e_312002

In the 4D Regge midpoint analysis, the mass-squared numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: sum the local contributions of each coupling term at a six-index slot $(a,b,c,d,i,j)\in(\mathrm{Fin},4)^6$. The companion table $Z$ is an explicit integer-valued kernel on the same index domain (sample values include $4$, $-2$, and other small integers on diagonal and off-diagonal patterns).

This module is chunk 13 of a 256-cell partition of that kernel. The local claim is only the numerical identity $m_2^{\mathrm{num}}=8Z$ at one concrete multi-index. The module setting is pure integer arithmetic on Fin-4 labels; no continuum limit or curvature hypothesis is in play here.

One-line computational proof: decide evaluates both sides at the closed Fin-4 tuple $(3,1,2,0,0,2)$ and checks integer equality. No lemmas are invoked beyond the definitions of the folded numerator and the explicit $Z$ table; the kernel is small enough for Lean’s decision procedure to discharge the equality directly.

Parent theorem m2Num_eq_eight_explicitZ assembles the full $\forall$-statement by fin_cases on all six indices; each chunk cell such as this one supplies one branch. Establishing $m_2^{\mathrm{num}}=8Z$ pointwise certifies that the midpoint TT mass-squared numerator is exactly eight copies of the tabulated coupling kernel, a structural identity in the 4D Regge gravity analysis stack.

Within Recognition Science this sits in the gravity domain (discrete curvature / Regge calculus side), not on the T0–T8 forcing chain itself. It closes a finite kernel obligation rather than an open physical conjecture: once all 256 cells and the assembler succeed, the midpoint identity is fully certified in Lean.