e_332201

In the Regge exact-midpoint analysis for 4D gravity, the TT-sector identity is reduced to an equality between two integer-valued kernels on $(\mathrm{Fin},4)^6$. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: each term contributes an integer depending on the six indices, and the fold starts from zero. The comparison object $Z_{\mathrm{explicit}}$ is a closed-form piecewise integer table on the same index domain (sample clauses assign $\pm 2$ or $4$ on selected patterns).

The local module is chunk 15 of the 256-case kernel certification that $m_2^{\mathrm{num}}=8,Z_{\mathrm{explicit}}$ pointwise. Upstream definitions supply both sides of the equality; no analytic continuum limit is taken here, only exact integer arithmetic on finite indices.

One-line computational proof: decide evaluates both sides at the concrete indices $(3,3,2,2,0,1)$ and checks integer equality in the kernel. No lemmas beyond the definitions of the numerator fold and the explicit table are required; the tactic discharges the goal by normalization of closed integer expressions.

This is one atomic cell in the exhaustive grid that builds m2Num_eq_eight_explicitZ, the parent theorem stating $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8,Z_{\mathrm{explicit}}$. That parent proceeds by nested fin_cases on all six $\mathrm{Fin},4$ indices and invokes the matching chunk theorem at each leaf. Within Recognition gravity, the identity certifies the discrete midpoint TT kernel used in the Regge analysis; it is bookkeeping infrastructure rather than a forcing-chain landmark (T5–T8), but it closes a concrete algebraic gap in the 4D assembly.