e_030102

This module is chunk 3 of a 256-way kernel certification that the folded numerator of the 4D Regge exact-midpoint $M_2$ TT identity agrees with eight times a closed-form integer table. Indices run over $\mathbb{F}_4$ (four discrete directions).

The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: start at $0$ and add each contribution term at those six indices. The explicit table $Z_{\mathrm{expl}}$ is a pattern-matched integer function on the same six $\mathbb{F}_4$ arguments (sample values include $4$, $-2$, and other small integers on the matched patterns).

The local claim is the equality at one concrete six-tuple inside that chunk. Upstream, only the two definitions are needed; no analytic lemma is invoked.

One-line computational proof: decide evaluates both sides at the concrete Fin-4 sextuple $(0,3,0,1,0,2)$. The left side reduces by unfolding the fold over the coupling list; the right side multiplies the matched explicitZ integer by eight. Equality of the resulting integers is discharged by the kernel decision procedure.

Feeds the assembler theorem m2Num_eq_eight_explicitZ, which states the identity for every $(a,b,c,d,i,j)\in(\mathbb{F}_4)^6$ and proves it by exhaustive fin_cases on all six indices. Each chunk cell such as this one closes one leaf of that split.

In the Recognition gravity stack, the identity certifies that the summed midpoint coupling numerator is exactly eight times the explicit kernel table, a discrete algebraic step toward the Regge exact-midpoint $M_2$ TT analysis in 4D. It is bookkeeping inside the gravity domain rather than a forcing-chain landmark (T5–T8), but it hardens the discrete curvature side of the RS gravity bridge.