e_213202

This module is chunk 9 of a 256-cell kernel certifying that the folded numerator of a 4D Regge midpoint mass-squared identity equals eight times an explicit integer table. Indices run over $\mathrm{Fin},4$, i.e. ${0,1,2,3}$.

The numerator $N=m2Num$ is defined by folding a fixed coupling list: start at 0 and add a contribution term for each coupling triple at the six indices. The table $Z=explicitZ$ is a sparse pattern-matched integer function on the same six indices (typical nonzero values $\pm 2,,4$).

The local claim is one concrete cell of $N=8Z$. Upstream only supplies those two definitions; no analytic lemma is needed beyond evaluation.

One-line computational proof: decide. Both sides reduce to concrete integers once the six $\mathrm{Fin},4$ arguments are substituted into the fold definition of the numerator and the pattern match of the explicit kernel, so the equality is a closed integer comparison discharged by the kernel decision procedure.

Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; N(a,b,c,d,i,j)=8Z(a,b,c,d,i,j)$ by exhaustive fin_cases over all six indices. Each chunk theorem such as this one is a leaf of that case split; without the pointwise cells the global identity does not close.

In the Recognition gravity stack this identity is bookkeeping for the exact midpoint Regge $M^2$ TT kernel in 4D: it converts a folded coupling sum into a sparse explicit integer table, enabling later closed-form or certified numerical work on the discrete curvature side. It is pure finite combinatorics, not a forcing-chain (T0–T8) step, but it is load-bearing scaffolding for the gravity analysis layer.