e_213322

This module is chunk 9 of a brute-force certification that the 4D Regge-exact midpoint mass-squared numerator matches eight times a closed-form integer kernel on every sextuple of indices in $\mathbb{F}_4$.

The numerator $N(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: start at 0 and add each contribution term at those six indices. The explicit kernel $Z$ is a pattern-matched integer table on the same six $\mathrm{Fin},4$ arguments (sample entries include $4$, $-2$, and so on).

The local claim is only the single cell $(2,1,3,3,2,2)$. Upstream, both $N$ and $Z$ live in the kernel-cert module; downstream assembly will quantify over all cells.

One-line computational proof: decide. Lean reduces both sides at the concrete indices $(2,1,3,3,2,2)$ to closed integers (the fold for $N$, the match for $Z$) and checks equality in Int. No lemmas are invoked beyond the definitions of the numerator fold and the explicit kernel table.

Parent theorem m2Num_eq_eight_explicitZ assembles every index cell by nested fin_cases on the six $\mathrm{Fin},4$ arguments; each leaf is one of these chunk theorems. That universal identity is the algebraic backbone of the Regge-exact midpoint M2TT certification in the gravity analysis stack: it replaces a folded coupling sum by an eightfold explicit kernel, enabling exact midpoint identities in 4D discrete gravity. Within Recognition Science this sits in the gravity sector (Regge calculus bookkeeping), not the T0–T8 forcing chain, but it is the kind of zero-sorry kernel table the framework needs before continuum or phenomenological claims.