e_003131

This module is chunk 0 of a 256-cell decide table proving that the folded coupling numerator equals eight times an explicit integer kernel on every sextuple of indices in $\mathbb{F}_4$. The local setting is the exact midpoint analysis of the 4D Regge M2/TT identity.

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 evaluated at the six indices. The explicit kernel $Z$ is a closed integer-valued table on $(\mathbb{F}_4)^6$, with sample entries such as $Z(0,0,1,1,2,2)=4$ and $Z(0,0,1,2,1,2)=-2$. Both objects live in the kernel-certificate module imported here.

The present declaration fixes one concrete cell of that table, namely the indices $(0,0,3,1,3,1)$.

One-line computational proof: decide evaluates both sides at the concrete Fin-4 sextuple $(0,0,3,1,3,1)$ and checks integer equality. No lemmas are invoked beyond the reducible definitions of the folded numerator and the explicit kernel table.

The parent theorem is the full assembly identity: for every $a,b,c,d,i,j\in\mathbb{F}_4$, the numerator equals eight times the explicit kernel. That proof runs nested fin_cases over all six indices and lands on cells such as this one. Without the cell-wise equalities the assembly cannot close.

In the Recognition gravity stack this identity is part of the exact midpoint M2/TT analysis in four dimensions, the spatial dimension forced by the T8 step of the unified forcing chain. It supplies a certified algebraic relation between the summed coupling data and the closed kernel used downstream in Regge curvature bookkeeping. The chunk layout (256 decides) is pure scaffolding for machine-checked exhaustion, not a physical hypothesis.