e_010031

This module sits in the 4D Regge exact-midpoint analysis of the M2–TT identity. The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is the integer obtained by folding a fixed coupling list: each term contributes an additive integer depending on the six Fin-4 indices. The companion kernel $Z$ is an explicit piecewise map $\mathrm{Fin},4^6\to\mathbb{Z}$ listing the closed-form values on the nonzero patterns (e.g. $4$, $-2$, and symmetric permutations).

The local goal, stated in the module header, is to certify $m_2^{\mathrm{num}}=8\cdot Z$ on a 256-case chunk by kernel decides. Upstream, both $m_2^{\mathrm{num}}$ and $Z$ are pure definitions in the KernelCert module; no analytic lemma is required beyond evaluating those definitions at concrete indices.

One-line computational proof: decide reduces both sides to concrete integers (the fold for the numerator and the pattern match for $Z$) and checks equality in $\mathbb{Z}$. No lemmas are invoked beyond the elaborator’s evaluation of the two definitions at $(0,1,0,0,3,1)$.

This is one atomic brick in the exhaustive case split that yields the parent theorem $m_2^{\mathrm{num}}=8,Z$ for all six Fin-4 indices. That assembly theorem runs nested fin_cases over $a,b,c,d,i,j$ and lands on chunk theorems such as this one. In the Recognition gravity stack the identity is the algebraic core of the exact midpoint M2–TT relation in 4D Regge calculus; certifying every kernel cell removes a scaffolding gap between the coupling-list definition and the closed-form $Z$ table. It does not itself touch the T0–T8 forcing chain, but it stabilizes the discrete gravity side that later interfaces with continuum limits.