e_132000

In the 4D Regge midpoint analysis, the M2 numerator is the integer obtained by folding a fixed coupling list: each term contributes an integer depending on six indices in $\mathrm{Fin},4$, and the fold starts at zero. The explicit $Z$ table is a closed-form integer function of the same six indices, with sparse nonzero values (for example $4$ or $-2$ on selected patterns).

This module is chunk 7 of the kernel certification that $m_2^{\mathrm{num}}=8\cdot Z$ pointwise. The ambient goal is an exact algebraic identity for the midpoint M2TT expression in four dimensions, reduced to finitely many integer equalities on a $4^6$ grid.

Both sides are fully concrete integers once the six indices are fixed to $1,3,2,0,0,0$. The proof is the single tactic decide, which evaluates $m_2^{\mathrm{num}}$ (the fold of contributions) and $8\cdot Z$ and checks equality in $\mathbb{Z}$. No lemmas beyond the definitions of the numerator fold and the explicit $Z$ table are required.

The parent theorem m2Num_eq_eight_explicitZ assembles every multi-index by nested fin_cases and invokes these chunk lemmas cell by cell. Establishing $m_2^{\mathrm{num}}=8Z$ everywhere is the numerical backbone of the exact midpoint M2TT identity in the gravity analysis stack. Within Recognition Science this sits in the discrete geometric side of the gravity sector (Regge-type curvature bookkeeping), not in the T0–T8 forcing chain itself, but it supplies a certified algebraic identity used downstream when matching continuum limits and coupling normalizations.