e_312212

In the Regge-exact midpoint analysis for the 4D M2 TT identity, two integer-valued kernels on six $\mathbb{F}_4$ indices are compared. The numerator $N$ is defined by folding a fixed coupling list: each term contributes an integer via a local contribution map, and the fold starts at zero. The companion table $Z$ is an explicit case-split function $\mathbb{F}_4^6\to\mathbb{Z}$ listing the closed-form values (e.g. $4$, $-2$, and so on on the listed patterns).

The module is chunk 13 of a 256-case kernel certification: each chunk discharges one concrete multi-index equality $N=8Z$ by decision procedure. The ambient claim is that this scaling holds for every six-tuple in $\mathbb{F}_4$.

One-line computational proof: decide evaluates both sides at the concrete indices $(3,1,2,2,1,2)$. The left side reduces by unfolding the fold over the coupling list; the right side multiplies the table lookup by eight. Both land on the same integer, so the equality is discharged with no further lemmas.

Feeds the assembled universal statement that $N(a,b,c,d,i,j)=8Z(a,b,c,d,i,j)$ for all six $\mathbb{F}_4$ indices, proved by exhaustive fin_cases over the six coordinates. That parent theorem is the algebraic backbone of the Regge-exact midpoint M2 TT identity in 4D gravity analysis inside the monolith. Without the pointwise chunks, the case split cannot close. The factor of eight is the discrete normalization linking the folded coupling sum to the explicit kernel table used downstream in the TT-sector identities.