e_213311

In the 4D Regge midpoint analysis, two integer-valued kernels on six Fin 4 indices are compared. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: each term contributes an integer contrib and the fold starts at zero. The comparison target is an explicit sparse table $Z$ on the same six indices, with nonzero pattern values such as $4$ and $-2$ on selected index patterns.

This module is chunk 9 of a 256-cell partition of the full $4^6$ index space. The local claim is the scalar identity $m_2^{\mathrm{num}}=8\cdot Z$ at one fixed multi-index. Upstream, both sides are pure definitions in the kernel certificate module; no analytic hypothesis is carried.

One-line computational proof: decide evaluates both sides at the concrete six-tuple $(2,1,3,3,1,1)$ and checks integer equality. The left side reduces by unfolding the fold over couplingZList; the right side multiplies the matched explicitZ clause by eight. No lemmas beyond kernel evaluation are required.

Feeds the assembler theorem m2Num_eq_eight_explicitZ, which states the identity for every six-tuple in Fin 4 by exhaustive fin_cases. That global equality is the algebraic core of the Regge exact midpoint M2 TT identity in 4D: it certifies that the folded numerator is exactly eight copies of the closed-form Z table, so continuum or continuum-limit arguments can replace the fold by the sparse explicit kernel.

Within Recognition gravity work this is bookkeeping infrastructure rather than a forcing-chain step (T0–T8). It closes one cell of the discrete certificate that underwrites the midpoint curvature bookkeeping used in the broader RS gravity analysis.