Equity Bridge Closure: Formal Proof

Abstract

This document formalizes the accounting equity bridge as a continuity equation with explicit source terms mandated by IFRS and US GAAP. It establishes that reported equity changes over a reporting period equal the sum of recognized sources and sinks: profit or loss, other comprehensive income (OCI), owner transactions, consolidation boundary flux, hyperinflation adjustments, and measurement adjustments. The proof complements the incidence-matrix derivation (Proof 1) and discrete Reynolds Transport Theorem (Proof 2) already present in the framework. Together, the three proofs demonstrate that double-entry bookkeeping enforces Kirchhoff-style continuity locally, that consolidation boundaries obey discrete Reynolds Transport, and that the taxonomy of source terms in docs/STANDARDS_CROSSWALK.md is complete, non-redundant, and necessary for closing the equity bridge.

Minimal Axiom Set

All subsequent proofs rest on this discrete, closed foundation:

Axiom 1 (Double-Entry Constraint). For every journal entry $j$ with postings $P_{i,j}$ to accounts $i \in A$:

$$\sum_{i \in A} P_{i,j} = 0.$$

Axiom 2 (Equity Residual Definition). Shareholders’ equity $E$ is defined as:

$$E = A - L,$$

where $A$ denotes total assets and $L$ denotes total liabilities, both measured at reporting date.

Axiom 3 (Comprehensive Income Partition). All income and expenses recognized in a period are partitioned into:

$$CI = P + O,$$

where $P$ is profit or loss (IAS 1.81A) and $O$ is other comprehensive income (IAS 1.82). The sets are mutually exclusive: $P \cap O = \emptyset$.

Axiom 4 (Owner Transaction Exclusion). Owner transactions (contributions and distributions) do not constitute income or expenses (IFRS Conceptual Framework §6.68, §6.74). They change equity directly without passing through comprehensive income:

$$\Phi_{\text{owner}} \notin CI.$$

Axiom 5 (Standards Completeness). All IFRS- and US GAAP-mandated equity-changing events are exhaustively covered by the disjoint union:

$$\{\text{NI}, \text{OCI}, \Phi_{\text{owner}}, \text{FX}, \text{Hyper}, \text{Meas}, \Delta \text{NCI}\},$$

where each category is defined by specific accounting standards (detailed in Parts 2-4 below).

Axiom 6 (Consolidation Boundary). At each reporting date $t$, the consolidated perimeter is a finite set of entities $S_t$. Changes in the perimeter $S_{t} \to S_{t+1}$ are governed by a mapping $M_{t \to t+1}$ that assigns entities to: - Retained (continuing consolidation), - Acquired (IFRS 3 business combinations), - Disposed (IFRS 10.25 loss of control), or - Unchanged (no boundary event).

Axiom 7 (Incidence Matrix Structure). At each period $t$, journal entries are represented by an incidence matrix:

$$P_t \in \mathbb{R}^{(|A| \cdot |S_t|) \times m_t},$$

where $|A|$ is the number of account types, $|S_t|$ is the number of consolidated entities, and $m_t$ is the number of journal entries in period $t$. Each column $P_t^{(j)}$ satisfies Axiom 1.

Discrete Equity-RTT (Final Form)

Theorem (Discrete Reynolds Transport for Equity). The change in parent shareholders’ equity over period $[t, t+1]$ satisfies:

$$\Delta E_{\text{parent}} = (\text{NI} + \text{OCI}) + \Phi_{\text{owner}} + \text{Meas} - \Delta \text{NCI},$$

where: - $\text{NI}$: Net income attributable to parent (IAS 1.81A), - $\text{OCI}$: Other comprehensive income attributable to parent (IAS 1.82), - $\Phi_{\text{owner}}$: Net owner transactions (issuances minus distributions/repurchases), - $\text{Meas}$: Measurement adjustments (fair value remeasurement, error corrections per IAS 8), - $\Delta \text{NCI}$: Change in non-controlling interest boundary (IFRS 10.B96).

Three Critical Corollaries:

Corollary 1 (M&A Acquisition with NCI). Upon acquiring a subsidiary with non-controlling interest at date $t_0$:

$$\Delta E_{\text{parent}} = -\text{Cash Paid} + (\text{FV of Net Assets} - \text{NCI at FV}) + \text{Goodwill}.$$

The NCI component is recorded as a boundary term, not as parent equity change (IFRS 3.19, 3.32).

Corollary 2 (Loss of Control). Upon disposing a subsidiary at date $t_1$:

$$\Delta E_{\text{parent}} = \text{Disposal Proceeds} + \text{FV of Retained Interest} - \text{Carrying Amount of Net Assets} + \text{Reclassified OCI}.$$

The gain/loss flows through $\text{NI}$ (IFRS 10.25); cumulative translation adjustments are recycled from OCI reserves (IAS 21.48).

Corollary 3 (Ownership Change Without Loss of Control). When parent’s ownership percentage changes but control is retained:

$$\Delta E_{\text{parent}} = \text{Consideration Paid/Received} - \Delta \text{NCI}.$$

No gain or loss is recognized in profit or loss; the difference is an equity transaction (IFRS 10.23, IFRS 10.B96).

Theorem 3 Statement

Let $E_t$ and $E_{t+1}$ denote reported shareholders’ equity at two consecutive reporting dates. Define the following period aggregates, measured over $[t, t+1]$:

$P\_t$: Profit or loss for the period (IFRS IAS 1.81A)
$O\_t$: Other comprehensive income for the period (IAS 1.82)
$Owner\_t$: Net owner transactions (contributions minus distributions) recorded directly in equity (IAS 1.106(c), IAS 32.33-40)
$FX\_t$: Cumulative translation adjustments and other foreign currency translation effects not already included in $O\_t$ (IAS 21.39-47)
$Hyper\_t$: Hyperinflation restatement impacts when IAS 29 applies
$Measure\_t$: Direct-to-equity adjustments arising from prior-period error corrections or changes in accounting policies (IAS 8.42)

Then the equity bridge identity holds:

$$E_{t+1} - E_t = P\_t + O\_t + Owner\_t + FX\_t + Hyper\_t + Measure\_t.$$

Furthermore: 1. Completeness: Every IFRS/GAAP-mandated equity-changing event is captured by exactly one of the six categories. 2. Non-redundancy: No category double-counts the effect of another; the taxonomy is mutually exclusive. 3. Necessity: Omitting any non-zero category yields a residual whose magnitude is material relative to equity, causing the bridge to fail.

Proof

Part 1: Foundations from IFRS Conceptual Framework

Claim. Equity changes arise from income, expenses, and owner transactions as defined in IFRS Conceptual Framework (CF) §6.

Key definitions (CF §6.61-6.89). - Equity ($E$): The residual interest in the assets after deducting liabilities. - Income: Increases in assets or decreases in liabilities that result in increases in equity, other than those relating to contributions from holders of equity claims (CF §6.68). - Expenses: Decreases in assets or increases in liabilities that result in decreases in equity, other than those relating to distributions to holders of equity claims (CF §6.74).

Implications. 1. Equity changes must arise from either income/expenses (thus affecting comprehensive income) or from owner transactions (contributions/distributions). No other mechanisms are permitted by definition. 2. The CF explicitly acknowledges that some income and expenses bypass profit or loss when another standard so requires. These items form OCI (CF §6.75-6.78). 3. Because owner transactions are excluded from income and expenses, they must be tracked separately as direct adjustments to equity.

Conclusion. The CF yields the generic continuity equation for equity:

$$\Delta E = \text{Income} - \text{Expenses} + \text{Owner Transactions}.$$

This is a continuity equation with source terms rather than a conservation law: sources are non-zero whenever income, expenses, or owner actions take place. QED Part 1.

Part 2: Decomposition of Comprehensive Income (IAS 1)

Claim. Profit or loss and OCI exhaustively partition income and expenses.

IAS 1.81A requires entities to present a statement of profit or loss and other comprehensive income, comprising: - Profit or loss (P/L): All income and expenses recognized in the period except those items required or permitted to be presented in OCI. - Other comprehensive income (OCI): Items of income and expense not recognized in P/L as required by other IFRS Standards. IAS 1.82(a)-(b) lists categories that will or will not be reclassified to P/L.

Detail. Examples mandated by IAS 1.82 include: - Foreign currency translation differences for foreign operations (IAS 21.39-47). - Fair value gains/losses on FVOCI financial instruments (IFRS 9.5.7.5 and 5.7.10). - Gains or losses on hedging instruments designated in cash flow hedges (IFRS 9.6.5.11). - Revaluation surplus on property, plant, and equipment (IAS 16.39-40) and intangible assets (IAS 38.85-87). - Remeasurement of defined benefit obligations (IAS 19.120).

Collectively, P/L and OCI account for all income and expenses. Therefore, the P/L + OCI pair forms the first two source categories in the theorem.

Conclusion. Comprehensive income $CI\_t = P\_t + O\_t$ captures every income/expense-driven change in equity. QED Part 2.

Part 3: Owner Transactions (IAS 1.106-110; IAS 32; ASC 505)

Claim. Owner transactions change equity without passing through comprehensive income.

IAS 1.106 requires a statement of changes in equity showing: 1. Total comprehensive income for the period. 2. Transaction effects with owners acting in their capacity as owners (contributions, distributions, and changes in ownership interests).

IAS 32.33-40 specifies presentation requirements for treasury shares (share buybacks), dividends, and share issuances. Under US GAAP, ASC 505-30 and ASC 230 classify the corresponding cash flows as financing activities and require direct equity presentation.

The framework aggregates these events into $Owner\_t$, capturing: - Cash dividends (negative source). - Share repurchases, including excise taxes and transaction costs (negative source). - Share issuances and capital injections (positive source). - Changes in non-controlling interests (NCI) without loss of control (IFRS 10.23; ASC 810-10-45-23).

Conclusion. Owner transactions are explicit source terms mandated by standards. QED Part 3.

Part 4: Special Source Terms (FX Translation, Hyperinflation, Measurement Adjustments)

Claim. Three categories capture equity changes not fully represented in P/L, OCI, or owner transactions.

(a) Foreign Currency Translation ($FX\_t$)

IAS 21.39-47 requires translation of financial statements for foreign operations with differences recognized in OCI until disposal. When translation balances are reported outside OCI (e.g., certain local GAAP reconciliations), the $FX\_t$ category preserves completeness.
US GAAP ASC 830-30 uses similar mechanics, accumulating translation adjustments in equity.

(b) Hyperinflation Adjustments ($Hyper\_t$)

IAS 29.27-28 requires restating financial statements in hyperinflationary economies. Gains or losses on net monetary position may be recognized in P/L or directly in equity. The $Hyper\_t$ term isolates these restatements.
US GAAP (ASC 830-10-45) forces remeasurement into USD using current rate; the concept is analogous even though US GAAP lacks a direct IAS 29 counterpart.

(c) Measurement Adjustments ($Measure\_t$)

IAS 8.42 mandates retrospective adjustments to opening equity for prior period errors or changes in accounting policies.
ASC 250 similarly requires cumulative-effect adjustments to retained earnings when accounting principles change.

These categories ensure the taxonomy covers restatements, translation peculiarities, and inflation adjustments that do not fall neatly into P/L, OCI, or owner buckets.

Conclusion. $FX\_t$, $Hyper\_t$, and $Measure\_t$ complete the list of explicit sources prescribed by standards. QED Part 4.

Part 5: Completeness and Non-Redundancy

Claim. The six categories cover all equity-changing events without overlap.

Proof by exhaustion. Consider any event that changes total equity in the consolidated financial statements:

Income/expense recognition: Must appear in P/L unless another standard requires OCI. Covered by $P\_t$ or $O\_t$ (Part 2).
Owner transactions: Includes dividends, buybacks, issuances, and NCI reallocations. Captured by $Owner\_t$ (Part 3).
Foreign currency translation: Either captured in OCI ($O\_t$) or, if separately disclosed, in $FX\_t$ (Part 4).
Hyperinflation restatement: Captured by $Hyper\_t$ (Part 4).
Measurement adjustments: Captured by $Measure\_t$ (Part 4).

No other IFRS or US GAAP guidance specifies direct-to-equity changes outside these categories. Items such as share-based compensation, fair value changes, and derivative settlements fall within P/L or OCI by design.

Non-redundancy. - IAS 1.81A ensures the split between P/L and OCI is mutually exclusive. - Owner transactions, by definition, exclude income and expenses (CF §6.68, §6.74). - $FX\_t$ is only invoked when translation effects are not already included in $O\_t$, preventing double counting. - IAS 29 gains/losses either flow through P/L (already counted) or are recorded directly in equity (counted once in $Hyper\_t$). - IAS 8 adjustments apply to opening balances and are recorded once in $Measure\_t$.

Conclusion. The taxonomy is complete and mutually exclusive. QED Part 5.

Part 6: Necessity – Omitting Terms Breaks Closure

Claim. Removing any non-zero source term yields a material residual in the equity bridge.

Let $Residual = (E_{t+1} - E_t) - (P\_t + O\_t + Owner\_t + FX\_t + Hyper\_t + Measure\_t)$. If one category is omitted, the residual equals the neglected amount (up to sign). We provide counterexamples for each case.

Omitting $P\_t$ (profit or loss). Company with $P = \$100M$, no OCI, no owner transactions: $\Delta E = \$100M$. If $P\_t$ is excluded, Residual $= \$100M$ → fails.
Omitting $O\_t$ (OCI). FVOCI equity gains of $50M, P/L of $100M, no owner transactions: $\Delta E = \$150M$. Without $O\_t$, Residual $= \$50M$ → fails.
Omitting $Owner\_t$. Company pays $30M dividend: $P = \$100M$, $O = \$10M$, Dividend $= -\$30M$. True $\Delta E = \$80M$. Without $Owner\_t$, Residual $= -\$30M$ → fails.
Omitting $FX\_t$. Consolidated entity with CTA of $50M recorded in equity (not OCI). If omitted, Residual $= \$50M$ → fails.
Omitting $Hyper\_t$. Entity in hyperinflationary environment records $20M restatement directly in equity. Removing $Hyper\_t$ leaves Residual $= \$20M$ → fails.
Omitting $Measure\_t$. Prior period error correction of $15M to opening retained earnings. True $\Delta E = \Delta E\_{\text{operational}} + 15M$. Without $Measure\_t$, Residual $= \$15M$ → fails.

In each scenario the residual equals the excluded category. Under audit materiality thresholds, such residuals are significant; therefore every category is necessary for closure. QED Part 6.

Part 7: Method-Invariance – Algebraic Equivalence of Policy Choices

Claim. When IFRS or US GAAP permit alternative accounting treatments for identical economic events, the resulting change in total equity $\Delta E$ is algebraically equivalent under both methods. The choice affects presentation within equity accounts but not the equity bridge identity.

This section provides formal ledger-posting proofs for five critical scenarios.

7.1 Treasury Stock vs. Retirement (IAS 32.33 / ASC 505-30-30-2 vs. ASC 505-30-30-3)

Economic event: Entity repurchases 100 shares at $50/share. Par value is $1/share; average issue price was $30/share.

Method A: Treasury Stock (IAS 32.33 / ASC 505-30-30-2)

Ledger entry:

Dr. Treasury Stock (contra-equity)  \$5,000
    Cr. Cash                                \$5,000

Method B: Retirement (ASC 505-30-30-3)

Ledger entry:

Dr. Common Stock (par)              \$100
Dr. Additional Paid-In Capital      \$2,900
Dr. Retained Earnings               \$2,000
    Cr. Cash                                \$5,000

Proof: Both methods result in $\Delta E = -\$5,000$ (cash outflow). Treasury method presents a single contra-equity line; retirement allocates across par, APIC, and retained earnings. Per IAS 32.33: “No gain or loss is recognised in profit or loss on the purchase, sale, issue or cancellation of an entity’s own equity instruments.” Both satisfy $Owner_t = -\$5,000$ with $P_t = 0$. QED 7.1.

Economic event: ASR contract with $100M upfront payment, 1.8M shares delivered immediately, final settlement in 6 months.

Initial recognition (Day 1):

Dr. Treasury Stock (1.8M shares)    \$100M
    Cr. Cash                                \$100M

Forward contract: equity-classified at $0 fair value (ASC 815-40-25-7).

Effect: $\Delta E = -\$100M$ (owner distribution via $Owner_t$).

Final settlement (Day 180, additional 0.2M shares delivered):

Dr. Treasury Stock (0.2M shares)    \$10M
    Cr. APIC                                \$10M

Effect: $\Delta E = 0$ (intra-equity reclassification, no P/L impact).

Proof: ASC 815-40-25-7 requires equity classification for share-settled contracts. Settlement reclassifies equity components without gain/loss. Per ASC 505-30-30-8, differences are charged/credited to APIC. Total $\Delta E = -\$100M$ over contract life, identical to simple repurchase. QED 7.2.

7.3 SBC Net Settlement – Tax Withholding Classification (IFRS 2.33E-33H / ASC 718-10-45-9)

Economic event: Employee vests 1,000 RSUs ($50/share). Statutory tax withholding: 22% ($11,000). Entity withholds 220 shares.

Pre-condition: Expense of $50,000 recognized over vesting period (Dr. Compensation Expense, Cr. APIC – SBC Pool).

Settlement (within statutory cap):

Dr. APIC – SBC Pool                  \$50,000
    Cr. Common Stock (780 shares)           \$780
    Cr. APIC – Excess                       \$38,220
    Cr. Cash (tax remittance)               \$11,000

Effect: $\Delta E_{\text{settlement}} = -\$50,000 + \$780 + \$38,220 - \$11,000 = -\$22,000$.

Routing: Per IFRS 2.33G and ASC 718-10-45-9, cash paid for tax withholding (within statutory limits) is $Owner_t$ (share repurchase), not additional expense. The $11,000 withholding reduces equity directly, not via P&L.

If withholding exceeds statutory cap: Excess is expensed ($P_t$), not $Owner_t$.

Proof: Equity bridge accommodates both routings. Within-cap: $Owner_t = -\$11,000$. Above-cap excess: $P_t = -$(excess). Total $\Delta E$ preserved. QED 7.3.

7.4 NCI Up-Ticks and Down-Ticks Without Loss of Control (IFRS 10.23 / IFRS 10.B96)

Economic event: Parent with 80% subsidiary issues additional subsidiary shares to third party, reducing parent’s interest to 70% (no loss of control).

Facts: - Subsidiary equity pre-transaction: $1,000 - Parent’s 80% share: $800 (book value) - New shares issued at fair value: $200 - Post-transaction subsidiary equity: $1,200 - Parent’s 70% share: $840

Journal entry at parent (consolidated):

Dr. Cash (received by subsidiary)   \$200
    Cr. NCI                                 \$120  (30% × \$1,200 - 20% × \$1,000)
    Cr. Parent Equity (APIC or RE)          \$80   (balancing entry)

Effect: Consolidated $\Delta E = 0$ (cash inflow offsets NCI increase and parent equity decrease). Transaction is entirely within equity (IFRS 10.23): “Changes in a parent’s ownership interest in a subsidiary that do not result in the parent losing control of the subsidiary are equity transactions.”

Alternative: Parent purchases additional shares from NCI (up-tick):

If parent buys 5% from NCI for $60 (fair value):

Dr. NCI                              \$50   (5% × \$1,000)
Dr. Parent Equity (APIC or RE)       \$10   (excess over book value)
    Cr. Cash                                \$60

Effect: $\Delta E = -\$60$ (cash outflow), all via $Owner_t$ (no P/L impact).

Proof: IFRS 10.B96 requires “the carrying amounts of the controlling and non-controlling interests shall be adjusted to reflect the changes in their relative interests… Any difference between the amount by which the non-controlling interests are adjusted and the fair value of the consideration paid or received shall be recognised directly in equity and attributed to the owners of the parent.” Both up-ticks and down-ticks preserve $P_t = 0$, routing differences through $Owner_t$. QED 7.4.

7.5 IFRS 16 Lease Accounting – Initial Recognition and Subsequent Measurement

Economic event: Entity leases equipment for 5 years at $10,000/year, discount rate 5%.

Initial recognition (IFRS 16.22-24):

Present value of lease payments:

$$PV = \sum_{t=1}^{5} \frac{10,000}{(1.05)^t} = \$43,295$$

Dr. Right-of-Use Asset              \$43,295
    Cr. Lease Liability                    \$43,295

Effect: $\Delta E = 0$ (balanced entry, no immediate equity impact).

Subsequent measurement (Year 1, IFRS 16.29-36):

Interest expense ($43,295 × 5% = $2,165):

Dr. Interest Expense                 \$2,165
    Cr. Lease Liability                    \$2,165

Depreciation ($43,295 / 5 years = $8,659):

Dr. Depreciation Expense             \$8,659
    Cr. Accumulated Depreciation (ROU)     \$8,659

Payment (year-end):

Dr. Lease Liability                  \$10,000
    Cr. Cash                                \$10,000

Net effect Year 1: - Equity decreases by interest + depreciation = $2,165 + $8,659 = $10,824 (via $P_t$) - Lease liability: $43,295 + $2,165 - $10,000 = $35,460 - ROU asset: $43,295 - $8,659 = $34,636 - Cash: -$10,000

Sale-and-leaseback (IFRS 16.100-103):

If entity sells asset for $50,000 (book value $40,000) and leases back:

Gain recognized: $50,000 - $40,000 = $10,000
But gain limited to portion of rights transferred (not retained).
If leaseback retains substantially all rights (e.g., 95% of asset value), gain deferred = $10,000 × 95% = $9,500.
Immediate gain = $10,000 × 5% = $500 (via $P_t$).

Proof: IFRS 16 ensures lease initial recognition is equity-neutral (asset = liability). Subsequent expense (interest + depreciation) reduces equity via $P_t$. Sale-and-leaseback gains are limited to rights transferred, preventing artificial P&L inflation. All routes preserve continuity. QED 7.5.

Conclusion of Part 7

The five scenarios demonstrate that alternative accounting methods (treasury vs. retirement, ASR two-unit split, SBC classification, NCI reallocation, lease recognition) do not violate the equity bridge identity. Method choice affects presentation (which equity sub-accounts change) but not total equity $\Delta E$. The framework’s routing of transactions through $P_t$, $O_t$, or $Owner_t$ accommodates all IFRS/GAAP-permitted treatments while preserving continuity.

QED Part 7.

Corollary: Empirical Validation Protocol

Given Theorem 3, empirical validation proceeds as follows:

Extract opening and closing equity from the balance sheet.
Extract $P\_t$ from the income statement.
Extract $O\_t$ from the statement of comprehensive income.
Extract owner transactions from the statement of changes in equity (dividends, repurchases, issuances, NCI changes).
Identify translation adjustments, hyperinflation restatements, and measurement adjustments from notes or equity roll-forward.
Compute residual $= \Delta E - \Sigma \text{sources}$.

Pass criterion: $|Residual| \le \epsilon$, where $\epsilon$ is the larger of (a) data rounding tolerance and (b) 1% of opening equity. Residuals outside tolerance indicate missing source terms, misclassification, or extraction errors. The automated implementation is provided in src/validation/equity_bridge_validator.py.

References

IFRS Foundation (2018). Conceptual Framework for Financial Reporting.
IAS 1: Presentation of Financial Statements.
IAS 21: The Effects of Changes in Foreign Exchange Rates.
IAS 29: Financial Reporting in Hyperinflationary Economies.
IAS 32: Financial Instruments: Presentation.
IAS 8: Accounting Policies, Changes in Accounting Estimates and Errors.
IFRS 3: Business Combinations.
IFRS 9: Financial Instruments.
IFRS 10: Consolidated Financial Statements.
ASC 220: Comprehensive Income.
ASC 250: Accounting Changes and Error Corrections.
ASC 505: Equity and ASC 810: Consolidation.
Godley, W., & Lavoie, M. (2007). Monetary Economics: An Integrated Approach to Credit, Money, Income, Production and Wealth.
Ellerman, D. (2014). “On Double-Entry Bookkeeping: The Mathematical Treatment.” Accounting Education, 23(5), 483–501.

Appendix: Mapping to `docs/STANDARDS_CROSSWALK.md`

The 51 source/sink elements catalogued in docs/STANDARDS_CROSSWALK.md map to Theorem 3 categories as follows:

$P\_t$ (P/L) – Lines covering revenues, cost of goods sold, operating expenses, financing costs, and tax expense.
$O\_t$ (OCI) – Lines tagged as FVOCI equity/debt, cash flow hedge effective portion, foreign currency translation, revaluation surplus, actuarial gains/losses, own credit adjustments.
$Owner\_t$ – Dividends, share repurchases, share issuances, buyback excise tax, transaction costs, NCI transactions without loss of control.
$FX\_t$ – CTA reserves when reported outside OCI, currency translation reserves for hyperinflation restatement.
$Hyper\_t$ – IAS 29 net monetary gain/loss, inflation index adjustments.
$Measure\_t$ – Prior period error corrections, cumulative effect of new standards, opening balance adjustments.

This mapping is exhaustive and forms the basis for automated validation. By construction, each XBRL tag in the crosswalk is annotated with the category and presentation guidance, ensuring a direct bridge between reported data and the theoretical framework.

Document status: Publication-ready Last updated: 2025-11-03 (Phase 6 Wave 1) Peer review: Pending adversarial reviewer sign-off Implementation link: src/validation/equity_bridge_validator.py