Multi-Entity Equity Continuity | Consolidation Framework

🎯 Key Insight: Internal Flows Cancel

When a parent owns subsidiaries, equity flows between entities (intercompany loans, management fees, dividends) cancel at the group level. This is the discrete analog of Kirchhoff's Law applied to multi-entity graphs.

Example: Parent lends $1M to Subsidiary. At consolidation:

Parent: -$1M (loan asset)
Subsidiary: +$1M (loan liability)
Group total: $0 (internal flows cancel via elimination entries)

Only external sources (group-level income, dividends to outside shareholders, M&A) change consolidated equity. This theorem formalizes IFRS 10 consolidation mechanics mathematically.

Theorem 4: Multi-Entity Continuity

Let $\mathcal{E}$ be the set of entities in a consolidation group. For each entity $i \in \mathcal{E}$, equity evolves as:

$E_{t+1}^{(i)} - E_t^{(i)} = (B F_t)^{(i)} + S_t^{(i)} - L_t^{(i)} + R_t^{(i)}$

Where:

$B F_t$: Internal flows (intercompany transactions)
$S_t$: External sources (P + O + Owner)
$L_t$: Liability reclassifications (written puts)
$R_t$: Boundary flux (M&A events)

At the consolidated group level, summing all entities:

$\sum_i (E_{t+1}^{(i)} - E_t^{(i)}) = \sum_i (S_t^{(i)} - L_t^{(i)} + R_t^{(i)})$

Internal flows $B F_t$ cancel due to Kirchhoff's Law ($\mathbf{1}^T B = 0$)

⭐ Corollary 1 (IFRS 10 Simplification)

Under full IFRS 10 compliance with purchase accounting (IFRS 3), boundary flux terms $R_t = 0$ for standard M&A transactions. Acquisitions are treated as asset swaps, and disposals flow through P&L. The equation simplifies to:

$\Delta E = S_t - L_t$

The analogy to Reynolds Transport Theorem (moving boundaries in fluid mechanics) provides intuition for M&A boundary effects but is not a rigorous mathematical derivation. The discrete accounting formulation is inspired by RTT structure, not derived via graph limits or continuum mechanics.

Mathematical Note: This is a structural analogy (pedagogical) not a formal limit theorem. A rigorous discrete-continuous correspondence would require graph limit theory (Lovász & Szegedy 2006) or discrete exterior calculus, which is beyond current scope.

Empirical Finding: Validation on 14 IFRS 10 scenarios (docs/proofs/CONSOLIDATION_ORACLES.md) shows boundary_flux ≈ 0 in all standard purchase accounting cases.

Full technical proof with incidence matrix formulation, Kirchhoff lemmas, and empirical validation protocol is available in the markdown version.
View Complete Proof (Markdown)

🏢 Multi-Entity Equity Continuity

🎯 Key Insight: Internal Flows Cancel

⭐ Corollary 1 (IFRS 10 Simplification)