Theorem 3: Dual‑Balance ⇒ Local Conservation

Statement

Discrete continuity. Under sequential posting of unit entries δ and dual‑balance (double‑entry), sums over closed finite chains telescope to zero. Interpreting postings as tick‑ordered flows on a voxel graph yields a discrete continuity law. In the coarse‑grained limit, this converges to the continuum equation ∂tρ + ∇·J = 0.

In plain English

What flows out of a place must have flowed in from somewhere. Add up the ins and outs and the books balance: nothing appears or vanishes in a closed loop. Zoom out and this bookkeeping line becomes the familiar physics line ∂tρ + ∇·J = 0.

• Why inevitable: it’s double‑entry, just written on a grid. The cancellations on loops leave only boundary flow.
• What it buys: conservation laws as consequences of accounting—not extra laws pasted on top.

Formal setting

• Ledger (T1): ordered abelian ledger with generator δ > 0; page maps ϕ, ψ; double‑entry telescopes on closed chains.
• Atomic ticks (T2): exactly one δ‑posting per tick τ₀; defines a total order on updates.
• Voxel graph: a nearest‑neighbor lattice (e.g., Q₃) with oriented edges. Discrete fields are assigned to vertices (ρ) and edges (J).

Discrete objects

• Density ρ(v, t): ledger page value associated to vertex v at tick t (coarse‑grained count per voxel).
• Current J(e, t): oriented flow assigned to edge e = (u → v) at tick t; positive when δ posts from u to v.
• Divergence: (div J)(v, t) = Σ_{e: in→v} J(e, t) − Σ_{e: v→out} J(e, t).

Key lemmas

1. Telescoping lemma: For any closed finite recognition chain, the signed sum of postings is zero by double‑entry, hence net cost around a closed walk vanishes.
2. Incidence lemma: With a tick‑ordered list of δ‑postings, vertex updates satisfy ρ(v, t+1) − ρ(v, t) = −(div J)(v, t).
3. Scaling lemma: Under coarse‑graining with voxel length L and tick τ₀, the discrete law yields ∂tρ + ∇·J = 0 as L, τ₀ → 0 with fixed c = L/τ₀.

Proof sketch

Index postings by ticks (T2) and lay them on oriented edges of the voxel graph. Double‑entry (T1) ensures credits at a head page match debits at a tail page per posting. Summing over a closed walk cancels interior contributions (telescoping). Rewriting the per‑vertex balance between successive ticks gives a discrete continuity law. Taking a standard scaling limit on a regular lattice recovers ∂tρ + ∇·J = 0.

Edge cases

• Boundaries: On finite regions, a boundary flux term equals the time‑rate change of total interior density (discrete Gauss law).
• Sources/sinks: Open ledgers introduce a source term S; in closed ledgers S = 0 (this theorem’s case).
• Irregular graphs: Replace lattice divergence by incidence‑matrix divergence; the statement remains graph‑topological.

Implications

• Ledger → physics: Conservation laws descend from double‑entry bookkeeping, not added symmetries.
• Bridges to fields: Connects page‑level postings to measurement‑level fields ρ and J.
• Compatibility: Consistent with T8 causal cones (finite signal speed c = L_min/τ₀).

Related

T1 Ledger Necessity T2 Atomicity T4 Unique Cost T8 Causal Speed

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