e_203121

This module is chunk 8 of a 256-way split of the kernel identity $N=8Z$ for the Regge exact-midpoint M2 numerator in four dimensions. Indices run over $\mathrm{Fin},4$ (four discrete directions).

The numerator $N(a,b,c,d,i,j)$ is defined by folding a fixed coupling list and summing a local contribution at each term. The comparison object $Z$ is an explicit integer-valued table on six $\mathrm{Fin},4$ arguments, given by a finite pattern match (sample clauses include $Z(0,0,1,1,2,2)=4$ and $Z(0,0,1,2,1,2)=-2$).

The local setting is pure finite arithmetic: both sides evaluate to concrete integers once the six indices are fixed, so each cell is independently decidable.

One-line computational proof: decide evaluates both the folded sum and the explicit table at the concrete sextuple $(2,0,3,1,2,1)$ and checks integer equality. No lemmas beyond the definitions of the numerator fold and the explicit table are required.

Feeds the assembler theorem that states $N(a,b,c,d,i,j)=8Z(a,b,c,d,i,j)$ for every sextuple in $(\mathrm{Fin},4)^6$. That assembler runs nested fin_cases over all six indices and dispatches each cell to a chunk theorem of this form.

In the gravity analysis stack, the identity certifies that the midpoint M2 numerator matches eight times a closed integer kernel, a bookkeeping step toward exact Regge-calculus identities in 4D. It is scaffolding arithmetic rather than a forcing-chain landmark (T5–T8), but it closes one of the 256 decidable obligations needed for the universal statement.