e_310310

In the 4D Regge exact-midpoint analysis two integer maps on sextuples from $\mathrm{Fin},4$ are compared. The numerator map folds a fixed coupling list, adding one contribution per entry; the explicit kernel is a sparse case table returning small integers (typically $0,\pm 2,\pm 4$).

This module is chunk 13 of a partitioned proof that the numerator equals eight times the explicit kernel on the whole finite domain. Each chunk discharges a block of concrete sextuples by kernel decision, so the global identity reduces to finitely many closed integer equalities.

Upstream, the numerator is the fold of contrib over couplingZList, and the explicit kernel is the case-analyzed integer table used as the closed form.

Both sides become closed Int terms once the six indices are fixed to $3,1,0,3,1,0$. The proof is the single tactic decide, which evaluates the two integers and closes propositional equality. No lemmas, rewrites, or case splits appear at this leaf.

The parent assembly theorem asserts the identity for every sextuple in $(\mathrm{Fin},4)^6$ and discharges the universal quantifier by exhaustive fin_cases on all six indices; each concrete leaf is one of these chunk theorems. Casewise equality $m_2^{\mathrm{num}}=8Z_{\mathrm{ex}}$ is the computational backbone of the exact midpoint $M_2$ TT identity in four dimensions inside the Recognition Science gravity/Regge analysis. It does not itself invoke the forcing chain (T0–T8) or the Recognition Composition Law; it is pure finite-kernel bookkeeping supporting that geometric identity.