e_312111

This module is chunk 13 of a 256-point kernel certification that the 4D Regge-exact midpoint numerator matches a closed integer table. Indices run over $\mathbb{F}_4$ (four discrete labels), six of them at a time.

The numerator $N(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: start at 0 and add each local contribution at those six indices. The table $Z$ is an explicit pattern-matched map $\mathbb{F}_4^6\to\mathbb{Z}$ (sample values include $4$, $-2$, and so on on the diagonal and off-diagonal slots).

The local claim is one concrete instance of $N=8Z$. The parent assembly theorem states the same equality for every six-tuple and discharges it by exhaustive case split on the six $\mathrm{Fin},4$ arguments.

One-line computational proof: decide evaluates both sides at the fixed indices $(3,1,2,1,1,1)$. The left side reduces by unfolding the fold over the coupling list; the right side reduces by pattern-matching the explicit table and multiplying by 8. Equality of the resulting integers is decided in the kernel. No lemmas beyond the two definitions are invoked.

Feeds the assembly theorem m2Num_eq_eight_explicitZ, which asserts $\forall a,b,c,d,i,j,, N=8Z$ by six nested fin_cases sweeps. Each chunk theorem such as this one closes one leaf of that case tree (here the leaf for $(3,1,2,1,1,1)$).

In the broader gravity analysis, the identity certifies that the midpoint M2TT numerator in 4D Regge calculus collapses to an eightfold multiple of a sparse integer kernel. That collapse is bookkeeping infrastructure for exact discrete curvature identities, not a new dynamical law. It sits downstream of the kernel certificate module and upstream of any global midpoint-identity packaging that quotes the assembled equality.