e_232102

In the 4D Regge exact-midpoint analysis, two integer-valued kernels on six $\mathbb{F}4$ indices are compared. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: each term contributes an integer via a local contribution map, and the fold starts at zero. The comparison target $Z{\mathrm{expl}}$ is an explicit pattern-matched table on the same six indices (sample clauses send $(0,0,1,1,2,2)$ to $4$ and $(0,0,1,2,1,2)$ to $-2$).

This module is chunk 11 of the 256-point kernel certification: each chunk discharges a batch of concrete index tuples so the global identity can be assembled by exhaustive fin_cases. The local claim is only the single tuple $(2,3,2,1,0,2)$.

One-line computational proof: decide. Both sides reduce to concrete Int values once the six Fin 4 arguments are substituted into the fold definition of the numerator and the pattern table for the explicit kernel; Lean’s decision procedure checks integer equality.

Parent theorem m2Num_eq_eight_explicitZ states the universal identity $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$ and proves it by nested fin_cases on all six indices. Each concrete leaf such as this one supplies the residual equality the case split needs. Closing the full kernel identity is a bookkeeping step inside the 4D Regge midpoint $M_2$ TT analysis used on the gravity side of the Recognition stack; it does not itself invoke the T0–T8 forcing chain, but it hardens the discrete curvature bookkeeping that later mass and coupling extractions rely on.