e_023303

In the 4D Regge exact-midpoint analysis, two integer kernels on six $\mathrm{Fin},4$ indices are compared. The folded numerator $N(a,b,c,d,i,j)$ is the sum of all coupling contributions obtained by folding a fixed coupling list against a local contrib term. The explicit table $Z$ is a sparse pattern-matched integer function on the same six indices (nonzero only on a short list of index patterns, with values such as $\pm 2,,4$).

This module is chunk 2 of a 256-way kernel certification: each declaration checks one concrete sextuple so that the global identity $N=8Z$ can be assembled by exhaustive case split. The local setting is pure integer arithmetic on a finite discrete index set; no continuum limit is taken here.

Both sides evaluate to concrete integers once the six indices are fixed. The proof is the single tactic decide, which reduces the equality of those closed integer terms in the kernel. No algebraic lemma is invoked beyond the definitions of the folded numerator and the explicit table.

The parent assembly theorem states $\forall a,b,c,d,i,j:,N=8Z$ and discharges the universal quantifier by nested fin_cases on all six $\mathrm{Fin},4$ coordinates; each leaf is one of these chunk identities. This entry covers the leaf $(0,2,3,3,0,3)$.

In the broader gravity stack the identity certifies that the midpoint Regge numerator matches the closed-form Z kernel used downstream in 4D curvature bookkeeping. It is infrastructure for the discrete geometric side of Recognition gravity, not a forcing-chain (T0–T8) step, but it keeps the exact midpoint calculus free of residual case obligations.