e_031202

This module is chunk 3 of a 256-way kernel certification that the 4D Regge exact-midpoint M2TT numerator equals eight times an explicit integer table. Indices run over $\mathrm{Fin},4$, matching a 4-simplex / 4D lattice labeling.

The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: start at 0 and add each contribution contrib t a b c d i j. The comparison target $Z_{\mathrm{explicit}}$ is a total function on six $\mathrm{Fin},4$ arguments with hard-coded integer values on the nonzero support (e.g. $4$, $-2$, and symmetric permutations).

The local claim is one concrete six-tuple equality inside that table identity, not the universal statement.

One-line computational proof: by decide. Both sides reduce to concrete Int values (the fold for m2Num on this fixed tuple, and the matching clause or default of explicitZ), and Lean’s decision procedure checks integer equality.

Parent theorem m2Num_eq_eight_explicitZ assembles the full universal identity $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8,Z_{\mathrm{explicit}}$ by exhausting all $4^6$ index tuples. This declaration discharges the single cell $(0,3,1,2,0,2)$ inside chunk 3 of that kernel cert.

In the gravity analysis stack it is bookkeeping infrastructure for the Regge exact-midpoint M2TT identity in 4D: once every cell matches, the closed form may replace the folded coupling sum in downstream curvature or mass-side arguments. It does not itself invoke the RS forcing chain (T5–T8) or the Recognition Composition Law; it is a finite combinatorial certificate under those larger gravity developments.