e_203232

In the 4D Regge midpoint analysis, two integer-valued maps on six Fin-4 indices are compared. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at 0 and add each local contribution contrib t a b c d i j. The comparison target explicitZ is a sparse pattern-matched table of small integers (entries such as $4$, $-2$, and so on) that is meant to be the closed form of that fold, up to a universal factor of 8.

This module is chunk 8 of a 256-cell kernel partition of the full $4^6$ index space. The local claim is only the single cell $(2,0,3,2,3,2)$. The surrounding certificate infrastructure lives in ReggeExactMidpointM2TTIdentity4DKernelCert, which supplies both definitions used here.

One-line kernel proof: by decide. Lean evaluates both the fold that defines $m_2^{\mathrm{num}}$ at $(2,0,3,2,3,2)$ and the pattern match for explicitZ at the same indices, then checks the integer equality against $8\cdot Z$. No lemmas are invoked beyond the two definitions.

The parent theorem m2Num_eq_eight_explicitZ assembles the universal identity $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8,Z$ by exhaustive fin_cases on all six indices. Each chunk cell such as this one discharges one concrete branch of that case split. Without the cell equalities, the midpoint M2TT numerator identity in 4D Regge calculus would remain an unchecked table claim rather than a machine-checked fact. The factor of 8 is the structural constant tying the folded coupling sum to the explicit sparse table across the whole kernel.