e_012330

The ambient module certifies, one index tuple at a time, that the 4D Regge midpoint numerator m2Num is identically eight times a sparse closed-form table explicitZ. Both maps take six Fin 4 arguments (four simplex labels and a pair of edge indices) and return an Int.

m2Num is defined by folding a fixed coupling list: each term contributes an integer weight via contrib, and the fold starts at 0. explicitZ is the matching piecewise table (nonzero only on a short list of index patterns such as $(0,0,1,1,2,2)\mapsto 4$ and $(0,0,1,2,1,2)\mapsto -2$). The module doc frames the work as chunk 1 of 256 kernel decides establishing $m_2^{\mathrm{num}}=8\cdot Z_{\mathrm{explicit}}$.

One-line computational proof: by decide. Lean evaluates both sides on the concrete six-tuple $(0,1,2,3,3,0)$ inside the integer kernel and checks equality. No lemmas are invoked beyond the definitions of m2Num and explicitZ.

This is one atomic cell in the exhaustive grid that feeds m2Num_eq_eight_explicitZ in the assemble module. That parent theorem states $\forall a,b,c,d,i,j:\mathrm{Fin},4,; m_2^{\mathrm{num}}=8,Z_{\mathrm{explicit}}$ and discharges the quantifiers by nested fin_cases. The identity is the algebraic core of the exact midpoint M2TT certificate for 4D Regge gravity in the Recognition Science stack: once numerator and closed form agree everywhere, curvature and mass-coupling bookkeeping can quote a single sparse table instead of a fold. It does not itself touch the T0–T8 forcing chain, but it hardens the gravity-side numerics that sit downstream of the eight-tick and $D=3$ landmarks.