e_120002

This module is one chunk of the 4D Regge exact-midpoint M2TT identity certification. The local claim is the pointwise relation $m_2^{\mathrm{num}}=8\cdot Z_{\mathrm{expl}}$ on a block of the $4^6$ index space, discharged by kernel decides.

The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: start at $0$ and add each contribution term at those six Fin-4 indices. The explicit kernel $Z_{\mathrm{expl}}$ is a closed integer table on the same six indices (sample entries include $4$, $-2$, and other small integers on distinguished patterns).

Upstream, both maps live in the kernel-certificate module that supplies the concrete arithmetic objects being compared. Downstream assembly will quantify over all Fin-4 sextuples by case splits that invoke these chunk equalities.

One-line proof: decide. Both sides reduce to concrete integers once the six Fin-4 arguments are fixed at $(1,2,0,0,0,2)$, so the kernel evaluates the fold defining the numerator and the matching table entry of the explicit kernel, then checks equality with the factor eight. No lemmas beyond the definitions of the two maps are required.

Parent theorem m2Num_eq_eight_explicitZ states the universal identity $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$ and proves it by exhaustive fin_cases on the six indices. Each chunk lemma such as this one discharges one concrete leaf of that case tree (here the leaf $(1,2,0,0,0,2)$).

In the gravity analysis stack this identity is the algebraic core of the Regge exact-midpoint M2TT comparison in four dimensions: it converts a folded coupling expression into an explicit sparse integer kernel that can be matched against continuum or discrete curvature bookkeeping. The module doc frames the work as chunk 6 of 256 kernel decides, so this declaration is pure certification scaffolding inside that finite check, not a new physical law.