e_033102

In the 4D Regge midpoint analysis, two integer-valued kernels on six Fin-4 indices are compared. The numerator $m_2^{\mathrm{num}}$ is the fold of a fixed coupling list: it sums a local contribution over every coupling triple. The comparison target $Z_{\mathrm{explicit}}$ is a sparse pattern table that returns a handful of small integers (typically $\pm 2$ or $4$) on selected index patterns and zero elsewhere.

The local module is chunk 3 of the exhaustive case split that asserts $m_2^{\mathrm{num}}=8,Z_{\mathrm{explicit}}$ at every multi-index. The ambient claim is the algebraic identity needed for the exact midpoint M2TT reduction in the gravity analysis stack; the chunk files discharge the 256 concrete evaluations by kernel decision rather than by a closed-form argument.

One-line computational proof: by decide. Lean evaluates both sides at the fixed indices $(0,3,3,1,0,2)$ and checks integer equality. No lemmas are invoked beyond the definitions of m2Num (fold of contributions) and explicitZ (pattern table).

This case is one brick in the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for all six Fin-4 indices by exhaustive fin_cases. That universal equality is the certified numerator step of the Regge exact-midpoint M2TT identity in 4D. Within Recognition Science gravity work it supplies a machine-checked algebraic reduction on the discrete curvature side, rather than a continuum approximation. It does not itself touch the forcing chain (T0–T8) or the J-cost; it is pure discrete-kernel bookkeeping downstream of those foundations.