromConvertibleDebt" xlink:to="us-gaap_ProceedsFromConvertibleDebt_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ProceedsFromConvertibleDebt_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Proceeds from Convertible Debt</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_ProceedsFromConvertibleDebt_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Proceeds from convertible promissory note - related party</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_PaymentsForRedemptionOfCommonStock" xlink:label="brkh_PaymentsForRedemptionOfCommonStock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_PaymentsForRedemptionOfCommonStock" xlink:to="brkh_PaymentsForRedemptionOfCommonStock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_PaymentsForRedemptionOfCommonStock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The cash outflow to redeem common stock during the period.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_PaymentsForRedemptionOfCommonStock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Payments for Redemption of Common Stock</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_PaymentsForRedemptionOfCommonStock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Redemption of Common Stock</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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">Net Cash Provided by (Used in) Financing Activities</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:label="us-gaap_NetCashProvidedByUsedInFinancingActivities_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Net cash used in financing activities</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalentsPeriodIncreaseDecreaseIncludingExchangeRateEffect" xlink:label="us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalentsPeriodIncreaseDecreaseIncludingExchangeRateEffect" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalentsPeriodIncreaseDecreaseIncludingExchangeRateEffect" xlink:to="us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalentsPeriodIncreaseDecreaseIncludingExchangeRateEffect_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalentsPeriodIncreaseDecreaseIncludingExchangeRateEffect_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cash, Cash Equivalents, Restricted Cash, and Restricted Cash Equivalents, Period Increase (Decrease), Including Exchange Rate Effect</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:label="us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalentsPeriodIncreaseDecreaseIncludingExchangeRateEffect_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Net change in cash and restricted cash</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalents" xlink:label="us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalents" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalents" xlink:to="us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalents_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalents_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cash, Cash Equivalents, Restricted Cash, and Restricted Cash Equivalents</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/periodEndLabel" xlink:label="us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalents_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cash and restricted cash, end of the period</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/periodStartLabel" xlink:label="us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalents_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cash and restricted cash, beginning of the period</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalents_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amount in operating bank accounts</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_SupplementalCashFlowElementsAbstract" xlink:label="us-gaap_SupplementalCashFlowElementsAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_SupplementalCashFlowElementsAbstract" xlink:to="us-gaap_SupplementalCashFlowElementsAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_SupplementalCashFlowElementsAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Supplemental disclosure of cash flow information:</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_IncomeTaxesPaidNet" xlink:label="us-gaap_IncomeTaxesPaidNet" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_IncomeTaxesPaidNet" xlink:to="us-gaap_IncomeTaxesPaidNet_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_IncomeTaxesPaidNet_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Income Taxes Paid, Net</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_IncomeTaxesPaidNet_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cash Paid for Income Taxes</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_CashFlowNoncashInvestingAndFinancingActivitiesDisclosureAbstract" xlink:label="us-gaap_CashFlowNoncashInvestingAndFinancingActivitiesDisclosureAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_CashFlowNoncashInvestingAndFinancingActivitiesDisclosureAbstract" xlink:to="us-gaap_CashFlowNoncashInvestingAndFinancingActivitiesDisclosureAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_CashFlowNoncashInvestingAndFinancingActivitiesDisclosureAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cash Flow, Noncash Investing and Financing Activities Disclosure [Abstract]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_CashFlowNoncashInvestingAndFinancingActivitiesDisclosureAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Supplemental disclosure of non-cash investing and financing activities:</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_ExciseTaxLiabilityArisingFromRedemptionOfClassShares" xlink:label="brkh_ExciseTaxLiabilityArisingFromRedemptionOfClassShares" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_ExciseTaxLiabilityArisingFromRedemptionOfClassShares" xlink:to="brkh_ExciseTaxLiabilityArisingFromRedemptionOfClassShares_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_ExciseTaxLiabilityArisingFromRedemptionOfClassShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amount of excise tax liability arising from redemption of Class A shares.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_ExciseTaxLiabilityArisingFromRedemptionOfClassShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Excise Tax Liability Arising From Redemption Of Class A shares</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_ExciseTaxLiabilityArisingFromRedemptionOfClassShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Excise tax liability arising from redemption of Class A shares</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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 of Class A common stock subject to possible redemption</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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 disclosure of reconciliation of Restricted Cash:</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">Reconciliation of Cash and Restricted Cash:</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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">Organization, Business Operation, Liquidity and Capital Resources</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_NatureOfOperations" xlink:label="us-gaap_NatureOfOperations" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_NatureOfOperations" xlink:to="us-gaap_NatureOfOperations_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_NatureOfOperations_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Nature of Operations [Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_NatureOfOperations_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Organization, Business Operation, Liquidity and Capital Resources</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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/2023/elts/us-gaap-2023.xsd#us-gaap_BasisOfPresentationAndSignificantAccountingPoliciesTextBlock" xlink:label="us-gaap_BasisOfPresentationAndSignificantAccountingPoliciesTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_BasisOfPresentationAndSignificantAccountingPoliciesTextBlock" xlink:to="us-gaap_BasisOfPresentationAndSignificantAccountingPoliciesTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_BasisOfPresentationAndSignificantAccountingPoliciesTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Basis of Presentation and Significant Accounting Policies [Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_BasisOfPresentationAndSignificantAccountingPoliciesTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">BASIS OF PRESENTATION AND SUMMARY OF SIGNIFICANT ACCOUNTING POLICIES</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_InitialPublicOfferingAbstract" xlink:label="brkh_InitialPublicOfferingAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_InitialPublicOfferingAbstract" xlink:to="brkh_InitialPublicOfferingAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_InitialPublicOfferingAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">No definition available.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_InitialPublicOfferingAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">INITIAL PUBLIC OFFERING</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_InitialPublicOfferingTextBlock" xlink:label="brkh_InitialPublicOfferingTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_InitialPublicOfferingTextBlock" xlink:to="brkh_InitialPublicOfferingTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_InitialPublicOfferingTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The entire disclosure on information about initial public offering.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_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="brkh_InitialPublicOfferingTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">INITIAL PUBLIC OFFERING</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_PrivatePlacementAbstract" xlink:label="brkh_PrivatePlacementAbstract" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_PrivatePlacementAbstract" xlink:to="brkh_PrivatePlacementAbstract_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_PrivatePlacementAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">No definition available.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_PrivatePlacementAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">PRIVATE PLACEMENT</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_PrivatePlacementTextBlock" xlink:label="brkh_PrivatePlacementTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_PrivatePlacementTextBlock" xlink:to="brkh_PrivatePlacementTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_PrivatePlacementTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The entire disclosure on information about private placement.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_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="brkh_PrivatePlacementTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">PRIVATE PLACEMENT</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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" />
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    <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/2023/elts/us-gaap-2023.xsd#us-gaap_StockholdersEquityNoteAbstract" xlink:label="us-gaap_StockholdersEquityNoteAbstract" 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_StockholdersEquityNoteAbstract_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">STOCKHOLDERS' DEFICIT</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_StockholdersEquityNoteDisclosureTextBlock" xlink:label="us-gaap_StockholdersEquityNoteDisclosureTextBlock" 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">STOCKHOLDERS' DEFICIT</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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/2023/elts/us-gaap-2023.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" />
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    <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/2023/elts/us-gaap-2023.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" />
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    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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/2023/elts/us-gaap-2023.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="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_ConsolidationPolicyTextBlock" xlink:label="us-gaap_ConsolidationPolicyTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ConsolidationPolicyTextBlock" xlink:to="us-gaap_ConsolidationPolicyTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ConsolidationPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Consolidation, Policy [Policy Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ConsolidationPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Principles of Consolidation</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_EmergingGrowthCompanyPolicyPolicyTextBlock" xlink:label="brkh_EmergingGrowthCompanyPolicyPolicyTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_EmergingGrowthCompanyPolicyPolicyTextBlock" xlink:to="brkh_EmergingGrowthCompanyPolicyPolicyTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_EmergingGrowthCompanyPolicyPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Disclosure of accounting policy for Emerging Growth Company.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_EmergingGrowthCompanyPolicyPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Emerging Growth Company Policy [Policy Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_EmergingGrowthCompanyPolicyPolicyTextBlock_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/2023/elts/us-gaap-2023.xsd#us-gaap_UseOfEstimates" xlink:label="us-gaap_UseOfEstimates" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_UseOfEstimates" xlink:to="us-gaap_UseOfEstimates_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_UseOfEstimates_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Use of Estimates, Policy [Policy Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_UseOfEstimates_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Use of Estimates</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_CashAndCashEquivalentsPolicyTextBlock" xlink:label="us-gaap_CashAndCashEquivalentsPolicyTextBlock" 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_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="brkh-20240331.xsd#brkh_AssetsHeldInTrustAccountPolicyPolicyTextBlock" xlink:label="brkh_AssetsHeldInTrustAccountPolicyPolicyTextBlock" 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="brkh_AssetsHeldInTrustAccountPolicyPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Assets Held In Trust Account, Policy [Policy Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_AssetsHeldInTrustAccountPolicyPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Investments Held in Trust Account</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_TemporaryEquityPolicyPolicyTextBlock" xlink:label="brkh_TemporaryEquityPolicyPolicyTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_TemporaryEquityPolicyPolicyTextBlock" xlink:to="brkh_TemporaryEquityPolicyPolicyTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_TemporaryEquityPolicyPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Disclosure of accounting policy for temporary equity.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_TemporaryEquityPolicyPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Temporary Equity, Policy [Policy Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_TemporaryEquityPolicyPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Common Stock Subject to Possible Redemption</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_DeferredChargesPolicyTextBlock" xlink:label="us-gaap_DeferredChargesPolicyTextBlock" 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_DeferredChargesPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Deferred Charges, Policy [Policy Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_DeferredChargesPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Offering Costs associated with the Initial Public Offering</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_IncomeTaxPolicyTextBlock" xlink:label="us-gaap_IncomeTaxPolicyTextBlock" 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_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>
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    <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 or Loss Per Common Stock</label>
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    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ConcentrationRiskCreditRisk_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Concentration Risk, Credit Risk, Policy [Policy Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ConcentrationRiskCreditRisk_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Concentration of Credit Risk</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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/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="brkh-20240331.xsd#brkh_WarrantClassificationPolicyTextBlock" xlink:label="brkh_WarrantClassificationPolicyTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_WarrantClassificationPolicyTextBlock" xlink:to="brkh_WarrantClassificationPolicyTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_WarrantClassificationPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Disclosure of accounting policy for warrant classification.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_WarrantClassificationPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Warrant Classification [Policy Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_WarrantClassificationPolicyTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Warrant Classification</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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/2023/elts/us-gaap-2023.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">Reconciliation of Net Loss per Common Share</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 reconciliation of basic and diluted net income per common stock</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_ScheduleOfCondensedBalanceSheetAreReconciledTableTextBlock" xlink:label="brkh_ScheduleOfCondensedBalanceSheetAreReconciledTableTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_ScheduleOfCondensedBalanceSheetAreReconciledTableTextBlock" xlink:to="brkh_ScheduleOfCondensedBalanceSheetAreReconciledTableTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_ScheduleOfCondensedBalanceSheetAreReconciledTableTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Tabular disclosure of condensed balance sheet are reconciled.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_ScheduleOfCondensedBalanceSheetAreReconciledTableTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Schedule Of Condensed Balance Sheet Are Reconciled [Table Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_ScheduleOfCondensedBalanceSheetAreReconciledTableTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Schedule of common stock subject to redemption reflected on the balance sheet</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_ScheduleOfFairValueAssetsAndLiabilitiesMeasuredOnRecurringBasisTableTextBlock" xlink:label="us-gaap_ScheduleOfFairValueAssetsAndLiabilitiesMeasuredOnRecurringBasisTableTextBlock" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ScheduleOfFairValueAssetsAndLiabilitiesMeasuredOnRecurringBasisTableTextBlock" xlink:to="us-gaap_ScheduleOfFairValueAssetsAndLiabilitiesMeasuredOnRecurringBasisTableTextBlock_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ScheduleOfFairValueAssetsAndLiabilitiesMeasuredOnRecurringBasisTableTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Schedule of Fair Value, Assets and Liabilities Measured on Recurring Basis [Table Text Block]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ScheduleOfFairValueAssetsAndLiabilitiesMeasuredOnRecurringBasisTableTextBlock_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Schedule of company's assets and liabilities that were measured at fair value on a recurring basis</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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 by Subsidiary or Equity Investee [Table]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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 Party [Axis]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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 Party [Domain]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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="brkh-20240331.xsd#brkh_SponsorMember" xlink:label="brkh_SponsorMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_SponsorMember" xlink:to="brkh_SponsorMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_SponsorMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">This member stands for sponsor.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_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="brkh_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/2023/elts/us-gaap-2023.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/2023/elts/us-gaap-2023.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="brkh-20240331.xsd#brkh_PromissoryNoteWithRelatedPartyMember" xlink:label="brkh_PromissoryNoteWithRelatedPartyMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_PromissoryNoteWithRelatedPartyMember" xlink:to="brkh_PromissoryNoteWithRelatedPartyMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_PromissoryNoteWithRelatedPartyMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">This member stands for promissory note with related party.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_PromissoryNoteWithRelatedPartyMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Promissory Note with Related Party [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_PromissoryNoteWithRelatedPartyMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Promissory Note with Related Party</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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/2023/elts/us-gaap-2023.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/2023/elts/us-gaap-2023.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">Initial Public Offering</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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 option</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_TypeOfArrangementAxis" xlink:label="us-gaap_TypeOfArrangementAxis" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_TypeOfArrangementAxis" xlink:to="us-gaap_TypeOfArrangementAxis_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_TypeOfArrangementAxis_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Collaborative Arrangement and Arrangement Other than Collaborative [Axis]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_ArrangementsAndNonarrangementTransactionsMember" xlink:label="us-gaap_ArrangementsAndNonarrangementTransactionsMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ArrangementsAndNonarrangementTransactionsMember" xlink:to="us-gaap_ArrangementsAndNonarrangementTransactionsMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ArrangementsAndNonarrangementTransactionsMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Collaborative Arrangement and Arrangement Other than Collaborative [Domain]</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_NonRedemptionAgreementsMember" xlink:label="brkh_NonRedemptionAgreementsMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_NonRedemptionAgreementsMember" xlink:to="brkh_NonRedemptionAgreementsMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_NonRedemptionAgreementsMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents the member information pertaining to non-redemption agreements.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_NonRedemptionAgreementsMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Non-Redemption Agreements [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_NonRedemptionAgreementsMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Non-Redemption Agreements</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_SecondTrustAmendmentMember" xlink:label="brkh_SecondTrustAmendmentMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_SecondTrustAmendmentMember" xlink:to="brkh_SecondTrustAmendmentMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_SecondTrustAmendmentMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents the member information pertaining to second trust amendment.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_SecondTrustAmendmentMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Second Trust Amendment [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_SecondTrustAmendmentMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Second Trust Amendment</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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/2023/elts/us-gaap-2023.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="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_WarrantMember" xlink:label="us-gaap_WarrantMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_WarrantMember" xlink:to="us-gaap_WarrantMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_WarrantMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Warrant [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_WarrantMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Warrants</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_PrivatePlacementWarrantsMember" xlink:label="brkh_PrivatePlacementWarrantsMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_PrivatePlacementWarrantsMember" xlink:to="brkh_PrivatePlacementWarrantsMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_PrivatePlacementWarrantsMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents a redeemable warrant (Private Placement Warrant) that entitles the holder to purchase shares of common stock if the underwriter's option is exercised in full.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_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="brkh_PrivatePlacementWarrantsMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Private Placement Units</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_PublicWarrantsMember" xlink:label="brkh_PublicWarrantsMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_PublicWarrantsMember" xlink:to="brkh_PublicWarrantsMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_PublicWarrantsMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents a redeemable warrant (Public Warrant) that entitles the holder to purchase shares of common stock subject to adjustment.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_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="brkh_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/srt/2023/elts/srt-2023.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/2023/elts/srt-2023.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="brkh-20240331.xsd#brkh_EfHuttonMember" xlink:label="brkh_EfHuttonMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_EfHuttonMember" xlink:to="brkh_EfHuttonMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_EfHuttonMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents the member information pertaining to EF Hutton.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_EfHuttonMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">EF Hutton [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_EfHuttonMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">EF Hutton</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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">BASIS OF PRESENTATION AND SUMMARY OF SIGNIFICANT ACCOUNTING POLICIES</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_ConditionForFutureBusinessCombinationNumberOfBusinessesMinimum" xlink:label="brkh_ConditionForFutureBusinessCombinationNumberOfBusinessesMinimum" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_ConditionForFutureBusinessCombinationNumberOfBusinessesMinimum" xlink:to="brkh_ConditionForFutureBusinessCombinationNumberOfBusinessesMinimum_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_ConditionForFutureBusinessCombinationNumberOfBusinessesMinimum_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The minimum number of businesses which the reporting entity must acquire with the net proceeds of the offering.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_ConditionForFutureBusinessCombinationNumberOfBusinessesMinimum_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Condition For Future Business Combination Number Of Businesses Minimum</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_ConditionForFutureBusinessCombinationNumberOfBusinessesMinimum_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Condition for future business combination number of businesses minimum</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_NumberOfSubsidiaries" xlink:label="brkh_NumberOfSubsidiaries" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_NumberOfSubsidiaries" xlink:to="brkh_NumberOfSubsidiaries_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_NumberOfSubsidiaries_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents the number of subsidiaries.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_NumberOfSubsidiaries_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number Of Subsidiaries</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_NumberOfSubsidiaries_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Wholly owned subsidiary</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_UnitsIssuedDuringPeriodSharesNewIssues" xlink:label="brkh_UnitsIssuedDuringPeriodSharesNewIssues" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_UnitsIssuedDuringPeriodSharesNewIssues" xlink:to="brkh_UnitsIssuedDuringPeriodSharesNewIssues_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_UnitsIssuedDuringPeriodSharesNewIssues_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="brkh_UnitsIssuedDuringPeriodSharesNewIssues_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Units Issued During Period, Shares, New Issues</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_UnitsIssuedDuringPeriodSharesNewIssues_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Sale of units, net of underwriting discounts (in shares)</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="brkh_UnitsIssuedDuringPeriodSharesNewIssues_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of units sold</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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">Representative shares per share</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_MinimumNetTangibleAssetsRequiredToConsummateBusinessCombination" xlink:label="brkh_MinimumNetTangibleAssetsRequiredToConsummateBusinessCombination" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_MinimumNetTangibleAssetsRequiredToConsummateBusinessCombination" xlink:to="brkh_MinimumNetTangibleAssetsRequiredToConsummateBusinessCombination_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_MinimumNetTangibleAssetsRequiredToConsummateBusinessCombination_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Minimum net tangible assets required to consummate business combination.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_MinimumNetTangibleAssetsRequiredToConsummateBusinessCombination_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Minimum Net Tangible Assets Required to Consummate Business Combination</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_MinimumNetTangibleAssetsRequiredToConsummateBusinessCombination_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Consummation of business combination</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_NumberOfSharesIssuedPerUnit" xlink:label="brkh_NumberOfSharesIssuedPerUnit" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_NumberOfSharesIssuedPerUnit" xlink:to="brkh_NumberOfSharesIssuedPerUnit_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_NumberOfSharesIssuedPerUnit_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents the number of shares in a unit.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_NumberOfSharesIssuedPerUnit_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of Shares Issued Per Unit</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_NumberOfSharesIssuedPerUnit_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of shares in a unit</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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 issuable per warrant</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_ClassOfWarrantOrRightNumberOfSecuritiesCalledByEachWarrantOrRight_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of shares per warrant</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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 of warrants</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_ClassOfWarrantOrRightExercisePriceOfWarrantsOrRights1_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Exercise price of warrant</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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/verboseLabel" xlink:label="us-gaap_ClassOfWarrantOrRightNumberOfSecuritiesCalledByWarrantsOrRights_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of warrants to purchase shares issued</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_ClassOfWarrantOrRightPriceOfWarrantsOrRights" xlink:label="brkh_ClassOfWarrantOrRightPriceOfWarrantsOrRights" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_ClassOfWarrantOrRightPriceOfWarrantsOrRights" xlink:to="brkh_ClassOfWarrantOrRightPriceOfWarrantsOrRights_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_ClassOfWarrantOrRightPriceOfWarrantsOrRights_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Price per share or per unit of warrants or rights outstanding.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_ClassOfWarrantOrRightPriceOfWarrantsOrRights_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class of Warrant or Right, Price of Warrants or Rights</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_ClassOfWarrantOrRightPriceOfWarrantsOrRights_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Price of warrant</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="brkh_ClassOfWarrantOrRightPriceOfWarrantsOrRights_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Price of warrants</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_ProceedsFromIssuanceOfWarrants" xlink:label="us-gaap_ProceedsFromIssuanceOfWarrants" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ProceedsFromIssuanceOfWarrants" xlink:to="us-gaap_ProceedsFromIssuanceOfWarrants_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ProceedsFromIssuanceOfWarrants_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Proceeds from Issuance of Warrants</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ProceedsFromIssuanceOfWarrants_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Proceeds allocated to public warrants</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_ProceedsFromIssuanceOfWarrants_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Aggregate purchase price</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_TransactionCosts" xlink:label="brkh_TransactionCosts" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_TransactionCosts" xlink:to="brkh_TransactionCosts_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_TransactionCosts_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents the amount of transaction costs incurred.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_TransactionCosts_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Transaction Costs</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_TransactionCosts_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Transaction costs</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:label="brkh_TransactionCosts_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Transaction costs</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_SaleOfStockUnderwritingFees" xlink:label="brkh_SaleOfStockUnderwritingFees" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_SaleOfStockUnderwritingFees" xlink:to="brkh_SaleOfStockUnderwritingFees_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_SaleOfStockUnderwritingFees_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents the amount of offering fees incurred and paid for underwriters.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_SaleOfStockUnderwritingFees_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Sale of Stock, Underwriting Fees</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_SaleOfStockUnderwritingFees_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Underwriting fees</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="brkh_SaleOfStockUnderwritingFees_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Underwriting commissions</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_FairValueOfSharesIssuedToUnderwriters" xlink:label="brkh_FairValueOfSharesIssuedToUnderwriters" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_FairValueOfSharesIssuedToUnderwriters" xlink:to="brkh_FairValueOfSharesIssuedToUnderwriters_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_FairValueOfSharesIssuedToUnderwriters_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents the fair value of shares issued to the underwriters.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_FairValueOfSharesIssuedToUnderwriters_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fair Value Of Shares Issued To Underwriters</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_FairValueOfSharesIssuedToUnderwriters_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fair value of class A shares issued to underwriters</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_SaleOfStockOtherOfferingCosts" xlink:label="brkh_SaleOfStockOtherOfferingCosts" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_SaleOfStockOtherOfferingCosts" xlink:to="brkh_SaleOfStockOtherOfferingCosts_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_SaleOfStockOtherOfferingCosts_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents the amount of other offering costs incurred.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_SaleOfStockOtherOfferingCosts_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Sale of Stock, Other Offering Costs</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_SaleOfStockOtherOfferingCosts_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Other offering costs</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_CashPaymentAgreedInLieuOfFullDeferredUnderwritingCommission" xlink:label="brkh_CashPaymentAgreedInLieuOfFullDeferredUnderwritingCommission" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_CashPaymentAgreedInLieuOfFullDeferredUnderwritingCommission" xlink:to="brkh_CashPaymentAgreedInLieuOfFullDeferredUnderwritingCommission_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_CashPaymentAgreedInLieuOfFullDeferredUnderwritingCommission_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amount of cash payment agreed in lieu of full deferred underwriting commissions.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_CashPaymentAgreedInLieuOfFullDeferredUnderwritingCommission_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cash Payment Agreed in Lieu of Full Deferred Underwriting Commission</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_CashPaymentAgreedInLieuOfFullDeferredUnderwritingCommission_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cash payment agreed in lieu of full deferred underwriting commission</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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">Assets 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">Cash held outside of Trust Account</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_ConditionForFutureBusinessCombinationUseOfProceedsPercentage" xlink:label="brkh_ConditionForFutureBusinessCombinationUseOfProceedsPercentage" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_ConditionForFutureBusinessCombinationUseOfProceedsPercentage" xlink:to="brkh_ConditionForFutureBusinessCombinationUseOfProceedsPercentage_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_ConditionForFutureBusinessCombinationUseOfProceedsPercentage_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The threshold percentage of the assets held in the trust account funded by proceeds from the offering which must be used for purposes of consummating a business combination.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_ConditionForFutureBusinessCombinationUseOfProceedsPercentage_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Condition for future business combination use of proceeds percentage</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_ConditionForFutureBusinessCombinationUseOfProceedsPercentage_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Condition for future business combination use of proceeds percentage</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_PaymentsForInvestmentOfCashInTrustAccount" xlink:label="brkh_PaymentsForInvestmentOfCashInTrustAccount" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_PaymentsForInvestmentOfCashInTrustAccount" xlink:to="brkh_PaymentsForInvestmentOfCashInTrustAccount_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_PaymentsForInvestmentOfCashInTrustAccount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The amount of cash outflow for investment of cash in trust account.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_PaymentsForInvestmentOfCashInTrustAccount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Payments for investment of cash in Trust Account</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="brkh_PaymentsForInvestmentOfCashInTrustAccount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cash deposited into trust account</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_InvestmentsMaximumMaturityTerm" xlink:label="brkh_InvestmentsMaximumMaturityTerm" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_InvestmentsMaximumMaturityTerm" xlink:to="brkh_InvestmentsMaximumMaturityTerm_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_InvestmentsMaximumMaturityTerm_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The maximum maturity term of investments, 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="brkh_InvestmentsMaximumMaturityTerm_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Investments Maximum Maturity Term</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_InvestmentsMaximumMaturityTerm_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Investments, maximum maturity term</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_MaximumAllowedDissolutionExpenses" xlink:label="brkh_MaximumAllowedDissolutionExpenses" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_MaximumAllowedDissolutionExpenses" xlink:to="brkh_MaximumAllowedDissolutionExpenses_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_MaximumAllowedDissolutionExpenses_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The maximum amount permitted to be paid for dissolution expenses if a business combination is not completed within the specified period.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_MaximumAllowedDissolutionExpenses_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Maximum Allowed Dissolution Expenses</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_MaximumAllowedDissolutionExpenses_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Maximum allowed dissolution expenses</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_PercentageObligationToRedeemPublicSharesIfEntityDoesNotCompleteBusinessCombination" xlink:label="brkh_PercentageObligationToRedeemPublicSharesIfEntityDoesNotCompleteBusinessCombination" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_PercentageObligationToRedeemPublicSharesIfEntityDoesNotCompleteBusinessCombination" xlink:to="brkh_PercentageObligationToRedeemPublicSharesIfEntityDoesNotCompleteBusinessCombination_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_PercentageObligationToRedeemPublicSharesIfEntityDoesNotCompleteBusinessCombination_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents the percentage of shares which the reporting entity is obligated to redeem if a business combination is not consummated using the offering proceeds within a specified period.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_PercentageObligationToRedeemPublicSharesIfEntityDoesNotCompleteBusinessCombination_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Percentage Obligation To Redeem Public Shares If Entity Does Not Complete A Business Combination</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_PercentageObligationToRedeemPublicSharesIfEntityDoesNotCompleteBusinessCombination_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Obligation to redeem public shares if entity does not complete a business combination (as a percent)</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_MonthsToCompleteAcquisitions" xlink:label="brkh_MonthsToCompleteAcquisitions" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_MonthsToCompleteAcquisitions" xlink:to="brkh_MonthsToCompleteAcquisitions_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_MonthsToCompleteAcquisitions_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents the anticipated business timeline in months for completing a business combination.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_MonthsToCompleteAcquisitions_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Months To Complete Acquisitions</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_MonthsToCompleteAcquisitions_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Months to complete acquisition</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_MaximumNumberOfSharesNotForRedemption" xlink:label="brkh_MaximumNumberOfSharesNotForRedemption" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_MaximumNumberOfSharesNotForRedemption" xlink:to="brkh_MaximumNumberOfSharesNotForRedemption_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_MaximumNumberOfSharesNotForRedemption_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Maximum number of stock not for redemption such as non-redeemed shares.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_MaximumNumberOfSharesNotForRedemption_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Maximum Number of Shares Not for Redemption</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_MaximumNumberOfSharesNotForRedemption_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Non-Redeemed shares</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_MaximumNumberOfSharesTransferredInExchangeForNotForRedemptionOfShares" xlink:label="brkh_MaximumNumberOfSharesTransferredInExchangeForNotForRedemptionOfShares" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_MaximumNumberOfSharesTransferredInExchangeForNotForRedemptionOfShares" xlink:to="brkh_MaximumNumberOfSharesTransferredInExchangeForNotForRedemptionOfShares_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_MaximumNumberOfSharesTransferredInExchangeForNotForRedemptionOfShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Maximum number of stock transferred to exchange for the foregoing commitments not to redeem such non-redeemed shares.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_MaximumNumberOfSharesTransferredInExchangeForNotForRedemptionOfShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Maximum Number of Shares Transferred In Exchange for Not for Redemption of Shares</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_MaximumNumberOfSharesTransferredInExchangeForNotForRedemptionOfShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Maximum number of shares transferred in exchange for non-redeemed shares</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_TemporaryEquityStockRedeemedDuringPeriodShares" xlink:label="brkh_TemporaryEquityStockRedeemedDuringPeriodShares" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_TemporaryEquityStockRedeemedDuringPeriodShares" xlink:to="brkh_TemporaryEquityStockRedeemedDuringPeriodShares_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_TemporaryEquityStockRedeemedDuringPeriodShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of temporary stock redeemed during the period.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_TemporaryEquityStockRedeemedDuringPeriodShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Temporary Equity, Stock Redeemed During Period, Shares</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_TemporaryEquityStockRedeemedDuringPeriodShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of shares redeemed</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_TemporaryEquityStockRedeemedDuringPeriodValue" xlink:label="brkh_TemporaryEquityStockRedeemedDuringPeriodValue" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_TemporaryEquityStockRedeemedDuringPeriodValue" xlink:to="brkh_TemporaryEquityStockRedeemedDuringPeriodValue_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_TemporaryEquityStockRedeemedDuringPeriodValue_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Value of temporary stock redeemed during the period.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_TemporaryEquityStockRedeemedDuringPeriodValue_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Temporary Equity, Stock Redeemed During Period, Value</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_TemporaryEquityStockRedeemedDuringPeriodValue_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Value of shares redeemed</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_NumberOfTimesEntityHasRightToExtendBusinessCombinationForAdditionalMonth" xlink:label="brkh_NumberOfTimesEntityHasRightToExtendBusinessCombinationForAdditionalMonth" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_NumberOfTimesEntityHasRightToExtendBusinessCombinationForAdditionalMonth" xlink:to="brkh_NumberOfTimesEntityHasRightToExtendBusinessCombinationForAdditionalMonth_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_NumberOfTimesEntityHasRightToExtendBusinessCombinationForAdditionalMonth_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of times the entity has right to extend business combination for additional month.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_NumberOfTimesEntityHasRightToExtendBusinessCombinationForAdditionalMonth_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of Times the Entity has Right to Extend Business Combination for Additional Month</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_NumberOfTimesEntityHasRightToExtendBusinessCombinationForAdditionalMonth_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of times the entity has right to extend business combination for additional month</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_AdditionalTermToCompleteBusinessCombination" xlink:label="brkh_AdditionalTermToCompleteBusinessCombination" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_AdditionalTermToCompleteBusinessCombination" xlink:to="brkh_AdditionalTermToCompleteBusinessCombination_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_AdditionalTermToCompleteBusinessCombination_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Additional time company can take to complete business combination.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_AdditionalTermToCompleteBusinessCombination_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Additional term to complete business combination</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_AdditionalTermToCompleteBusinessCombination_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Additional term to complete business combination</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_AmountPerUnredeemedShareDepositedIntoTrustAccount" xlink:label="brkh_AmountPerUnredeemedShareDepositedIntoTrustAccount" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_AmountPerUnredeemedShareDepositedIntoTrustAccount" xlink:to="brkh_AmountPerUnredeemedShareDepositedIntoTrustAccount_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_AmountPerUnredeemedShareDepositedIntoTrustAccount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amount to be deposited per unredeemed share into the trust account.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_AmountPerUnredeemedShareDepositedIntoTrustAccount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amount per Unredeemed Share Deposited into Trust Account</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_AmountPerUnredeemedShareDepositedIntoTrustAccount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amount per unredeemed share deposited into trust account</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_DepositToTrustAccountExtensionPaymentPerMonth" xlink:label="brkh_DepositToTrustAccountExtensionPaymentPerMonth" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_DepositToTrustAccountExtensionPaymentPerMonth" xlink:to="brkh_DepositToTrustAccountExtensionPaymentPerMonth_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_DepositToTrustAccountExtensionPaymentPerMonth_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The amount to be paid each month for extension as per the second trust amendment.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_DepositToTrustAccountExtensionPaymentPerMonth_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Deposit to Trust Account, extension payment per month</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_DepositInTrustAccount" xlink:label="brkh_DepositInTrustAccount" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_DepositInTrustAccount" xlink:to="brkh_DepositInTrustAccount_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_DepositInTrustAccount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amount of deposit in the trust account .</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_DepositInTrustAccount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Deposit In Trust Account</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_DepositInTrustAccount_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Deposit in trust account</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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">Aggregate purchase price</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_LineOfCredit" xlink:label="us-gaap_LineOfCredit" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_LineOfCredit" xlink:to="us-gaap_LineOfCredit_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_LineOfCredit_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Long-Term Line of Credit</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_LineOfCredit_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Outstanding balance</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_ProceedsFromTrustAccountUsedToPayFranchiseAndIncomeTaxes" xlink:label="brkh_ProceedsFromTrustAccountUsedToPayFranchiseAndIncomeTaxes" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_ProceedsFromTrustAccountUsedToPayFranchiseAndIncomeTaxes" xlink:to="brkh_ProceedsFromTrustAccountUsedToPayFranchiseAndIncomeTaxes_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_ProceedsFromTrustAccountUsedToPayFranchiseAndIncomeTaxes_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The amount of cash inflow associated with proceeds from trust account used to pay franchise and income taxes.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_ProceedsFromTrustAccountUsedToPayFranchiseAndIncomeTaxes_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Proceeds From Trust Account Used To Pay Franchise And Income Taxes</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="brkh_ProceedsFromTrustAccountUsedToPayFranchiseAndIncomeTaxes_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Withdrawn from the interest earned in the Trust Account</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_CashWithdrawnFromTrustAccountUsedForOperatingExpenses" xlink:label="brkh_CashWithdrawnFromTrustAccountUsedForOperatingExpenses" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_CashWithdrawnFromTrustAccountUsedForOperatingExpenses" xlink:to="brkh_CashWithdrawnFromTrustAccountUsedForOperatingExpenses_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_CashWithdrawnFromTrustAccountUsedForOperatingExpenses_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amount of cash withdrawn from trust account used to pay operating expenses.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_CashWithdrawnFromTrustAccountUsedForOperatingExpenses_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cash Withdrawn from Trust Account Used for Operating Expenses</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_CashWithdrawnFromTrustAccountUsedForOperatingExpenses_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cash withdrawn from trust account</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_DisclosureOfAccountingPoliciesTable" xlink:label="brkh_DisclosureOfAccountingPoliciesTable" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_DisclosureOfAccountingPoliciesTable" xlink:to="brkh_DisclosureOfAccountingPoliciesTable_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_DisclosureOfAccountingPoliciesTable_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Schedule of accounting policies.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_DisclosureOfAccountingPoliciesTable_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Disclosure Of Accounting Policies [Table]</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_SummaryOfSignificantAccountingPoliciesLineItems" xlink:label="brkh_SummaryOfSignificantAccountingPoliciesLineItems" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_SummaryOfSignificantAccountingPoliciesLineItems" xlink:to="brkh_SummaryOfSignificantAccountingPoliciesLineItems_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_SummaryOfSignificantAccountingPoliciesLineItems_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Line items represent financial concepts included in a table. These concepts are used to disclose reportable information associated with domain members defined in one or many axes to the table.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_SummaryOfSignificantAccountingPoliciesLineItems_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Summary of Significant Accounting Policies [Line Items]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_SummaryOfSignificantAccountingPoliciesLineItems_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">BASIS OF PRESENTATION AND SUMMARY OF SIGNIFICANT ACCOUNTING POLICIES</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_CashEquivalentsAtCarryingValue" xlink:label="us-gaap_CashEquivalentsAtCarryingValue" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_CashEquivalentsAtCarryingValue" xlink:to="us-gaap_CashEquivalentsAtCarryingValue_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_CashEquivalentsAtCarryingValue_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cash Equivalents, at Carrying Value</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_CashEquivalentsAtCarryingValue_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Cash equivalents</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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/terseLabel" xlink:label="us-gaap_UnrecognizedTaxBenefits_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Unrecognized tax benefits</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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/terseLabel" xlink:label="us-gaap_UnrecognizedTaxBenefitsIncomeTaxPenaltiesAndInterestAccrued_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Unrecognized tax benefits accrued for interest and penalties</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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 securities attributable to warrants (in shares)</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_EffectiveIncomeTaxRateContinuingOperations" xlink:label="us-gaap_EffectiveIncomeTaxRateContinuingOperations" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_EffectiveIncomeTaxRateContinuingOperations" xlink:to="us-gaap_EffectiveIncomeTaxRateContinuingOperations_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_EffectiveIncomeTaxRateContinuingOperations_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Effective Income Tax Rate Reconciliation, Percent</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_EffectiveIncomeTaxRateContinuingOperations_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Effective income tax rate (in percent)</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_EffectiveIncomeTaxRateReconciliationAtFederalStatutoryIncomeTaxRate" xlink:label="us-gaap_EffectiveIncomeTaxRateReconciliationAtFederalStatutoryIncomeTaxRate" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_EffectiveIncomeTaxRateReconciliationAtFederalStatutoryIncomeTaxRate" xlink:to="us-gaap_EffectiveIncomeTaxRateReconciliationAtFederalStatutoryIncomeTaxRate_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_EffectiveIncomeTaxRateReconciliationAtFederalStatutoryIncomeTaxRate_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Effective Income Tax Rate Reconciliation, at Federal Statutory Income Tax Rate, Percent</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="us-gaap_EffectiveIncomeTaxRateReconciliationAtFederalStatutoryIncomeTaxRate_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Statutory federal income tax rate (in percent)</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_ClassBCommonStockNotSubjectToRedemptionMember" xlink:label="brkh_ClassBCommonStockNotSubjectToRedemptionMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_ClassBCommonStockNotSubjectToRedemptionMember" xlink:to="brkh_ClassBCommonStockNotSubjectToRedemptionMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_ClassBCommonStockNotSubjectToRedemptionMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">This member stands for classification of common stock representing ownership interest in a corporation that is not subject to redemption.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_ClassBCommonStockNotSubjectToRedemptionMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class B Common Stock Not Subject to Redemption [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_ClassBCommonStockNotSubjectToRedemptionMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Non-redeemable Class B common stock</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_TemporaryEquityNetIncome" xlink:label="us-gaap_TemporaryEquityNetIncome" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_TemporaryEquityNetIncome" xlink:to="us-gaap_TemporaryEquityNetIncome_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_TemporaryEquityNetIncome_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Temporary Equity, Net Income</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_TemporaryEquityNetIncome_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Allocation of net income (loss), common stock subject to redemption</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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/terseLabel" xlink:label="us-gaap_NetIncomeLossAvailableToCommonStockholdersBasic_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Allocation of net income (loss)</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_InitialPublicOfferingTable" xlink:label="brkh_InitialPublicOfferingTable" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_InitialPublicOfferingTable" xlink:to="brkh_InitialPublicOfferingTable_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_InitialPublicOfferingTable_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Schedule of initial public offering.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_InitialPublicOfferingTable_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">INITIAL PUBLIC OFFERING [Table]</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_InitialPublicOfferingLineItems" xlink:label="brkh_InitialPublicOfferingLineItems" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_InitialPublicOfferingLineItems" xlink:to="brkh_InitialPublicOfferingLineItems_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_InitialPublicOfferingLineItems_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Line items represent financial concepts included in a table. These concepts are used to disclose reportable information associated with domain members defined in one or many axes to the table.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_InitialPublicOfferingLineItems_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">INITIAL PUBLIC OFFERING [Line Items]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_InitialPublicOfferingLineItems_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">INITIAL PUBLIC OFFERING</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_UnitPrice" xlink:label="brkh_UnitPrice" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_UnitPrice" xlink:to="brkh_UnitPrice_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_UnitPrice_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Price of a single unit.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_UnitPrice_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Unit Price</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_UnitPrice_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Price per share</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_NumberOfWarrantsIssuedPerUnit" xlink:label="brkh_NumberOfWarrantsIssuedPerUnit" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_NumberOfWarrantsIssuedPerUnit" xlink:to="brkh_NumberOfWarrantsIssuedPerUnit_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_NumberOfWarrantsIssuedPerUnit_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents the number of warrants in a unit.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_NumberOfWarrantsIssuedPerUnit_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of Warrants Issued Per Unit</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_NumberOfWarrantsIssuedPerUnit_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="brkh-20240331.xsd#brkh_AccretionOfCarryingValueToRedemptionValue" xlink:label="brkh_AccretionOfCarryingValueToRedemptionValue" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_AccretionOfCarryingValueToRedemptionValue" xlink:to="brkh_AccretionOfCarryingValueToRedemptionValue_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_AccretionOfCarryingValueToRedemptionValue_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Accretion of carrying value to redemption value.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_AccretionOfCarryingValueToRedemptionValue_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Accretion Of Carrying Value To Redemption Value</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_AccretionOfCarryingValueToRedemptionValue_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Accretion of carrying value to redemption value</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_TemporaryEquityStockRedeemedOrCalledDuringPeriodValue" xlink:label="brkh_TemporaryEquityStockRedeemedOrCalledDuringPeriodValue" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_TemporaryEquityStockRedeemedOrCalledDuringPeriodValue" xlink:to="brkh_TemporaryEquityStockRedeemedOrCalledDuringPeriodValue_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_TemporaryEquityStockRedeemedOrCalledDuringPeriodValue_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The impact of the value of stock classified as temporary equity bought back by the entity at the exercise price or redemption price.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_TemporaryEquityStockRedeemedOrCalledDuringPeriodValue_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Temporary Equity, Stock Redeemed Or Called During Period, Value</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_TemporaryEquityStockRedeemedOrCalledDuringPeriodValue_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Redemptions</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_PrivatePlacementTable" xlink:label="brkh_PrivatePlacementTable" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_PrivatePlacementTable" xlink:to="brkh_PrivatePlacementTable_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_PrivatePlacementTable_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Schedule of private placement.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_PrivatePlacementTable_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">PRIVATE PLACEMENT [Table]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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="brkh-20240331.xsd#brkh_PrivatePlacementLineItems" xlink:label="brkh_PrivatePlacementLineItems" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_PrivatePlacementLineItems" xlink:to="brkh_PrivatePlacementLineItems_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_PrivatePlacementLineItems_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Line items represent financial concepts included in a table. These concepts are used to disclose reportable information associated with domain members defined in one or many axes to the table.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_PrivatePlacementLineItems_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Private placement [Line Items]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_PrivatePlacementLineItems_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">PRIVATE PLACEMENT</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_ThresholdPeriodForNotToTransferAssignOrSellAnySharesOrWarrantsAfterCompletionOfInitialBusinessCombination" xlink:label="brkh_ThresholdPeriodForNotToTransferAssignOrSellAnySharesOrWarrantsAfterCompletionOfInitialBusinessCombination" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_ThresholdPeriodForNotToTransferAssignOrSellAnySharesOrWarrantsAfterCompletionOfInitialBusinessCombination" xlink:to="brkh_ThresholdPeriodForNotToTransferAssignOrSellAnySharesOrWarrantsAfterCompletionOfInitialBusinessCombination_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_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="brkh_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="brkh_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>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="brkh_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 units after the completion of the initial business combination</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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">Schedule of Related Party Transactions, by Related Party [Table]</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_FounderSharesMember" xlink:label="brkh_FounderSharesMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_FounderSharesMember" xlink:to="brkh_FounderSharesMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_FounderSharesMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The member representing the founder shares.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_FounderSharesMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Founder Shares [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_FounderSharesMember_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/2023/elts/us-gaap-2023.xsd#us-gaap_RelatedPartyTransactionLineItems" xlink:label="us-gaap_RelatedPartyTransactionLineItems" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_RelatedPartyTransactionLineItems" xlink:to="us-gaap_RelatedPartyTransactionLineItems_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_RelatedPartyTransactionLineItems_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Related Party Transaction [Line Items]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_RelatedPartyTransactionLineItems_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">RELATED PARTY TRANSACTIONS</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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">Stock issued (in shares)</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_StockholdersEquityNoteStockSplitConversionRatio1" xlink:label="us-gaap_StockholdersEquityNoteStockSplitConversionRatio1" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_StockholdersEquityNoteStockSplitConversionRatio1" xlink:to="us-gaap_StockholdersEquityNoteStockSplitConversionRatio1_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_StockholdersEquityNoteStockSplitConversionRatio1_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Stockholders' Equity Note, Stock Split, Conversion Ratio</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_StockholdersEquityNoteStockSplitConversionRatio1_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Stock split</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_AggregateNumberOfSharesOwned" xlink:label="brkh_AggregateNumberOfSharesOwned" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_AggregateNumberOfSharesOwned" xlink:to="brkh_AggregateNumberOfSharesOwned_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_AggregateNumberOfSharesOwned_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The number of shares owned by the founders after the impact of the stock dividend.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_AggregateNumberOfSharesOwned_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Aggregate Number Of Shares Owned</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_AggregateNumberOfSharesOwned_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Aggregate number of shares owned</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_NumberOfSharesSubjectToForfeiture" xlink:label="brkh_NumberOfSharesSubjectToForfeiture" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_NumberOfSharesSubjectToForfeiture" xlink:to="brkh_NumberOfSharesSubjectToForfeiture_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_NumberOfSharesSubjectToForfeiture_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The number of shares owned by the founders subject to forfeiture if the underwriter overallotment option is not exercised in the proposed public offering.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_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="brkh_NumberOfSharesSubjectToForfeiture_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Shares subject to forfeiture</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_PercentageOfIssuedAndOutstandingSharesAfterInitialPublicOfferingCollectivelyHeldByInitialStockholders" xlink:label="brkh_PercentageOfIssuedAndOutstandingSharesAfterInitialPublicOfferingCollectivelyHeldByInitialStockholders" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_PercentageOfIssuedAndOutstandingSharesAfterInitialPublicOfferingCollectivelyHeldByInitialStockholders" xlink:to="brkh_PercentageOfIssuedAndOutstandingSharesAfterInitialPublicOfferingCollectivelyHeldByInitialStockholders_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_PercentageOfIssuedAndOutstandingSharesAfterInitialPublicOfferingCollectivelyHeldByInitialStockholders_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The expected ownership percentage by the founders after completion of the proposed public offering.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_PercentageOfIssuedAndOutstandingSharesAfterInitialPublicOfferingCollectivelyHeldByInitialStockholders_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Percentage Of Issued And Outstanding Shares After The Initial Public Offering Collectively Held By Initial Stockholders</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_PercentageOfIssuedAndOutstandingSharesAfterInitialPublicOfferingCollectivelyHeldByInitialStockholders_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Percentage of issued and outstanding shares after the Initial Public Offering collectively held by initial stockholders</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_TransferAssignOrSellAnySharesOrWarrantsAfterCompletionOfInitialBusinessCombinationStockPriceTrigger" xlink:label="brkh_TransferAssignOrSellAnySharesOrWarrantsAfterCompletionOfInitialBusinessCombinationStockPriceTrigger" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_TransferAssignOrSellAnySharesOrWarrantsAfterCompletionOfInitialBusinessCombinationStockPriceTrigger" xlink:to="brkh_TransferAssignOrSellAnySharesOrWarrantsAfterCompletionOfInitialBusinessCombinationStockPriceTrigger_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_TransferAssignOrSellAnySharesOrWarrantsAfterCompletionOfInitialBusinessCombinationStockPriceTrigger_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The share price threshold that must be achieved in order to waive the restriction on transfer of shares during a restricted period after a business combination.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_TransferAssignOrSellAnySharesOrWarrantsAfterCompletionOfInitialBusinessCombinationStockPriceTrigger_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Transfer, Assign Or Sell Any Shares Or Warrants After Completion Of Initial Business Combination, Stock Price Trigger</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_TransferAssignOrSellAnySharesOrWarrantsAfterCompletionOfInitialBusinessCombinationStockPriceTrigger_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Stock price trigger to transfer, assign or sell any shares or warrants of the company, after the completion of the initial business combination (in dollars per share)</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_ThresholdTradingDaysForTransferAssignOrSaleOfSharesOrWarrantsAfterCompletionOfInitialBusinessCombination" xlink:label="brkh_ThresholdTradingDaysForTransferAssignOrSaleOfSharesOrWarrantsAfterCompletionOfInitialBusinessCombination" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_ThresholdTradingDaysForTransferAssignOrSaleOfSharesOrWarrantsAfterCompletionOfInitialBusinessCombination" xlink:to="brkh_ThresholdTradingDaysForTransferAssignOrSaleOfSharesOrWarrantsAfterCompletionOfInitialBusinessCombination_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_ThresholdTradingDaysForTransferAssignOrSaleOfSharesOrWarrantsAfterCompletionOfInitialBusinessCombination_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold trading days for transfer, assign or sale of shares or warrants, after the completion of the initial business combination.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_ThresholdTradingDaysForTransferAssignOrSaleOfSharesOrWarrantsAfterCompletionOfInitialBusinessCombination_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold Trading Days For Transfer Assign Or Sale Of 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="brkh_ThresholdTradingDaysForTransferAssignOrSaleOfSharesOrWarrantsAfterCompletionOfInitialBusinessCombination_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold trading days for transfer, assign or sale of shares or warrants, after the completion of the initial business combination</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_ThresholdConsecutiveTradingDaysForTransferAssignOrSaleOfSharesOrWarrantsAfterCompletionOfInitialBusinessCombinations" xlink:label="brkh_ThresholdConsecutiveTradingDaysForTransferAssignOrSaleOfSharesOrWarrantsAfterCompletionOfInitialBusinessCombinations" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_ThresholdConsecutiveTradingDaysForTransferAssignOrSaleOfSharesOrWarrantsAfterCompletionOfInitialBusinessCombinations" xlink:to="brkh_ThresholdConsecutiveTradingDaysForTransferAssignOrSaleOfSharesOrWarrantsAfterCompletionOfInitialBusinessCombinations_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_ThresholdConsecutiveTradingDaysForTransferAssignOrSaleOfSharesOrWarrantsAfterCompletionOfInitialBusinessCombinations_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold consecutive trading days for transfer, assign or sale of shares or warrants, after the completion of the initial business combination.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_ThresholdConsecutiveTradingDaysForTransferAssignOrSaleOfSharesOrWarrantsAfterCompletionOfInitialBusinessCombinations_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold Consecutive Trading Days For Transfer Assign Or Sale Of Shares Or Warrants After Completion Of Initial Business Combinations</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_ThresholdConsecutiveTradingDaysForTransferAssignOrSaleOfSharesOrWarrantsAfterCompletionOfInitialBusinessCombinations_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold consecutive trading days for transfer, assign or sale of shares or warrants, after the completion of the initial business combination</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_AdministrativeSupportAgreementMember" xlink:label="brkh_AdministrativeSupportAgreementMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_AdministrativeSupportAgreementMember" xlink:to="brkh_AdministrativeSupportAgreementMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_AdministrativeSupportAgreementMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">This member stands for Administrative Support Agreement.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_AdministrativeSupportAgreementMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Administrative Support Agreement [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_AdministrativeSupportAgreementMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Administrative Support Agreement</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_RelatedPartyLoansMember" xlink:label="brkh_RelatedPartyLoansMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_RelatedPartyLoansMember" xlink:to="brkh_RelatedPartyLoansMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_RelatedPartyLoansMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">This member stands for related party loans.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_RelatedPartyLoansMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Related Party Loans [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_RelatedPartyLoansMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Related Party Loans</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_WorkingCapitalLoansWarrantMember" xlink:label="brkh_WorkingCapitalLoansWarrantMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_WorkingCapitalLoansWarrantMember" xlink:to="brkh_WorkingCapitalLoansWarrantMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_WorkingCapitalLoansWarrantMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Working capital loans warrant.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_WorkingCapitalLoansWarrantMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Working capital loans warrant</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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/2023/elts/us-gaap-2023.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="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_ConvertibleNotesPayableMember" xlink:label="us-gaap_ConvertibleNotesPayableMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_ConvertibleNotesPayableMember" xlink:to="us-gaap_ConvertibleNotesPayableMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_ConvertibleNotesPayableMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Convertible Notes Payable [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_ConvertibleNotesPayableMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Unsecured convertible promissory note</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_RelatedPartyTransactionExpensesFromTransactionsWithRelatedPartyPerMonth" xlink:label="brkh_RelatedPartyTransactionExpensesFromTransactionsWithRelatedPartyPerMonth" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_RelatedPartyTransactionExpensesFromTransactionsWithRelatedPartyPerMonth" xlink:to="brkh_RelatedPartyTransactionExpensesFromTransactionsWithRelatedPartyPerMonth_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_RelatedPartyTransactionExpensesFromTransactionsWithRelatedPartyPerMonth_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The contractual monthly amount to be paid for support services.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_RelatedPartyTransactionExpensesFromTransactionsWithRelatedPartyPerMonth_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Related Party Transaction, Expenses from Transactions with Related Party Per Month</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_RelatedPartyTransactionExpensesFromTransactionsWithRelatedPartyPerMonth_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Expenses per month</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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">Expenses incurred and paid</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_MaximumLoansConvertibleIntoWarrants" xlink:label="brkh_MaximumLoansConvertibleIntoWarrants" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_MaximumLoansConvertibleIntoWarrants" xlink:to="brkh_MaximumLoansConvertibleIntoWarrants_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_MaximumLoansConvertibleIntoWarrants_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The maximum amount which a potential loan could have repaid through issuance of warrants.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_MaximumLoansConvertibleIntoWarrants_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">maximum Loans Convertible Into Warrants</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_MaximumLoansConvertibleIntoWarrants_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Loan conversion agreement warrant</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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">Price per which the outstanding principal amount can be converted</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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">Amount borrowed</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_LossContingenciesTable" xlink:label="us-gaap_LossContingenciesTable" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_LossContingenciesTable" xlink:to="us-gaap_LossContingenciesTable_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_LossContingenciesTable_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Loss Contingencies [Table]</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_LockUpAgreementMember" xlink:label="brkh_LockUpAgreementMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_LockUpAgreementMember" xlink:to="brkh_LockUpAgreementMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_LockUpAgreementMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The member representing the lock up agreement.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_LockUpAgreementMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Lock-up Agreement [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_LockUpAgreementMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Lock up Agreement</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_MergerAgreementMember" xlink:label="brkh_MergerAgreementMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_MergerAgreementMember" xlink:to="brkh_MergerAgreementMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_MergerAgreementMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">This member stands for merger Agreement.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_MergerAgreementMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Merger Agreement [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_MergerAgreementMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Merger Agreement</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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/2023/elts/us-gaap-2023.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="brkh-20240331.xsd#brkh_ProbabilityOfSuccessfulBusinessCombinationMember" xlink:label="brkh_ProbabilityOfSuccessfulBusinessCombinationMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_ProbabilityOfSuccessfulBusinessCombinationMember" xlink:to="brkh_ProbabilityOfSuccessfulBusinessCombinationMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_ProbabilityOfSuccessfulBusinessCombinationMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Measurement input using probability of a successful Business Combination.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_ProbabilityOfSuccessfulBusinessCombinationMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Probability of a Successful Business Combination [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_ProbabilityOfSuccessfulBusinessCombinationMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Probability of a successful business combination</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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">Implied volatility</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_MeasurementInputExercisePriceMember" xlink:label="us-gaap_MeasurementInputExercisePriceMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <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">Value per shares as of the valuation date</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_LossContingenciesLineItems" xlink:label="us-gaap_LossContingenciesLineItems" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_LossContingenciesLineItems" xlink:to="us-gaap_LossContingenciesLineItems_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_LossContingenciesLineItems_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Loss Contingencies [Line Items]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_LossContingenciesLineItems_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">COMMITMENTS AND CONTINGENCIES</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_MaximumNumberOfDemandsForRegistrationOfSecurities" xlink:label="brkh_MaximumNumberOfDemandsForRegistrationOfSecurities" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_MaximumNumberOfDemandsForRegistrationOfSecurities" xlink:to="brkh_MaximumNumberOfDemandsForRegistrationOfSecurities_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_MaximumNumberOfDemandsForRegistrationOfSecurities_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents the maximum number of demands for registration of securities.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_MaximumNumberOfDemandsForRegistrationOfSecurities_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Maximum Number Of Demands For Registration Of Securities</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_MaximumNumberOfDemandsForRegistrationOfSecurities_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Maximum number of demands for registration of securities</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_UnderwritingCashDiscountPercentagePerUnit" xlink:label="brkh_UnderwritingCashDiscountPercentagePerUnit" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_UnderwritingCashDiscountPercentagePerUnit" xlink:to="brkh_UnderwritingCashDiscountPercentagePerUnit_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_UnderwritingCashDiscountPercentagePerUnit_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents the underwriting cash discount percentage per unit.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_UnderwritingCashDiscountPercentagePerUnit_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Underwriting Cash Discount Percentage Per Unit</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_UnderwritingCashDiscountPercentagePerUnit_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Underwriting cash discount per unit (in percentage)</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_DeferredUnderwritingDiscountPercentagePerUnit" xlink:label="brkh_DeferredUnderwritingDiscountPercentagePerUnit" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_DeferredUnderwritingDiscountPercentagePerUnit" xlink:to="brkh_DeferredUnderwritingDiscountPercentagePerUnit_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_DeferredUnderwritingDiscountPercentagePerUnit_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents the deferred underwriting discount percentage per unit.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_DeferredUnderwritingDiscountPercentagePerUnit_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Deferred Underwriting Discount Percentage Per Unit</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_DeferredUnderwritingDiscountPercentagePerUnit_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Deferred underwriting discount per unit (in percentage)</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_AgreementFairValuePerShare" xlink:label="brkh_AgreementFairValuePerShare" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_AgreementFairValuePerShare" xlink:to="brkh_AgreementFairValuePerShare_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_AgreementFairValuePerShare_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The amount of fair value of equity transferrable to the Non-Redeeming Stockholders pursuant to the Non-Redemption Agreement.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_AgreementFairValuePerShare_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Agreement, Fair Value Per Share</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_AgreementFairValuePerShare_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/2023/elts/us-gaap-2023.xsd#us-gaap_EquitySecuritiesFvNiMeasurementInput" xlink:label="us-gaap_EquitySecuritiesFvNiMeasurementInput" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_EquitySecuritiesFvNiMeasurementInput" xlink:to="us-gaap_EquitySecuritiesFvNiMeasurementInput_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_EquitySecuritiesFvNiMeasurementInput_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Equity Securities, FV-NI, Measurement Input</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_EquitySecuritiesFvNiMeasurementInput_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Measurement input</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_PercentageOfSharesRedeemed" xlink:label="brkh_PercentageOfSharesRedeemed" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_PercentageOfSharesRedeemed" xlink:to="brkh_PercentageOfSharesRedeemed_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_PercentageOfSharesRedeemed_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents the percentage of shares redeemed.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_PercentageOfSharesRedeemed_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Percentage of Shares Redeemed</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_PercentageOfSharesRedeemed_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Percentage of shares redeemed</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_PercentageOfBeneficialOwnershipHeld" xlink:label="brkh_PercentageOfBeneficialOwnershipHeld" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_PercentageOfBeneficialOwnershipHeld" xlink:to="brkh_PercentageOfBeneficialOwnershipHeld_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_PercentageOfBeneficialOwnershipHeld_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Percentage of beneficial ownership held.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_PercentageOfBeneficialOwnershipHeld_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Percentage Of beneficial Ownership Held</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_PercentageOfBeneficialOwnershipHeld_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Percentage of beneficial ownership held</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_ThresholdPeriodAfterBusinessCombinationInWhichSpecifiedTradingDaysWithinAnySpecifiedTradingDayPeriodCommences" xlink:label="brkh_ThresholdPeriodAfterBusinessCombinationInWhichSpecifiedTradingDaysWithinAnySpecifiedTradingDayPeriodCommences" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_ThresholdPeriodAfterBusinessCombinationInWhichSpecifiedTradingDaysWithinAnySpecifiedTradingDayPeriodCommences" xlink:to="brkh_ThresholdPeriodAfterBusinessCombinationInWhichSpecifiedTradingDaysWithinAnySpecifiedTradingDayPeriodCommences_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_ThresholdPeriodAfterBusinessCombinationInWhichSpecifiedTradingDaysWithinAnySpecifiedTradingDayPeriodCommences_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The period of time after a business combination which must elapse before consideration of the share price condition for transfer of shares.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_ThresholdPeriodAfterBusinessCombinationInWhichSpecifiedTradingDaysWithinAnySpecifiedTradingDayPeriodCommences_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold Period After Business Combination In Which Specified Trading Days Within Any Specified Trading Day Period Commences</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_ThresholdPeriodAfterBusinessCombinationInWhichSpecifiedTradingDaysWithinAnySpecifiedTradingDayPeriodCommences_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold period after the business combination in which the 20 trading days within any 30 trading day period commences</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_NumberOfSharesAgreedToForfeitBySponsorToBeEffectiveImmediatelyPriorToBusinessCombination" xlink:label="brkh_NumberOfSharesAgreedToForfeitBySponsorToBeEffectiveImmediatelyPriorToBusinessCombination" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_NumberOfSharesAgreedToForfeitBySponsorToBeEffectiveImmediatelyPriorToBusinessCombination" xlink:to="brkh_NumberOfSharesAgreedToForfeitBySponsorToBeEffectiveImmediatelyPriorToBusinessCombination_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_NumberOfSharesAgreedToForfeitBySponsorToBeEffectiveImmediatelyPriorToBusinessCombination_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of shares agreed to forfeit by sponsor to be effective immediately prior to the closing of the Business Combination.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_NumberOfSharesAgreedToForfeitBySponsorToBeEffectiveImmediatelyPriorToBusinessCombination_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of Shares Agreed to Forfeit by Sponsor to be Effective Immediately Prior to Business Combination</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_NumberOfSharesAgreedToForfeitBySponsorToBeEffectiveImmediatelyPriorToBusinessCombination_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of shares agreed to forfeit by sponsor to be effective immediately prior to business combination</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_AggregateReserveSizedUnderEquityIncentivePlanAndEsppPercentage" xlink:label="brkh_AggregateReserveSizedUnderEquityIncentivePlanAndEsppPercentage" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_AggregateReserveSizedUnderEquityIncentivePlanAndEsppPercentage" xlink:to="brkh_AggregateReserveSizedUnderEquityIncentivePlanAndEsppPercentage_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_AggregateReserveSizedUnderEquityIncentivePlanAndEsppPercentage_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Percentage of aggregate reserve size under Equity Incentive Plan and ESPP.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_AggregateReserveSizedUnderEquityIncentivePlanAndEsppPercentage_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Aggregate Reserve Sized under Equity Incentive Plan and ESPP, Percentage</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_AggregateReserveSizedUnderEquityIncentivePlanAndEsppPercentage_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Percentage of aggregate reserve size under Equity Incentive Plan and ESPP</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_EvergreenPercentageForEquityIncentivePlan" xlink:label="brkh_EvergreenPercentageForEquityIncentivePlan" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_EvergreenPercentageForEquityIncentivePlan" xlink:to="brkh_EvergreenPercentageForEquityIncentivePlan_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_EvergreenPercentageForEquityIncentivePlan_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Evergreen percentage for equity incentive plan.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_EvergreenPercentageForEquityIncentivePlan_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Evergreen Percentage for Equity Incentive Plan</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_EvergreenPercentageForEquityIncentivePlan_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Evergreen percentage for equity incentive plan</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_BasePurchasePrice" xlink:label="brkh_BasePurchasePrice" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_BasePurchasePrice" xlink:to="brkh_BasePurchasePrice_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_BasePurchasePrice_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Amount of base purchase price.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_BasePurchasePrice_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Base Purchase Price</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_BasePurchasePrice_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Base purchase price</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_SharesConsideredForCalculatingProportionateShares" xlink:label="brkh_SharesConsideredForCalculatingProportionateShares" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_SharesConsideredForCalculatingProportionateShares" xlink:to="brkh_SharesConsideredForCalculatingProportionateShares_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_SharesConsideredForCalculatingProportionateShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Number of shares considered for calculating proportionate shares.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_SharesConsideredForCalculatingProportionateShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Shares Considered for Calculating Proportionate Shares</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_SharesConsideredForCalculatingProportionateShares_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Shares considered for calculating proportionate shares</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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">Schedule of Stock by Class [Table]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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">STOCKHOLDERS' DEFICIT</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_CommonStockNumberOfVotesPerShare" xlink:label="brkh_CommonStockNumberOfVotesPerShare" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_CommonStockNumberOfVotesPerShare" xlink:to="brkh_CommonStockNumberOfVotesPerShare_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_CommonStockNumberOfVotesPerShare_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The number of votes that each common share is entitled.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_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="brkh_CommonStockNumberOfVotesPerShare_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Common shares, votes per share</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_StockIssuedDuringPeriodSharesConversionFromOneClassOfStockToAnother" xlink:label="brkh_StockIssuedDuringPeriodSharesConversionFromOneClassOfStockToAnother" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_StockIssuedDuringPeriodSharesConversionFromOneClassOfStockToAnother" xlink:to="brkh_StockIssuedDuringPeriodSharesConversionFromOneClassOfStockToAnother_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_StockIssuedDuringPeriodSharesConversionFromOneClassOfStockToAnother_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The number of shares issued as a conversion from one class of stock to another.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_StockIssuedDuringPeriodSharesConversionFromOneClassOfStockToAnother_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Stock Issued During Period, Shares, Conversion From One Class of Stock to Another</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_StockIssuedDuringPeriodSharesConversionFromOneClassOfStockToAnother_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Conversion of Class B to Class A Shares (in shares)</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_ConvertibleStockConversionRatio" xlink:label="brkh_ConvertibleStockConversionRatio" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_ConvertibleStockConversionRatio" xlink:to="brkh_ConvertibleStockConversionRatio_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_ConvertibleStockConversionRatio_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The ratio to be applied to the stock in a conversion of convertible stock.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_ConvertibleStockConversionRatio_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Convertible Stock Conversion Ratio</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_ConvertibleStockConversionRatio_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Ratio to be applied to the stock in the conversion</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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/srt/2023/elts/srt-2023.xsd#srt_StatementScenarioAxis" xlink:label="srt_StatementScenarioAxis" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="srt_StatementScenarioAxis" xlink:to="srt_StatementScenarioAxis_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="srt_StatementScenarioAxis_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Scenario [Axis]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/srt/2023/elts/srt-2023.xsd#srt_ScenarioUnspecifiedDomain" xlink:label="srt_ScenarioUnspecifiedDomain" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="srt_ScenarioUnspecifiedDomain" xlink:to="srt_ScenarioUnspecifiedDomain_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="srt_ScenarioUnspecifiedDomain_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Scenario [Domain]</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_RedemptionOfWarrantsWhenPricePerShareOfClassCommonStockEqualsOrExceeds18.00Member" xlink:label="brkh_RedemptionOfWarrantsWhenPricePerShareOfClassCommonStockEqualsOrExceeds18.00Member" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_RedemptionOfWarrantsWhenPricePerShareOfClassCommonStockEqualsOrExceeds18.00Member" xlink:to="brkh_RedemptionOfWarrantsWhenPricePerShareOfClassCommonStockEqualsOrExceeds18.00Member_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_RedemptionOfWarrantsWhenPricePerShareOfClassCommonStockEqualsOrExceeds18.00Member_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">This member stands for the scenario, where redemption of warrants when the price per share of class a common stock equals or exceeds $18.00.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_RedemptionOfWarrantsWhenPricePerShareOfClassCommonStockEqualsOrExceeds18.00Member_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Redemption of Warrants When the Price per Class A Ordinary Share Equals or Exceeds $18.00</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_RedemptionOfWarrantsWhenPricePerShareOfClassCommonStockEqualsOrExceeds18.00Member_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Redemption of warrants when the price per class A ordinary share equals or exceeds $18.00</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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">STOCKHOLDERS' DEFICIT</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_SharePriceTriggerUsedToMeasureDilutionOfWarrants" xlink:label="brkh_SharePriceTriggerUsedToMeasureDilutionOfWarrants" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_SharePriceTriggerUsedToMeasureDilutionOfWarrants" xlink:to="brkh_SharePriceTriggerUsedToMeasureDilutionOfWarrants_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_SharePriceTriggerUsedToMeasureDilutionOfWarrants_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The cutoff price used to measure whether dilution of the warrant has occurred. Shares issued below this price will cause the exercise price of the warrant to be adjusted.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_SharePriceTriggerUsedToMeasureDilutionOfWarrants_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Share Price Trigger Used To Measure Dilution Of Warrants</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_SharePriceTriggerUsedToMeasureDilutionOfWarrants_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Share price trigger used to measure dilution of warrant</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.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">Public Warrants expiration term</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_WarrantsAndRightsOutstandingExercisableTermAfterBusinessCombination" xlink:label="brkh_WarrantsAndRightsOutstandingExercisableTermAfterBusinessCombination" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_WarrantsAndRightsOutstandingExercisableTermAfterBusinessCombination" xlink:to="brkh_WarrantsAndRightsOutstandingExercisableTermAfterBusinessCombination_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_WarrantsAndRightsOutstandingExercisableTermAfterBusinessCombination_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The warrants exercisable term after the completion of a business combination, 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="brkh_WarrantsAndRightsOutstandingExercisableTermAfterBusinessCombination_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Warrants And Rights Outstanding Exercisable Term After Business Combination</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_WarrantsAndRightsOutstandingExercisableTermAfterBusinessCombination_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Public Warrants exercisable term after the completion of a business combination</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_WarrantsAndRightsOutstandingExercisableTermFromClosingOfInitialPublicOffering" xlink:label="brkh_WarrantsAndRightsOutstandingExercisableTermFromClosingOfInitialPublicOffering" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_WarrantsAndRightsOutstandingExercisableTermFromClosingOfInitialPublicOffering" xlink:to="brkh_WarrantsAndRightsOutstandingExercisableTermFromClosingOfInitialPublicOffering_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_WarrantsAndRightsOutstandingExercisableTermFromClosingOfInitialPublicOffering_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The warrants exercisable term from the closing of the 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="brkh_WarrantsAndRightsOutstandingExercisableTermFromClosingOfInitialPublicOffering_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Warrants And Rights Outstanding Exercisable Term From Closing Of Initial Public Offering</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_WarrantsAndRightsOutstandingExercisableTermFromClosingOfInitialPublicOffering_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Public Warrants exercisable term from the closing of the initial public offering</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_ThresholdPeriodForRegistrationStatementToBeEffectiveAfterWhichWarrantsCanBeExercisedOnCashlessBasis" xlink:label="brkh_ThresholdPeriodForRegistrationStatementToBeEffectiveAfterWhichWarrantsCanBeExercisedOnCashlessBasis" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_ThresholdPeriodForRegistrationStatementToBeEffectiveAfterWhichWarrantsCanBeExercisedOnCashlessBasis" xlink:to="brkh_ThresholdPeriodForRegistrationStatementToBeEffectiveAfterWhichWarrantsCanBeExercisedOnCashlessBasis_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_ThresholdPeriodForRegistrationStatementToBeEffectiveAfterWhichWarrantsCanBeExercisedOnCashlessBasis_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold period for registration statement to be effective after which warrants can be exercised on a cashless basis.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_ThresholdPeriodForRegistrationStatementToBeEffectiveAfterWhichWarrantsCanBeExercisedOnCashlessBasis_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold Period For Registration Statement To Be Effective After Which Warrants Can Be Exercised On Cashless Basis</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_ThresholdPeriodForRegistrationStatementToBeEffectiveAfterWhichWarrantsCanBeExercisedOnCashlessBasis_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold period for registration statement to be effective after which warrants can be exercised on a cashless basis</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_ClassOfWarrantOrRightRedemptionPriceOfWarrantsOrRights" xlink:label="brkh_ClassOfWarrantOrRightRedemptionPriceOfWarrantsOrRights" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_ClassOfWarrantOrRightRedemptionPriceOfWarrantsOrRights" xlink:to="brkh_ClassOfWarrantOrRightRedemptionPriceOfWarrantsOrRights_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_ClassOfWarrantOrRightRedemptionPriceOfWarrantsOrRights_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Redemption price per share or per unit of warrants or rights outstanding.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_ClassOfWarrantOrRightRedemptionPriceOfWarrantsOrRights_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class Of Warrant Or Right, Redemption Price Of Warrants Or Rights</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_ClassOfWarrantOrRightRedemptionPriceOfWarrantsOrRights_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Redemption price per public warrant (in dollars per share)</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_RedemptionPeriod" xlink:label="brkh_RedemptionPeriod" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_RedemptionPeriod" xlink:to="brkh_RedemptionPeriod_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_RedemptionPeriod_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Redemption period.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_RedemptionPeriod_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Redemption Period</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_RedemptionPeriod_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Redemption period</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_ClassOfWarrantOrRightRedemptionOfWarrantsOrRightsStockPriceTrigger" xlink:label="brkh_ClassOfWarrantOrRightRedemptionOfWarrantsOrRightsStockPriceTrigger" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_ClassOfWarrantOrRightRedemptionOfWarrantsOrRightsStockPriceTrigger" xlink:to="brkh_ClassOfWarrantOrRightRedemptionOfWarrantsOrRightsStockPriceTrigger_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_ClassOfWarrantOrRightRedemptionOfWarrantsOrRightsStockPriceTrigger_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 to trigger the redemption of warrants.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_ClassOfWarrantOrRightRedemptionOfWarrantsOrRightsStockPriceTrigger_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class Of Warrant Or Right Redemption Of Warrants Or Rights Stock Price Trigger</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_ClassOfWarrantOrRightRedemptionOfWarrantsOrRightsStockPriceTrigger_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Stock price trigger for redemption of public warrants (in dollars per share)</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_ClassOfWarrantOrRightAdjustmentOfRedemptionPriceOfWarrantsOrRightsPercentBasedOnMarketValueAndNewlyIssuedPrice2" xlink:label="brkh_ClassOfWarrantOrRightAdjustmentOfRedemptionPriceOfWarrantsOrRightsPercentBasedOnMarketValueAndNewlyIssuedPrice2" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_ClassOfWarrantOrRightAdjustmentOfRedemptionPriceOfWarrantsOrRightsPercentBasedOnMarketValueAndNewlyIssuedPrice2" xlink:to="brkh_ClassOfWarrantOrRightAdjustmentOfRedemptionPriceOfWarrantsOrRightsPercentBasedOnMarketValueAndNewlyIssuedPrice2_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_ClassOfWarrantOrRightAdjustmentOfRedemptionPriceOfWarrantsOrRightsPercentBasedOnMarketValueAndNewlyIssuedPrice2_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Percentage of adjustment of redemption price of stock based on market value and newly issued price.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_ClassOfWarrantOrRightAdjustmentOfRedemptionPriceOfWarrantsOrRightsPercentBasedOnMarketValueAndNewlyIssuedPrice2_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class of Warrant or Right, Adjustment of Redemption Price of Warrants or Rights, Percent, Based On Market Value And Newly Issued Price 2</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:label="brkh_ClassOfWarrantOrRightAdjustmentOfRedemptionPriceOfWarrantsOrRightsPercentBasedOnMarketValueAndNewlyIssuedPrice2_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Adjustment two of redemption price of stock based on market value and newly issued price (as a percent)</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_ClassOfWarrantOrRightRedemptionOfWarrantsOrRightsThresholdTradingDays" xlink:label="brkh_ClassOfWarrantOrRightRedemptionOfWarrantsOrRightsThresholdTradingDays" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_ClassOfWarrantOrRightRedemptionOfWarrantsOrRightsThresholdTradingDays" xlink:to="brkh_ClassOfWarrantOrRightRedemptionOfWarrantsOrRightsThresholdTradingDays_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_ClassOfWarrantOrRightRedemptionOfWarrantsOrRightsThresholdTradingDays_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold number of specified trading days for stock price trigger considered for redemption of warrants.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_ClassOfWarrantOrRightRedemptionOfWarrantsOrRightsThresholdTradingDays_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class Of Warrant Or Right, Redemption Of Warrants Or Rights, , Threshold Trading Days</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_ClassOfWarrantOrRightRedemptionOfWarrantsOrRightsThresholdTradingDays_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold trading days for redemption of public warrants</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_ClassOfWarrantOrRightRedemptionOfWarrantsOrRightsThresholdConsecutiveTradingDays" xlink:label="brkh_ClassOfWarrantOrRightRedemptionOfWarrantsOrRightsThresholdConsecutiveTradingDays" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_ClassOfWarrantOrRightRedemptionOfWarrantsOrRightsThresholdConsecutiveTradingDays" xlink:to="brkh_ClassOfWarrantOrRightRedemptionOfWarrantsOrRightsThresholdConsecutiveTradingDays_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_ClassOfWarrantOrRightRedemptionOfWarrantsOrRightsThresholdConsecutiveTradingDays_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold number of specified consecutive trading days for stock price trigger considered for redemption of warrants.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_ClassOfWarrantOrRightRedemptionOfWarrantsOrRightsThresholdConsecutiveTradingDays_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class Of Warrant Or Right, Redemption Of Warrants Or Rights, , Threshold Consecutive Trading Days</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_ClassOfWarrantOrRightRedemptionOfWarrantsOrRightsThresholdConsecutiveTradingDays_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold consecutive trading days for redemption of public warrants</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_ThresholdNumberOfBusinessDaysBeforeSendingNoticeOfRedemptionToWarrantHolders" xlink:label="brkh_ThresholdNumberOfBusinessDaysBeforeSendingNoticeOfRedemptionToWarrantHolders" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_ThresholdNumberOfBusinessDaysBeforeSendingNoticeOfRedemptionToWarrantHolders" xlink:to="brkh_ThresholdNumberOfBusinessDaysBeforeSendingNoticeOfRedemptionToWarrantHolders_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_ThresholdNumberOfBusinessDaysBeforeSendingNoticeOfRedemptionToWarrantHolders_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold  number of business days before sending notice of redemption to warrant holders.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_ThresholdNumberOfBusinessDaysBeforeSendingNoticeOfRedemptionToWarrantHolders_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold Number of Business Days Before Sending Notice of Redemption to Warrant Holders</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_ThresholdNumberOfBusinessDaysBeforeSendingNoticeOfRedemptionToWarrantHolders_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold number of business days before sending notice of redemption to warrant holders</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_PercentageOfGrossProceedsOnTotalEquityProceeds" xlink:label="brkh_PercentageOfGrossProceedsOnTotalEquityProceeds" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_PercentageOfGrossProceedsOnTotalEquityProceeds" xlink:to="brkh_PercentageOfGrossProceedsOnTotalEquityProceeds_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_PercentageOfGrossProceedsOnTotalEquityProceeds_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Represents the percentage of gross proceeds on total equity proceeds.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_PercentageOfGrossProceedsOnTotalEquityProceeds_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Percentage Of Gross Proceeds On Total Equity Proceeds</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_PercentageOfGrossProceedsOnTotalEquityProceeds_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Percentage of gross proceeds on total equity proceeds</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_ThresholdTradingDaysForCalculatingMarketValue" xlink:label="brkh_ThresholdTradingDaysForCalculatingMarketValue" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_ThresholdTradingDaysForCalculatingMarketValue" xlink:to="brkh_ThresholdTradingDaysForCalculatingMarketValue_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_ThresholdTradingDaysForCalculatingMarketValue_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold trading days for calculating Market Value.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_ThresholdTradingDaysForCalculatingMarketValue_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold Trading Days For Calculating Market Value</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_ThresholdTradingDaysForCalculatingMarketValue_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold trading days for calculating Market Value</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_ThresholdIssuePriceForCapitalRaisingPurposesInConnectionWithClosingOfBusinessCombination" xlink:label="brkh_ThresholdIssuePriceForCapitalRaisingPurposesInConnectionWithClosingOfBusinessCombination" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_ThresholdIssuePriceForCapitalRaisingPurposesInConnectionWithClosingOfBusinessCombination" xlink:to="brkh_ThresholdIssuePriceForCapitalRaisingPurposesInConnectionWithClosingOfBusinessCombination_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_ThresholdIssuePriceForCapitalRaisingPurposesInConnectionWithClosingOfBusinessCombination_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold issue price for capital raising purposes in connection with the closing of a Business Combination.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_ThresholdIssuePriceForCapitalRaisingPurposesInConnectionWithClosingOfBusinessCombination_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold Issue Price For Capital Raising Purposes In Connection With Closing Of Business Combination</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_ThresholdIssuePriceForCapitalRaisingPurposesInConnectionWithClosingOfBusinessCombination_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Threshold issue price for capital raising purposes in connection with the closing of a Business Combination</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_ClassOfWarrantOrRightAdjustmentOfExercisePriceOfWarrantsOrRightsPercentBasedOnMarketValueAndNewlyIssuedPrice" xlink:label="brkh_ClassOfWarrantOrRightAdjustmentOfExercisePriceOfWarrantsOrRightsPercentBasedOnMarketValueAndNewlyIssuedPrice" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_ClassOfWarrantOrRightAdjustmentOfExercisePriceOfWarrantsOrRightsPercentBasedOnMarketValueAndNewlyIssuedPrice" xlink:to="brkh_ClassOfWarrantOrRightAdjustmentOfExercisePriceOfWarrantsOrRightsPercentBasedOnMarketValueAndNewlyIssuedPrice_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_ClassOfWarrantOrRightAdjustmentOfExercisePriceOfWarrantsOrRightsPercentBasedOnMarketValueAndNewlyIssuedPrice_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Percentage of adjustment of exercise price of warrants based on market value and newly issued price.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_ClassOfWarrantOrRightAdjustmentOfExercisePriceOfWarrantsOrRightsPercentBasedOnMarketValueAndNewlyIssuedPrice_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class Of Warrant Or Right Adjustment Of Exercise Price Of Warrants Or Rights Percent Based On Market Value And Newly Issued Price</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_ClassOfWarrantOrRightAdjustmentOfExercisePriceOfWarrantsOrRightsPercentBasedOnMarketValueAndNewlyIssuedPrice_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Adjustment of exercise price of warrants based on market value and newly issued price (as a percent)</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_ClassOfWarrantOrRightMinimumThresholdWrittenNoticePeriodForRedemptionOfWarrants" xlink:label="brkh_ClassOfWarrantOrRightMinimumThresholdWrittenNoticePeriodForRedemptionOfWarrants" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_ClassOfWarrantOrRightMinimumThresholdWrittenNoticePeriodForRedemptionOfWarrants" xlink:to="brkh_ClassOfWarrantOrRightMinimumThresholdWrittenNoticePeriodForRedemptionOfWarrants_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_ClassOfWarrantOrRightMinimumThresholdWrittenNoticePeriodForRedemptionOfWarrants_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">The minimum threshold period during which a written notice is required for redemption of warrants, 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="brkh_ClassOfWarrantOrRightMinimumThresholdWrittenNoticePeriodForRedemptionOfWarrants_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Class Of Warrant Or Right, Minimum Threshold Written Notice Period For Redemption Of Warrants</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_ClassOfWarrantOrRightMinimumThresholdWrittenNoticePeriodForRedemptionOfWarrants_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Minimum threshold written notice period for redemption of public warrants</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_FairValueAssetsAndLiabilitiesMeasuredOnRecurringAndNonrecurringBasisTable" xlink:label="us-gaap_FairValueAssetsAndLiabilitiesMeasuredOnRecurringAndNonrecurringBasisTable" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_FairValueAssetsAndLiabilitiesMeasuredOnRecurringAndNonrecurringBasisTable" xlink:to="us-gaap_FairValueAssetsAndLiabilitiesMeasuredOnRecurringAndNonrecurringBasisTable_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_FairValueAssetsAndLiabilitiesMeasuredOnRecurringAndNonrecurringBasisTable_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fair Value, Recurring and Nonrecurring [Table]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_FairValueByFairValueHierarchyLevelAxis" xlink:label="us-gaap_FairValueByFairValueHierarchyLevelAxis" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_FairValueByFairValueHierarchyLevelAxis" xlink:to="us-gaap_FairValueByFairValueHierarchyLevelAxis_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_FairValueByFairValueHierarchyLevelAxis_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fair Value Hierarchy and NAV [Axis]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_FairValueMeasurementsFairValueHierarchyDomain" xlink:label="us-gaap_FairValueMeasurementsFairValueHierarchyDomain" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_FairValueMeasurementsFairValueHierarchyDomain" xlink:to="us-gaap_FairValueMeasurementsFairValueHierarchyDomain_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_FairValueMeasurementsFairValueHierarchyDomain_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Fair Value Hierarchy and NAV [Domain]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_FairValueInputsLevel1Member" xlink:label="us-gaap_FairValueInputsLevel1Member" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_FairValueInputsLevel1Member" xlink:to="us-gaap_FairValueInputsLevel1Member_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_FairValueInputsLevel1Member_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Level 1 [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="us-gaap_FairValueInputsLevel1Member_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Level 1</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_FinancialInstrumentAxis" xlink:label="us-gaap_FinancialInstrumentAxis" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_FinancialInstrumentAxis" xlink:to="us-gaap_FinancialInstrumentAxis_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_FinancialInstrumentAxis_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Financial Instrument [Axis]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_TransfersAndServicingOfFinancialInstrumentsTypesOfFinancialInstrumentsDomain" xlink:label="us-gaap_TransfersAndServicingOfFinancialInstrumentsTypesOfFinancialInstrumentsDomain" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_TransfersAndServicingOfFinancialInstrumentsTypesOfFinancialInstrumentsDomain" xlink:to="us-gaap_TransfersAndServicingOfFinancialInstrumentsTypesOfFinancialInstrumentsDomain_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_TransfersAndServicingOfFinancialInstrumentsTypesOfFinancialInstrumentsDomain_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Financial Instruments [Domain]</label>
    <loc xlink:type="locator" xlink:href="brkh-20240331.xsd#brkh_U.s.MoneyMarketFundHeldInTrustAccountMember" xlink:label="brkh_U.s.MoneyMarketFundHeldInTrustAccountMember" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="brkh_U.s.MoneyMarketFundHeldInTrustAccountMember" xlink:to="brkh_U.s.MoneyMarketFundHeldInTrustAccountMember_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:label="brkh_U.s.MoneyMarketFundHeldInTrustAccountMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">This member is stands for U.S. Money Market Fund held in Trust Account.</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="brkh_U.s.MoneyMarketFundHeldInTrustAccountMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">U.S. Money Market Fund held in Trust Account [Member]</label>
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:label="brkh_U.s.MoneyMarketFundHeldInTrustAccountMember_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">U.S. Money Market Funds held in Trust Account</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_FairValueByMeasurementFrequencyAxis" xlink:label="us-gaap_FairValueByMeasurementFrequencyAxis" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_FairValueByMeasurementFrequencyAxis" xlink:to="us-gaap_FairValueByMeasurementFrequencyAxis_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_FairValueByMeasurementFrequencyAxis_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Measurement Frequency [Axis]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_FairValueMeasurementFrequencyDomain" xlink:label="us-gaap_FairValueMeasurementFrequencyDomain" xmlns="http://www.xbrl.org/2003/linkbase" />
    <labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="us-gaap_FairValueMeasurementFrequencyDomain" xlink:to="us-gaap_FairValueMeasurementFrequencyDomain_lbl" xmlns="http://www.xbrl.org/2003/linkbase" />
    <label xlink:type="resource" xlink:role="http://www.xbrl.org/2003/role/label" xlink:label="us-gaap_FairValueMeasurementFrequencyDomain_lbl" xml:lang="en-US" xmlns="http://www.xbrl.org/2003/linkbase">Measurement Frequency [Domain]</label>
    <loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2023/elts/us-gaap-2023.xsd#us-gaap_FairValueMeasurementsRecurringMember" xlink:label="us-gaap_FairValueMeasurementsRecurringMember" xmlns="http://www.xbrl.org/2003/linkbase" />
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