e_320113

This module is chunk 14 of a 256-cell kernel certification that the midpoint mass-squared numerator equals eight times an explicit integer table on all of $(\mathbb{F}_4)^6$. The setting is the 4D Regge exact-midpoint TT identity analysis in the Gravity stack.

The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at $0$ and add each contribution term at the six indices. The explicit kernel $Z$ is a closed integer table on the same six $\mathrm{Fin},4$ arguments (sample values include $4$, $-2$, and other small integers on the listed patterns).

The global claim is $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8Z$. Because both sides are computable on a finite domain, the proof is split into per-cell decides and later assembled by exhaustive fin_cases.

One-line computational proof: decide evaluates both sides at the concrete six-tuple $(3,2,0,1,1,3)$ and checks integer equality. No lemmas are invoked beyond the definitions of the folded numerator and the explicit $Z$ table; the kernel reduces the equality in the integer decidable instance.

Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for every six-tuple in $(\mathbb{F}_4)^6$ by nested fin_cases and discharge of each cell. Without the full 256-cell cover, the midpoint mass-squared numerator cannot be replaced by the closed $8Z$ form in the 4D Regge TT identity pipeline.

In the Recognition gravity analysis this is bookkeeping infrastructure, not a new physical law: it certifies that a discrete coupling fold matches an explicit sparse integer kernel, so later curvature or propagator identities can quote $8Z$ instead of the fold. It does not itself invoke the forcing chain (T5–T8), RCL, or the $\varphi$-ladder; those enter only if the TT identity is later tied to continuum RS gravity.