An exact finite weak conservation identity

Summary

Discrete models of spacetime need to connect to continuum physics somewhere, and conservation is the usual place. The temptation is to write down a discrete balance equation, take a limit, and announce that a continuum conservation law has emerged. Almost all of the content is hidden in the limit.

This result separates the two halves honestly. The finite half is exact and unconditional: a properly declared fixed-incidence conservation ledger has an exact finite weak identity, derivable with no choices made anywhere. The continuum half is conditional: it is an interface lemma resting on six hypotheses that are stated and left undischarged. It is a bridge with one end firmly attached and the other end honestly labelled as not yet attached.

1. The master identity

Strip every normalisation. A finite locality structure supplies only cells i, oriented interfaces e, and an incidence B_ie that is +1 if e leaves i, −1 if it enters, and 0 otherwise. The ledger supplies four extensive processes — each an amount rather than a density:

E_i(t)    total content of cell i
Phi_e(t)  total transport through interface e up to time t
S_i(t)    total volumetric source delivered into cell i
W_i(t)    total external port work delivered into cell i

and the ledger itself is dE_i + Σ_e B_ie dPhi_e = dS_i + dW_i. Pairing with an arbitrary test function φ on cells and summing, using only the elementary identity Σ_i B_ie φ_i = −(dφ)_e, gives

Σ_i φ_i dE_i  −  Σ_e (dPhi)_e (dφ)_e  =  Σ_i φ_i dS_i + Σ_i φ_i dW_i

with no choices made anywhere. The two ds are different objects: one is a time increment of the integrated transport, the other a raw spatial difference of the test function. Conflating them is the standard way this calculation goes wrong.

The point of stripping the measures out is that it makes visible which later statements are consequences of the structure and which are consequences of a convention. Once normalisations are reintroduced they can be checked against this identity rather than assumed.

2. Exactly what is established

The accepted result is an exact finite weak identity plus a conditional discrete-to-continuum interface lemma, with:

• fixed incidence over each time-integrated finite bridge;
• a general source-retaining total balance;
• source-free reduction only under a declared local-exchange closure;
• static normalisation as the default;
• moving normalisation on fixed incidence supported only conditionally;
• topology-changing incidence explicitly outside scope;
• six continuum hypotheses undischarged.

That last line is the result's real content. It would have been easy to discharge the hypotheses by assumption and claim a continuum conservation law; the lemma instead names them and stops.

3. Claim ceiling

The lemma does not establish any of:

convergence of a microscopic model     continuum emergence
stress-energy                          spacetime
dimension                              metric
Lorentz invariance                     gravity

No conservation law, no current, no stress tensor, no geometry, no dimension, no Lorentz invariance and no gravity is established, claimed or implied.

4. Why it was derived, not patched

This result is worth reading as a methodological case study, because the version before it was wrong in an instructive way.

An earlier repair had flipped a sign and appended a boundary term. Independent review found the appended term spurious, found the sign question misdiagnosed — the conflict was not internal to one section but between two sections' hypotheses — and found a third hole nobody had stated at all: the edge inner-product weight was undeclared.

The accepted version therefore does not patch anything. It declares the spaces and their inner products, takes the genuine adjoint, and reads off what the identity must be. Every finding then arrives as a consequence rather than as an adjustment. That is the difference between a result and a repair, and it is why three separate defects closed at once.

Several claims from the superseded version were withdrawn outright in the process — including a universal weighting claim that turned out to be true only under one convention, a justification whose stated reason was wrong even though its conclusion survived, and a frozen test class that failed for deterministic tests that jump. Those withdrawals are recorded in the source alongside what replaced them.

5. How it was reviewed

This result passed a fresh review by an agent that did not author it. The reviewer ran 119 checks with zero failures — 53 core, 34 on the conditional branch, 32 cross-checks — in exact rational arithmetic, on a reviewer-constructed 6-cell / 8-edge complex disjoint from both the author's complex and the prior reviewer's. Constructing a fresh complex matters: a check that reuses the author's example can only confirm the author's arithmetic.

Author consistency suites at the accepted commit all exit zero, and both figures were independently reproduced before integration.

6. Known imperfections

The reviewer recorded four non-blocking observations and explicitly instructed that the accepted commit must not be cleaned up. There is no revision 5. Any later editorial cleanup must be separated, classified, and freshly reviewed if it changes accepted scientific text.