e_200313

In the 4D Regge exact-midpoint M2 TT identity work, two integer kernels on six Fin 4 indices are compared. m2Num is the folded sum of coupling contributions over a fixed list: it accumulates contrib t a b c d i j for each term t in couplingZList. explicitZ is a sparse case table of the same arity, returning small integers such as $4$, $-2$, and (by default) $0$ on unmatched patterns.

The local module is chunk 8 of a 256-way partition of kernel decides. Each chunk theorem pins one concrete six-tuple so the global assembly can case-split without re-evaluating the fold at every leaf.

Upstream, only the two definitions are needed: the fold that builds m2Num and the table that defines explicitZ. No analytic lemma about Regge calculus is invoked at this leaf.

One-line proof by decide. Both sides reduce to concrete Int values once the six indices are literals: the left-hand side evaluates the fold of contrib over couplingZList at $(2,0,0,3,1,3)$; the right-hand side looks up explicitZ and multiplies by $8$. The kernel closes the integer equality with no tactics beyond decision.

This leaf is one of the 256 decides that feed m2Num_eq_eight_explicitZ in the assemble module, whose statement is the full universal claim $\forall a,b,c,d,i,j:\mathrm{Fin},4,; m_2^{\mathrm{num}}=8,Z$. That parent is proved by exhaustive fin_cases on the six indices; each residual goal is exactly one chunk identity of this form.

In the broader gravity analysis, equating the folded numerator to eight times the explicit Z table is the algebraic certificate that the midpoint M2 TT kernel matches its closed-form coupling table on the 4D index set. The factor $8$ is bookkeeping from the contribution list, not a new physical constant. The result sits downstream of the kernel definitions and upstream of any theorem that quotes the assembled identity rather than re-folding.