1.1 Continuity Equations in Nature

The fundamental continuity equation with source terms governing all physical, biological, and economic systems is the continuity equation. This equation describes how a stock variable satisfying discrete continuity—mass, energy, organisms, information, or capital—flows through space and time. Its differential form is:

where $\rho(x, y, z, t)$ is the density of the stock variable satisfying discrete continuity at position $(x, y, z)$ and time $t$, and $\mathbf{J}(x, y, z, t)$ is the flux vector representing the flow of that quantity per unit area per unit time. This equation states a profound truth: the rate at which density accumulates at a point ($\partial \rho / \partial t$) equals the negative divergence of the flux ($-\nabla \cdot \mathbf{J}$). In plain language: if stuff is piling up somewhere, it must be flowing in from somewhere else; if it's draining away, it must be flowing out.

This is not merely a convenient mathematical description—it is the only way to describe systems where quantity is neither created nor destroyed. To see why this form is necessary, consider a small volume element $\Delta V$ in space. The total amount of the stock variable satisfying discrete continuity within this volume is $\int\int\int_{\Delta V} \rho \, dV$. By conservation, the rate of change of this total must equal the net flux through the surface bounding the volume. By the divergence theorem, this surface integral of flux equals the volume integral of $\nabla \cdot \mathbf{J}$, yielding equation (1.1) in the limit $\Delta V \to 0$.

The universality of this equation stems from a single axiom: quantity cannot spontaneously appear or disappear. All other properties of the system—its material constitution, its spatial dimension, its governing forces—are irrelevant. Conservation transcends the specifics of the medium. Mass obeys (1.1) whether the medium is water, air, or plasma. Organisms obey (1.1) whether they are bacteria, fish, or birds. Capital obeys (1.1) whether the entity is a corporation, a government, or a household.

This universal applicability has a profound implication: systems governed by (1.1) are isomorphic in their mathematical structure. They may differ in their physical interpretation, their units of measure, their boundary conditions, but they share the same underlying dynamics. A physicist studying fluid flow, a biologist studying population dispersal, and an accountant studying equity changes are all solving the same equation in different coordinate systems.

1.2 Mathematical Foundations

Before proceeding to applications, we establish the mathematical framework rigorously. The continuity equation (1.1) can be expressed in multiple equivalent forms, each suited to different analytical purposes.

The integral form (1.2) has a clear physical interpretation: the rate of change of the total quantity inside $V$ equals the negative of the net outward flux through the boundary $S$. If more flows out than in, the total decreases; if more flows in than out, the total increases. This form is particularly useful when dealing with discrete entities (e.g., companies, nations, individual organisms) where the concept of "density" at a point is less natural than "total quantity in a region."

A second equivalent form arises by expanding the divergence in Cartesian coordinates. Denoting $\mathbf{J} = (J_x, J_y, J_z)$, the continuity equation becomes:

This form makes explicit that conservation involves four dimensions: three spatial ($x, y, z$) and one temporal ($t$). The quantity $\rho$ is a scalar field on this four-dimensional manifold, and the continuity equation with source terms is a constraint on how this field evolves. In the language of differential geometry, (1.1) states that the four-divergence of the current four-vector $(\rho c, \mathbf{J})$ vanishes, where $c$ is an appropriate dimensional constant. This observation hints at the relativistic generalization of conservation, though we will not pursue that direction here.

For systems with sources or sinks—where the stock variable satisfying discrete continuity can be created or destroyed—the continuity equation generalizes to:

where $S(x, y, z, t)$ is the source term (positive for creation, negative for destruction). In accounting, $S$ might represent external capital injections (equity issuance) or extractions (buybacks). In ecology, $S$ represents births and deaths. The presence of $S \neq 0$ does not invalidate conservation—it merely redefines what we mean by "conserved." If we include the source mechanism in our accounting (e.g., by tracking the origin of new capital), an expanded continuity equation with source terms with $S = 0$ can always be recovered.

Three Formal Proofs

1. Proof 1: Incidence Matrix → Kirchhoff's Law (incidence.py): Balanced double-entry implies 1ᵀP = 0 (graph conservation)
2. Proof 2: Discrete Reynolds Transport Theorem (RTT_FORMAL_PROOF.html): Equity continuity for moving boundaries (M&A, consolidation)
3. Proof 3: Equity Bridge Closure (EQUITY_BRIDGE_PROOF.html): ΔE = P&L + OCI + Owner + Translation + Hyperinflation + Measurement. Proves STANDARDS_CROSSWALK taxonomy is complete and necessary.

1.3 The Universality Claim

We now state the central claim of this work, which will be proven rigorously in subsequent sections:

The proof of Theorem 1.2 requires establishing two directions: (a) that conservation implies the continuity equation, and (b) that structure-preserving mappings retain the continuity equation. Direction (a) was sketched in Section 1.1; direction (b) follows because both $\nabla \cdot \mathbf{J}$ and $\partial \rho / \partial t$ transform covariantly under either continuous changes of variables or discrete aggregations. The details are standard in differential geometry and graph theory and will not be repeated here.

The significance of Theorem 1.2 is this: accounting shares mathematical structure with physics (Kirchhoff's Current Law, continuity equations), expressed on a discrete entity graph. The equation $A = L + E$ is not merely an empirical regularity; it follows from the control-volume aggregation of (1.1). The leverage ratio $A/E$, the sustainable growth rate $g = \text{ROE}(1 - d)$, and the allocation of earnings to non-controlling interests are not arbitrary heuristics—they are theorems that follow from the continuity structure. Understanding accounting deeply means recognizing it can be formalized using discrete conservation principles, providing a mathematical foundation for automated validation.

This perspective transforms accounting education. Traditionally, accounting is taught as a system of rules: debits on the left, credits on the right; assets equal liabilities plus equity; revenue minus expenses equals net income. Students memorize these rules without understanding their necessity. But if accounting is recognized as applied conservation, the rules cease to be arbitrary. The double-entry system is not a convention—it is the only consistent way to track flows in a conserved system. The balance sheet is not a report format—it is a snapshot of the density field $\rho(e, t)$ at a fixed time $t$. The income statement is not a list of transactions—it is the temporal derivative $\partial \rho / \partial t$.

In the sections that follow, we develop this perspective rigorously. Section 2 demonstrates that the same equation (1.1) governs physics, biology, network theory, and accounting, differing only in the choice of coordinates and the interpretation of variables. Section 3 shows that the fundamental accounting identity $A = L + E$ is equivalent to (1.1) after aggregating the continuum equation over discrete entity control volumes. Section 4 derives standard financial identities from first principles, showing they are mathematical consequences of conservation. Section 5 provides empirical validation on real financial data, confirming that deviations from the theoretical predictions are small and attributable to measurement error. Section 6 concludes by discussing implications for accounting practice, education, and research.

2.1 Cross-Domain Comparison

To establish that equation (1.1) is truly universal, we now present its manifestation in four distinct domains: classical mechanics (fluid dynamics), population biology, network theory, and accounting. The key observation is that the mathematical form is identical across all four—only the interpretation of $\rho$ and $\mathbf{J}$ changes. This is not metaphor or analogy; it is structural conservation.

Several observations emerge from Table 2.1. First, the continuity equation with source terms can be written in structurally similar forms across domains. This similarity is pedagogically useful and suggests underlying mathematical patterns, though each domain has distinct measurement rules. Second, the density $\rho$ generally represents "how much per unit of the relevant coordinate" (spatial volume in physics, area in biology, nodes in networks, entities in accounting). Third, the flux $\mathbf{J}$ conceptually represents "flow across a boundary," though the nature of boundaries differs substantially (physical surfaces vs. accounting consolidation perimeters). The analogy illuminates accounting structure but should not be interpreted as physical equivalence.

Fourth, and most subtly, the coordinate systems differ in dimensionality and topology. Physics uses continuous three-dimensional Euclidean space $(x, y, z)$ plus time. Biology often uses two-dimensional spatial domains (e.g., latitude-longitude on Earth's surface) plus time. Network theory uses discrete graph topology (nodes and edges) plus time. Accounting uses discrete entity labels plus time. Despite these topological differences, the continuity equation with source terms (1.1) holds in all cases because it is a local statement about infinitesimal volume elements (or their discrete analogues). The divergence operator $\nabla \cdot$ adapts to the coordinate system: in Cartesian coordinates it is $\partial/\partial x + \partial/\partial y + \partial/\partial z$; on a graph it becomes a sum over adjacent nodes; in accounting it becomes differences across entity boundaries.

Fifth, the source term $S$ generalizes conservation to allow creation/destruction of the quantity, but this does not undermine the universality of (1.1)—it merely shifts what we mean by "the stock variable satisfying discrete continuity." In biology, for example, if we define $\rho$ as organism count (not biomass), then births and deaths act as sources and sinks. But if we instead define $\rho$ as total mass (organisms plus their environment), conservation is restored with $S = 0$. Similarly, in accounting, if equity issuance is treated as a source, conservation holds for "original equity"; if issuance is included in the definition of equity, conservation holds for "total equity." The choice of accounting boundary determines whether $S = 0$ or $S \neq 0$, but the form of the equation remains (1.1) or (1.4).

Sixth, the typical partial differential equations (PDEs) shown in the last row of Table 2.1 are derived from (1.1) by specifying the functional form of $\mathbf{J}$. In physics, Fick's law $\mathbf{J} = -D \nabla \rho$ (diffusion) leads to the heat/diffusion equation. In biology, adding a logistic source term leads to the Fisher-KPP equation. In networks, specifying routing rules gives graph dynamical systems. In accounting, defining $\mathbf{J}$ as earnings minus dividends leads to the retained earnings equation. All of these are special cases of (1.1), obtained by choosing particular constitutive relations for $\mathbf{J}$.

This last point is crucial: the continuity equation with source terms (1.1) is more fundamental than any specific PDE. The heat equation, the diffusion equation, the Fisher-KPP equation, the retained earnings equation—all are consequences of (1.1) plus auxiliary assumptions about the nature of the flux. Accounting textbooks often present the retained earnings equation as a starting point, but this obscures its derivation from conservation. By recognizing (1.1) as primary, we unify disparate accounting identities under a single framework.

2.2 Structural Mapping Across Scales

The claim that accounting shares continuity structure with physics requires more than observing that both satisfy (1.1). We must show that one can be transformed into the other by a coordinate change, and that this transformation preserves the mathematical structure. This is the content of the next theorem.

Accounting as Aggregation, Not Coordinate Transformation

Accounting aggregates spatial distributions into entity-level totals. This is not a smooth change of variables for three reasons:

1. Discrete Entity Index: The entity identifier $e$ is a discrete label (ticker symbol, legal entity ID), not a continuous coordinate. There is no smooth manifold structure.
2. Aggregation, Not Bijection: Multiple physical locations $(x_1, y_1, z_1), (x_2, y_2, z_2), \ldots$ map to the same entity $e$. This is a many-to-one projection, not an invertible transformation.
3. Measurement Regimes: IFRS/GAAP measurement rules (fair value, amortized cost, impairment) are policy choices, not physical laws. The same physical inventory can have different balance sheet values under different standards.

The Correct Analogy: Structural Isomorphism

Physics and accounting share the mathematical structure of continuity equation with source termss:

Both domains satisfy a continuity equation:

• Physics: $\nabla \cdot \mathbf{J} = 0$ for internal flows (no sources/sinks)
• Accounting: $\mathbf{1}^T \mathbf{P} = 0$ for internal postings (debits = credits)

This is a discrete-continuous analogy (graph theory ↔ differential geometry), not a literal change of coordinates. See Ellerman (2014) and Liang (2020) for graph-theoretic foundations of double-entry bookkeeping.

4.3 Reynolds Transport Theorem for Moving Boundaries (M&A)

When the entity boundary changes (mergers, acquisitions, spin-offs), the consolidation perimeter $\Omega(t)$ moves over time. The Reynolds Transport Theorem decomposes equity change into internal operations and boundary flux.

Discrete Form (Accounting)

For a consolidation group $\mathcal{G}_t$ observed at discrete reporting dates, total equity change decomposes as:

Where:

• $\Delta E_i^{\text{internal}} = \text{NI}_i + \text{OCI}_i - \text{Div}_i$ for entity $i$ present in both periods
• $E_{\text{acquired}}$ = book equity of newly acquired subsidiaries (IFRS 3 / ASC 805)
• $E_{\text{divested}}$ = book equity of divested subsidiaries (IFRS 10.B97-B99 / ASC 810-10-40)
• $\Phi_{\text{owner}} = -\text{Div}_{\text{parent}} - \text{Repurch}_{\text{parent}} + \text{Issue}_{\text{parent}} + \Delta \text{NCI}_{(\text{no loss of control})}$

Standards Mapping: NCI Transactions Without Loss of Control

An essential component of the discrete RTT is capturing NCI equity transactions per IFRS 10.23 and ASC 810-10-45-23.

Dr. Cash                         $15M
    Cr. Noncontrolling Interest          $15M

IFRS/GAAP Mapping

Implementation: See src/core/reynolds_transport.py for the discrete decomposition algorithm and M\&A event detection.

For a comprehensive mapping of every source and sink term to authoritative standards and XBRL tags, consult the Standards Crosswalk.

References:

• Reynolds, O. (1903). The Sub-Mechanics of the Universe. Cambridge University Press.
• Truesdell, C., & Toupin, R. (1960). The classical field theories. In Handbuch der Physik, Vol. III/1. Springer.
• Cimbala, J. M. (2012). The Reynolds Transport Theorem. Penn State Lecture Notes. Link

3.1 The Accounting Coordinate System

We now define the accounting coordinate system rigorously. Let $(x, y, z, t)$ denote standard physical coordinates: three spatial dimensions plus time. An entity is a region $V_e \subset \mathbb{R}^3$ in physical space, typically defined by legal boundaries (e.g., property owned or controlled by a corporation). The entity label $e$ is a discrete coordinate taking values in some index set $\mathcal{E}$. Time $t$ remains continuous (or discretized uniformly, as in quarterly reporting).

Define the equity density in accounting coordinates as:

where $\rho_{\text{equity}}(x, y, z, t)$ is the physical equity density (in dollars per cubic meter, for example). This definition treats equity as a spatially distributed quantity that is aggregated over the entity's physical extent. In practice, $\rho_{\text{equity}}$ is concentrated at certain locations (corporate headquarters, bank accounts, equipment) and zero elsewhere, but mathematically we can extend it to all of space by setting it to zero outside $V_e$.

Similarly, define the equity flux in accounting coordinates as:

where $S_e$ is the boundary surface of region $V_e$, and $\mathbf{J}_{\text{equity}}$ is the physical equity flux vector (in dollars per square meter per time). The outward normal $\hat{\mathbf{n}}$ ensures that positive $\mathbf{J}_E$ represents net outflow from the entity.

With these definitions, we can state the accounting version of the continuity equation:

Proposition 3.1 shows that equity in accounting coordinates obeys a simple first-order ordinary differential equation (ODE), which is the discrete analogue of the PDE (1.1). The flux $\mathbf{J}_E$ represents net equity flows out of the entity per unit time. In accounting terms, $-\mathbf{J}_E$ is "net income" (positive for income, negative for losses), and $S$ represents external capital transactions (positive for injections, negative for buybacks). Thus, equation (3.3) is precisely the retained earnings equation:

This is not a new accounting principle—it is a restatement of (3.3) using familiar terminology. The insight is that (3.4) is not an empirical regularity or a convention; it is a mathematical theorem obtained by integrating the continuity equation (1.1) over discrete entity control volumes.

3.2 The Fundamental Accounting Identity

We now prove the central result of this work: the accounting identity $A = L + E$ is equivalent to the continuity equation with source terms.

3.3 The Discrete Bridge: Double-Entry as Graph Divergence

The continuity equation operates on continuous fields. Accounting operates on discrete ledgers. The bridge is incidence matrix algebra, treating each journal entry as a directed edge in a graph. This is precisely the mathematical formalization advocated by Ellerman (2014).

             E1   E2
    Cash   [  1    0 ]
    Revenue[ -1    1 ]
    Equity [  0   -1 ]
    

Connection to the PDE

The continuous PDE ($\nabla \cdot \mathbf{J} = 0$) becomes discrete ($\mathbf{1}^T \mathbf{P} = \mathbf{0}$ for internal postings). Both express: "Net flow into any region equals zero." Graph divergence is therefore the discrete avatar of $\nabla \cdot \mathbf{J}$, and double-entry bookkeeping enforces this continuity equation with source terms exactly.

Source Term Classification: Physics Analogy

The stock-flow equation decomposes balance changes into internal flows and external sources:

Where $\mathbf{P} \mathbf{a}$ represents internal postings (revenue, expenses) and $\mathbf{s}$ represents external sources/sinks. The physical analogy is:

This mapping makes explicit that net income arises from internal postings (not a source term), whereas dividends, OCI, and M&A cross the entity boundary or remeasure the state vector.

3.4 Limitations of the Physical Analogy

While the continuity equation with source terms framework provides powerful insights, we must acknowledge where the physics analogy breaks down. Accounting is not identically equivalent to fluid mechanics or mass conservation - it is structurally analogous but with important differences.

Limitation 1: Negative Equity

In physics, density $\rho$ (mass per unit volume) cannot be negative. Negative mass is non-physical. However, in accounting, equity can be negative when liabilities exceed assets. Example: Boeing (BA) reported parent equity of $E^P = -\$3.3$ billion in Q2 2025 due to aggressive share buybacks. The accounting identity $A = L + E$ still holds ($\$155B = \$158B - \$3B$), but the physical interpretation of "equity density" breaks down. There is no physical system with negative density that remains stable.

Implication: The continuity structure (change = inflows minus outflows) remains valid, but equity is better understood as a signed measure rather than a physical density. Negative equity represents a state where all assets are claimed by creditors with obligations exceeding available resources - economically precarious but mathematically well-defined.

Limitation 2: Discrete Entities vs. Continuous Fields

The continuity equation $\nabla \cdot \mathbf{J} + \frac{\partial \rho}{\partial t} = 0$ assumes continuous spatial coordinates $(x, y, z)$ where calculus applies. Accounting uses discrete entity labels $(e)$ representing legally distinct corporations. The divergence operator $\nabla \cdot$ does not apply to discrete coordinates in the standard sense - we require a discrete analogue (finite differences over entity boundaries).

Furthermore, entities are not fixed volumes. Mergers and acquisitions change entity boundaries over time, violating the "fixed region $V$" assumption of Theorem 1.1. When Company A acquires Company B, the pre-merger continuity equation with source termss for A and B separately must be combined, accounting for the flux across the newly merged boundary. This requires careful treatment of purchase accounting adjustments (fair value step-ups, goodwill recognition) that have no direct physical analogue.

Limitation 3: Fraud and Measurement Error

In physics, if a measurement violates conservation (e.g., energy appears to be created), we conclude the measurement is wrong or we've discovered new physics. In accounting, if measured values satisfy $A = L + E$ perfectly, this does not guarantee the numbers are correct - they may be fraudulent.

Example: Enron's balance sheets satisfied $A = L + E$ at the time of publication, but the values were falsified through off-balance-sheet Special Purpose Entities (SPEs). The continuity equation with source terms held for the reported numbers but not the true numbers. Physics conservation is ontological (mass cannot be created); accounting conservation is epistemological (reported claims must balance, but reports can lie).

Limitation 4: Topology Changes (Mergers, Spin-offs)

Physical systems maintain topological continuity - a volume of fluid cannot instantaneously split or merge. Corporate entities undergo discrete topology changes: mergers (two entities → one), spin-offs (one entity → two), and liquidations (entity ceases to exist). These events violate the smooth-boundary assumptions underlying the divergence theorem.

Conclusion: The continuity framework is a powerful structural analogy that illuminates accounting dynamics, but it is not a perfect isomorphism. The core insight - that equity flows satisfy continuity with explicit source terms (change = sources minus uses) - remains valid. However, claims that "accounting IS physics" must be tempered with recognition of discrete coordinates, signed measures (negative equity), and the possibility of fraudulent reporting.

4.1 The Leverage Identity

4.2 The Sustainable Growth Formula

4.3 Non-Controlling Interest Dynamics

4.4 The Dividend Distribution Formula

4.5 Formal Proofs of Core Theorems

This section provides rigorous mathematical proofs of the framework's foundational claims, demonstrating that accounting method choices (treasury vs. retirement, APIC vs. Retained Earnings allocation) are gauge transformations that preserve total equity.

Dr. Additional Paid-In Capital (APIC)   $50,000
    Cr. Retained Earnings                       $50,000
    

Dr. Treasury Stock (contra-equity)   $P
    Cr. Cash                                 $P
        

Dr. Common Stock (par)                 $N \cdot v
Dr. Additional Paid-In Capital         $N \cdot a
Dr. Retained Earnings (plug)           $P - N(v + a)  [if P > N(v+a)]
    Cr. Cash                                       $P
        

Dr. Common Stock (par)                 $N \cdot v
Dr. Additional Paid-In Capital         $N \cdot a
    Cr. Cash                                       $P
    Cr. Additional Paid-In Capital (gain)          $N(v+a) - P
        

Dr. Treasury Stock              $50,000
    Cr. Cash                            $50,000
    

Dr. Common Stock ($1 x 1,000)    $1,000
Dr. APIC ($10 x 1,000)          $10,000
Dr. Retained Earnings (plug)    $39,000
    Cr. Cash                            $50,000
    

Dr. Treasury Stock (or equity accounts)   $P + 0.01 * P
    Cr. Cash                                      $P
    Cr. Excise Tax Payable                        $0.01 * P
    

Dr. Excise Tax Payable            $0.01 * P
    Cr. Cash                              $0.01 * P
    

6.4 Buyback Excise Tax (IRA 2022 §4501)

The Inflation Reduction Act of 2022 introduced a 1% excise tax on net stock repurchases by publicly traded US corporations, effective January 1, 2023.

Accounting Treatment

Per industry practice (PwC, Deloitte guidance), the excise tax is typically treated as:

• Direct equity cost of the repurchase (not income tax expense)
• Reduces equity along with the cash paid for shares
• NOT recorded in P&L (aligns with IAS 32.33 / ASC 505-30 principles)

Example: Lowe's Companies (LOW) Q3 2024

Conservation equation:
ΔE = NI + OCI - Div - Repurchases - Excise Tax
ΔE = NI + OCI - Div - (4,000M + 40M)

Netting with Issuances

Per §4501(a)(1)(B), excise tax applies to net repurchases:

• If entity repurchases $100M and issues $30M in same year → Net $70M
• Excise tax = 1% × $70M = $700K

Code reference: src/validation/own_share_accounting.py::extract_excise_tax()
Tests: tests/test_own_share_accounting.py::test_extract_excise_tax_*
Documentation: OWN_SHARE_ACCOUNTING.md Section 2

Note: Tax policy and accounting treatment may evolve. Always consult Deloitte DART §7.16 or PwC Viewpoint (Treasury Stock) for latest guidance.