e_200023

In the Regge exact-midpoint 4D analysis, two integer-valued kernels on six Fin-4 indices are compared. The numerator m2Num folds a fixed coupling list, summing a local contribution at each table entry. The companion explicitZ is a pattern-matched closed form that returns a small integer (typically $\pm 2$ or $4$) on selected index patterns and zero elsewhere.

The module is chunk 8 of a 256-way partition of the Fin-4^6 domain. Each chunk theorem asserts the scalar identity m2Num = 8·explicitZ at one concrete multi-index. The factor eight is the global normalization that makes the folded sum match the closed form.

Upstream, both m2Num and explicitZ are pure definitions in the kernel certificate module; no analytic hypotheses are carried.

One-line kernel proof: by decide. Lean evaluates both sides at the fixed indices (2,0,0,0,2,3) and checks integer equality. No lemmas are invoked beyond the decidable equality instance on Int.

This atom is consumed by m2Num_eq_eight_explicitZ in the assemble module, which runs fin_cases on all six Fin-4 arguments and dispatches each case to the matching chunk theorem. That global identity is the algebraic backbone of the Regge exact-midpoint M2TT certificate in 4D gravity analysis.

Within Recognition Science gravity work, the identity confirms that the discrete curvature/coupling numerator is exactly eight times the explicit kernel, so later continuum or continuum-limit arguments can quote the closed form without re-folding the coupling list. It does not itself touch the T0–T8 forcing chain; it is infrastructure under the gravity side of the monolith.