e_203210

This module is one chunk of a 256-case kernel certification that the 4D Regge midpoint numerator equals eight times a tabulated integer kernel. Indices run over $\mathrm{Fin},4$, i.e. coordinate labels ${0,1,2,3}$ for a 4D simplex edge/face pattern.

The numerator $N=\texttt{m2Num}$ is defined by folding a fixed coupling list: start at 0 and add each local contribution $\texttt{contrib},t,a,b,c,d,i,j$. The closed form $Z=\texttt{explicitZ}$ is a pure pattern match on the six indices, returning small integers such as $4$, $-2$, and so on for the nonzero support.

The local claim is the single hex-tuple $(2,0,3,2,1,0)$ inside chunk 8 of that decide grid. Upstream only supplies the two definitions being compared; no analytic lemma is required beyond evaluation.

One-line proof by decide. Both sides are closed integer expressions once the six Fin-4 indices are fixed: the left folds a finite coupling list of contributions, the right is a single match arm of the explicit kernel. Lean reduces both to concrete Int values and checks equality. No lemmas beyond the definitions of m2Num and explicitZ are invoked.

Feeds the assembler m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,,N=8Z$ by exhaustive fin_cases over all six indices. Each chunk theorem such as this one discharges one concrete cell so the universal identity is a pure case split rather than a fresh computation.

In the gravity stack this identity is the algebraic backbone of the exact midpoint $M_2$ TT kernel in 4D Regge calculus: once $N=8Z$ is certified, curvature and mass-side comparisons can quote the compact kernel $Z$ instead of the folded sum. It is bookkeeping inside the Recognition gravity analysis, not a forcing-chain (T0–T8) step, but it hardens the discrete geometric side that later couples to continuum limits.