e_031111

In the 4D Regge exact-midpoint analysis, two integer-valued kernels on six $\mathrm{Fin},4$ indices are compared. The numerator $N(a,b,c,d,i,j)$ is defined by folding a fixed coupling list, accumulating a local contribution at each triple; it is a pure integer sum. The comparison object $Z$ is an explicit pattern-matched table of small integers (entries such as $4$, $-2$, and so on) that is intended to be the closed form of that sum up to a universal factor of eight.

This module is chunk 3 of a 256-way partition of the finite index space. The local claim is only the single cell $(0,3,1,1,1,1)$. The ambient goal, stated in the module header, is the identity $N=8Z$ on the whole chunk.

Both sides evaluate to concrete integers once the six indices are fixed. The tactic decide runs the kernel decision procedure on the resulting numeral equality and closes the goal. No lemmas beyond the definitions of the numerator fold and the explicit table are required.

The parent theorem m2Num_eq_eight_explicitZ asserts the identity for every six-tuple in $(\mathrm{Fin},4)^6$, discharging the goal by exhaustive case split on each coordinate. Each chunk cell such as this one supplies one decided equality that the assembly depends on. In the broader Recognition gravity stack, the identity certifies that the midpoint Regge coupling numerator collapses to an explicit sparse integer kernel, which is the algebraic input to later continuum and continuum-limit comparisons. It does not itself touch the forcing chain (T0–T8) or the J-cost; it is pure discrete kernel bookkeeping inside the gravity analysis layer.