e_201233

In the Regge exact-midpoint analysis for 4D gravity, one compares a folded numerator built from a fixed coupling list against a sparse explicit integer table. The numerator $m_2^{\mathrm{num}}$ is defined by folding couplingZList and summing a local contribution at each six-index slot. The table explicitZ assigns a small integer (typically $\pm 2$ or $4$, else implicitly $0$ on unmatched patterns) to each $(a,b,c,d,i,j)\in(\mathrm{Fin},4)^6$.

This module is chunk 8 of a 256-way kernel split: each chunk discharges a block of pointwise equalities $m_2^{\mathrm{num}}=8\cdot Z$ by pure decision procedures on concrete Int values. The factor $8$ is the global normalization relating the folded sum to the closed-form kernel used downstream in the M2TT identity.

One-line proof by decide. Both sides reduce to concrete integers once the six Fin 4 arguments are fixed: the left-hand side evaluates the fold that defines $m_2^{\mathrm{num}}$, the right-hand side multiplies the matching explicitZ clause (or zero) by eight. No lemmas beyond kernel evaluation are required.

Feeds the assembler m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8,Z$ by exhaustive fin_cases over all $4^6$ slots and invokes one chunk theorem per case. That global identity is the algebraic backbone of the Regge exact-midpoint M2TT certification in 4D: it converts a list-folded coupling sum into a sparse closed form suitable for curvature and mass-term bookkeeping in the Recognition gravity stack.

Within RS, this sits in the gravity-analysis layer that supports continuum limits and discrete curvature identities; it does not itself touch the T0–T8 forcing chain, but it is part of the certified discrete geometry used when matching continuum constants and ladder masses.