e_310111

In the 4D Regge midpoint analysis, the m2 numerator is the integer obtained by folding a fixed coupling list: each term adds a local contribution at six Fin-4 indices. The explicit Z kernel is the matching closed-form piecewise map $\mathrm{Fin},4^6\to\mathbb{Z}$ (sample values include $4$, $-2$, and so on on the diagonal and off-diagonal patterns).

This module is chunk 13 of the exhaustive check that the numerator is identically eight times that kernel. The local setting is pure integer equality on a $4^6$ grid, split so each chunk holds 256 kernel-decidable cells.

Upstream, both sides are already defined: the fold m2Num and the pattern-matched explicitZ. No continuum limit or physical units enter at this layer.

One-line proof: decide. Lean reduces both the fold that defines the m2 numerator at $(3,1,0,1,1,1)$ and the corresponding explicit-Z clause to concrete integers, then confirms the equality $n=8z$ by kernel computation. No lemmas beyond the two definitions are invoked.

The cell feeds the assembler m2Num_eq_eight_explicitZ, which states the identity for every six-tuple in $\mathrm{Fin},4$ and discharges the grid by nested fin_cases. Each chunk theorem such as this one closes one concrete goal in that case split (chunk 13 covers 256 of the decides).

Inside Recognition gravity, the identity is bookkeeping for the exact midpoint m2–TT kernel in 4D Regge calculus: once numerator and explicit kernel match by the factor eight everywhere, later curvature and continuum comparisons can quote a single closed form rather than the fold. It does not itself touch the forcing chain (T0–T8) or the J-cost; it is infrastructure under the discrete gravity side.