e_201121

This module is chunk 8 of a 256-way split of the 4D Regge exact-midpoint identity that asserts $m_2^{\mathrm{num}}=8\cdot Z$ pointwise on $(\mathrm{Fin},4)^6$. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at 0 and add each triple's contribution at the six indices. The right-hand side is the sparse explicit integer table $Z$, which returns a handful of nonzero values (such as $\pm 2,,4$) on distinguished index patterns and is otherwise determined by that table.

The local setting is pure integer kernel certification inside the gravity analysis layer: no continuum limit, no physical units, only exact equality of two Int-valued combinatorial expressions on a finite index set. Upstream, both sides are the certified defs m2Num and explicitZ from the kernel certificate module.

One-line computational proof: decide evaluates both sides as closed Int expressions at the concrete six-tuple $(2,0,1,1,2,1)$ and checks equality. No lemmas are invoked beyond the unfoldings of m2Num (the fold over couplingZList with contrib) and explicitZ (the pattern table). The goal is propositional equality of integers, so the kernel decision procedure finishes it.

Parent theorem m2Num_eq_eight_explicitZ assembles every index tuple by nested fin_cases and needs each atomic equality such as this one. That global identity is the algebraic core of the 4D Regge exact-midpoint $M_2$ TT certificate: the folded numerator is identically eight times the explicit sparse kernel. In the Recognition gravity stack this is bookkeeping infrastructure for discrete curvature/mass-squared identities, not a forcing-chain landmark (T5–T8) by itself. It closes one cell of the 256-chunk decide grid so the assemble theorem can stay a pure case split.