e_211102

This module is chunk 9 of a 256-case kernel certification that the 4D Regge exact-midpoint numerator coincides with eight times a closed-form integer table. The ambient setting is discrete gravity analysis: curvature and coupling contributions on a 4-simplex skeleton are reduced to integer arithmetic on six Fin-4 indices.

The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at 0 and add each local contribution contrib t a b c d i j. The comparison target explicitZ is a pattern-matched integer table on the same six indices (sample entries include 4 on diagonal-like pairs and -2 on crossed pairs).

The local claim is only the equality at one concrete tuple; the surrounding chunks cover the rest of the $4^6$ grid.

One-line computational proof: by decide. Lean evaluates both sides of the integer equality (the fold that defines the numerator versus eight times the table lookup) and closes the goal by the decidable equality instance on Int. No lemmas are invoked beyond the definitions of the two sides.

The parent assembly theorem m2Num_eq_eight_explicitZ states the identity for every six-tuple in (Fin 4)^6. It is built by exhaustive case split; each chunk such as this one discharges a concrete cell so the global statement is a pure certificate rather than an analytic derivation.

In the Recognition gravity stack this identity is bookkeeping infrastructure: it certifies that the midpoint-mass numerator used in the 4D Regge analysis is exactly eight times an explicit sparse integer kernel. That kernel is what later steps quote when reducing continuum limits or matching continuum Einstein–Hilbert structure. It does not itself invoke the forcing chain (T5–T8) or the Recognition Composition Law; it is a finite combinatorial lemma inside the gravity analysis layer.