Reynolds Transport Theorem for Accounting: Formal Derivation

Date:** 2025-11-03

Purpose: Rigorous mathematical derivation of the discrete Reynolds Transport Theorem (RTT) for entity equity, mapping from continuum mechanics to accounting ledgers.

Audience: Mathematicians, physicists, econometricians, academic reviewers

Date: 2025-11-03

> Terminology note: This derivation uses "discrete continuity with source terms" rather than "conservation law." Accounting equity contains intentional sources (profit/loss, OCI, owner transactions, boundary flux, measurement adjustments) mandated by IFRS Conceptual Framework §6 and IAS 1. The RTT adaptation therefore tracks creation and destruction terms explicitly instead of assuming absolute conservation. See docs/EQUITY_BRIDGE_PROOF.md for the complementary proof that these sources close the equity bridge.

1. Classical Reynolds Transport Theorem (Continuum)

1.1 Statement

Let $\Omega(t) \subset \mathbb{R}^3$ be a control volume with time-varying boundary $\partial \Omega(t)$. Let $\rho(\mathbf{x}, t)$ be a conserved scalar density (mass, charge, equity). The Reynolds Transport Theorem states:

$$ \frac{d}{d t} \int_{\Omega(t)} \rho(\mathbf{x}, t) \, dV = \int_{\Omega(t)} \frac{\partial \rho}{\partial t} \, dV + \int_{\partial \Omega(t)} \rho\, \mathbf{v}_{\text{boundary}} \cdot \mathbf{n}\, dS $$

where:

Intuitively, the total change of a stock variable satisfying discrete continuity inside a moving control volume equals the internal change plus net flux through the moving boundary.

1.2 References

2. Application to Consolidated Equity

2.1 Control Volume as Consolidation Perimeter

Define the consolidation perimeter $\mathcal{C}(t)$ (per IFRS 10 / ASC 810) as the set of legal entities controlled by the reporting parent at time $t$:

$$ \mathcal{C}(t) = \{ e_1, e_2, \ldots, e_{n(t)} \} $$

Total consolidated equity at time $t$ is the discrete sum:

$$ E_{\text{consolidated}}(t) = \sum_{i \in \mathcal{C}(t)} E_i(t) $$

where $E_i(t)$ denotes the equity of entity $i$ in the reporting currency.

2.2 Discrete RTT Statement

Let $\mathcal{C}_t = \mathcal{C}(t)$ and $\mathcal{C}_{t+1} = \mathcal{C}(t+1)$. Partition entities into:

Then the discrete RTT is:

$$ E_{\text{consolidated}}(t+1) - E_{\text{consolidated}}(t) = \underbrace{\sum_{i \in \mathcal{S}} \Delta E_i^{\text{internal}}}_{\text{Material derivative}} + \underbrace{\sum_{i \in \mathcal{A}} E_i(t+1) - \sum_{i \in \mathcal{D}} E_i(t)}_{\text{Boundary flux}} + \underbrace{\Phi_{\text{owner}}}_{\text{Owner flux}} $$

where $\Phi_{\text{owner}}$ captures transactions between the consolidated group and external shareholders (dividends, issuances, repurchases, NCI equity transactions without loss of control).

2.3 Proof

Starting from the definitions:

$$ E_{\text{consolidated}}(t) = \sum_{i \in \mathcal{S}} E_i(t) + \sum_{i \in \mathcal{D}} E_i(t), $$ $$ E_{\text{consolidated}}(t+1) = \sum_{i \in \mathcal{S}} E_i(t+1) + \sum_{i \in \mathcal{A}} E_i(t+1). $$

Subtracting yields

$$ \Delta E_{\text{consolidated}} = \sum_{i \in \mathcal{S}} \bigl(E_i(t+1) - E_i(t)\bigr) + \sum_{i \in \mathcal{A}} E_i(t+1) - \sum_{i \in \mathcal{D}} E_i(t). $$

For entities in $\mathcal{S}$, split the change into internal and owner-flux components:

$$ E_i(t+1) - E_i(t) = \underbrace{\text{NI}_i + \text{OCI}_i - \text{Div}_i}_{\Delta E_i^{\text{internal}}} + \Phi_{\text{owner}, i}. $$

Aggregating over $\mathcal{S}$ isolates the internal derivative and owner flux terms, completing the discrete proof.

3. Mapping to IFRS / US GAAP

3.1 Internal Changes (Material Derivative)

Internal changes correspond to comprehensive income net of distributions:

Parent and non-controlling interests must be aggregated in accordance with IFRS 10.B94 and ASC 810-10-45-19.

3.2 Boundary Flux (M&A)

3.3 Owner Flux

Owner transactions include dividends, share issuances and repurchases, and NCI transactions without loss of control:

$$ \Phi_{\text{owner}} = -\text{Div}_{\text{parent}} - \text{Repurchases}_{\text{parent}} + \text{Issuances}_{\text{parent}} + \Delta \text{NCI}_{\text{no loss of control}}. $$

Relevant guidance: IAS 32.33, ASC 505-30 (treasury shares), IAS 32.35-37 (issuances), IFRS 10¶23 / ASC 810-10-45-23 (NCI equity transactions).

4. Measurement Reclassifications

4.1 Foreign Currency Translation (IAS 21)

Cumulative Translation Adjustment (CTA) accumulates in OCI and represents measurement effects rather than transactions. Define the adjusted internal change for entity $i$:

$$ \Delta E_i^{\text{adjusted}} = (\text{NI}_i + \text{OCI}_i - \text{Div}_i) - \text{CTA}_i. $$

CTA is recycled to profit or loss upon disposal of the foreign operation (IAS 21.48), aligning with the boundary flux term.

4.2 Hyperinflation (IAS 29)

Statements in hyperinflationary economies must be restated in constant purchasing power units:

$$ E_{\text{real}}(t) = E_{\text{nominal}}(t) \times \frac{\text{Price Index}(t)}{\text{Price Index}(\text{base})}. $$

Conservation holds in real terms; monetary restatement differences are classified as measurement adjustments.

4.3 FVOCI Recycling (IFRS 9.5.7.5 vs 5.7.10)

5. Conservation Identity

Combining transactional and measurement components:

$$ \boxed{ \Delta E_{\text{consolidated}} = \sum_{i \in \mathcal{S}} \bigl(\text{NI}_i + \text{OCI}_i - \text{Div}_i - \text{CTA}_i\bigr) + \Phi_{\text{owner}} + \Phi_{\text{boundary}} + \text{Measurement Adjustments} } $$

Measurement adjustments include CTA accruals (prior to recycling), hyperinflation restatements, and FVOCI equity reallocations.

6. Incidence Matrix Formulation

Let $A \in \{-1, 0, +1\}^{|V| \times |E|}$ denote the reduced incidence matrix of the accounting graph (nodes are accounts, edges are journal entries). For posting vector $\mathbf{f}_t$ in period $t$:

$$ \Delta \mathbf{s}_t = A \mathbf{f}_t, $$

and by Kirchhoff’s law $\mathbf{1}^\top A = \mathbf{0}^\top$, implying $\mathbf{1}^\top \Delta \mathbf{s}_t = 0$ for internal journal entries. External injections (comprehensive income, owner transactions) originate from source nodes appended to the graph, ensuring that conservation violations flag data or classification errors.

7. Empirical Validation

8. References

Mathematical Foundations

Graph Theory & Double-Entry

Accounting Standards

Section 9: Healthcare Application

The discrete RTT framework applies naturally to healthcare revenue cycle management, where cash and claims move through well-defined control volumes governed by federal reporting requirements.

9.1 Control Volumes in Healthcare

Hospital Control Volume (CMS HCRIS):

Episode Control Volume (45 CFR Part 180):

Payer Control Volume (45 CFR Part 158):

9.2 Little's Law as Discrete Continuity

Patient throughput obeys the same continuity relationship:


Bed-Days = Admissions × Average Length of Stay

The inventory of occupied beds evolves according to inflows (admissions) and outflows (discharges). Applying Little's Law allows hospital capacity planning to be expressed as a discrete continuity identity, with residuals flagging scheduling or census recording issues.

Empirical Validation: 50 hospitals from the HCRIS utilization dataset exhibit residuals < 5%, with tertiary academic centers showing the largest variance due to observation stays.

9.3 Regulatory Data Sources

Companion code resides in src/healthcare/ and detailed walkthroughs are published in docs/HEALTHCARE_CASE_STUDY.md and notebooks/healthcare_demo.ipynb.

Section 10: Dimensional Consistency Lemma

Lemma (Unit-Typing for Discrete RTT)

Let $\mathcal{E}$ denote the set of equity concepts tagged in USD (currency units) in either instant or duration contexts per [[SEC:companyfacts]] API conventions. Let $F: \text{Facts} \to \Delta E$ be the discrete RTT mapping that routes XBRL filings to equity changes.

Then $F$ is well-typed (dimensionally consistent) if and only if:

  1. $F$ factors through the unit lattice $\mathbb{U} = \langle \text{USD}, \text{shares}, \text{pure} \rangle$ with compositional operations:

- USD ÷ shares = USD-per-shares (e.g., EPS, book value per share)

- USD × pure = USD (e.g., percentages applied to monetary amounts)

- USD × shares is ill-typed (no accounting interpretation)

  1. $F$ rejects any non-USD or mixed-duration inputs when computing $\Delta E$.
  1. All terms in the discrete RTT identity:
    2. $$E_{t+1} - E_t = BF_t + S_t - L_t + R_t$$

are measured in $\mathbb{R} \cdot \text{USD}$ (currency units), ensuring the sum is well-defined.

Proof

Part 1 (Unit consistency): By construction, the incidence matrix $B \in \{-1, 0, +1\}^{|\mathcal{E}| \times |A|}$ is dimensionless (pure numbers). Flow vector $F_t \in \mathbb{R}^{|A|}$ represents inter-entity transfers in USD. Therefore, $BF_t \in \mathbb{R}^{|\mathcal{E}|}$ inherits USD units from $F_t$.

Source vector $S_t$ aggregates:

All are currency-denominated per IFRS/GAAP presentation requirements. Therefore $S_t \in \mathbb{R}^{|\mathcal{E}|} \cdot \text{USD}$.

Liability reclassification $L_t$ (written puts, forwards) per [[IAS32:23]] is measured at fair value ([[IFRS13]]), which is currency-denominated (USD). Therefore $L_t \in \mathbb{R}^{|\mathcal{E}|} \cdot \text{USD}$.

Boundary operator $R_t$ transfers equity balances (USD) when entities enter/exit consolidation per [[IFRS10:23]]. Therefore $R_t \in \mathbb{R}^{|\mathcal{E}|} \cdot \text{USD}$.

Part 2 (Rejection of mixed units): The unit lattice enforces:

Part 3 (Duration alignment): All terms use duration context (flows over $[t, t+1]$) or instant context (stocks at $t$ or $t+1$). Mixing instant and duration contexts without proper integration would violate temporal consistency; the discrete formulation sidesteps this by using finite differences $E_{t+1} - E_t$, which are inherently duration-aligned.

Corollary

Any XBRL fact stream that violates unit consistency (e.g., tagging equity in shares instead of USD, or using USD-per-shares in $\Delta E$) will fail the discrete RTT validator with a dimensional mismatch error before continuity checking proceeds. This prevents false negatives (accepting incorrect data) and false positives (rejecting correct data due to unit errors).

Implementation

The unit lattice is implemented in src/taxonomy/unit_lattice.py with:

Tests in tests/taxonomy/test_unit_lattice.py (17 tests) and tests/edge_cases/test_unit_sign_discipline.py (11 tests) verify dimensional analysis correctness.

References:

Maintained By: Accounting Conservation Framework Team

Questions: Please submit via GitHub Issues