e_303131

In the 4D Regge midpoint M₂ TT-identity analysis, two integer kernels on six Fin-4 indices are compared. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: each term contributes an integer contrib and the fold starts at 0. The companion explicitZ is a closed pattern match on the six indices, returning small integers such as $4$, $-2$, and so on for the nonzero cells.

This module is chunk 12 of the 256-decide kernel that checks $m_2^{\mathrm{num}}=8\cdot Z$ pointwise. The local setting is pure finite enumeration: every index lives in $\mathrm{Fin},4$, so each concrete sextuple is a closed integer equality.

One-line computational proof: decide evaluates both sides at the fixed sextuple $(3,0,3,1,3,1)$. The left side runs the fold that defines the numerator; the right side multiplies the matched explicitZ entry by 8. No lemmas are invoked beyond kernel reduction of concrete Int arithmetic.

Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, m_2^{\mathrm{num}}=8\cdot Z$ and discharges the universal claim by fin_cases on all six indices. Each chunk theorem such as this one closes one cell of that case split.

In the Recognition gravity stack this identity is bookkeeping for the exact midpoint Regge M₂ TT relation in 4D: once numerator and explicit kernel agree everywhere up to the factor 8, later analytic steps can quote the closed form $Z$ instead of the fold. It does not itself touch T0–T8 or the RCL; it is infrastructure inside the gravity analysis layer.