e_110102

This module is chunk 5 of a 256-cell kernel certification that the folded numerator of the 4D Regge midpoint M2 TT identity equals eight times an explicit integer table on every sextuple of indices in $\mathbb{F}_4$.

The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: it sums a contribution functional over that list and returns an integer. The companion table $Z_{\mathrm{expl}}$ is a total function $\mathbb{F}_4^6\to\mathbb{Z}$ given by exhaustive pattern match (sample clauses include $(0,0,1,1,2,2)\mapsto 4$ and $(0,0,1,2,1,2)\mapsto -2$).

The local claim is the equality of these two integers at one fixed cell of the $4^6$ grid.

Both sides evaluate to concrete integers once the six $\mathbb{F}_4$ arguments are substituted. The proof is the single tactic decide, which asks the kernel to check integer equality after unfolding the fold that defines the numerator and the pattern match that defines the explicit table. No lemmas are invoked beyond definitional reduction.

The parent theorem m2Num_eq_eight_explicitZ asserts the identity for every sextuple in $\mathbb{F}_4^6$. Its proof is nested fin_cases on all six indices; each leaf is one of these chunk theorems. This declaration discharges the leaf $(1,1,0,1,0,2)$ inside chunk 5.

In the broader gravity analysis, the factor-of-eight relation between the folded numerator and the explicit kernel is the algebraic content needed to certify the exact midpoint M2 TT identity in 4D Regge calculus. Closing all 256 cells removes case-analysis debt from that certification path.