e_110312

In the 4D Regge midpoint analysis, the mass-squared numerator is defined by folding a fixed coupling list: each term contributes an integer depending on six $\mathbb{F}_4$ indices, and the fold starts at zero. The companion object is an explicit integer table $Z$ on the same six indices, given by a finite pattern match (typical values $\pm 2,,4$, and defaults).

This module is chunk 5 of the 256 kernel decides that check $m_2^{\mathrm{num}}=8Z$ pointwise. The local setting is pure integer arithmetic on a finite index set; no continuum limit or physical units enter the statement.

Upstream, $m_2^{\mathrm{num}}$ is the fold of contrib over couplingZList, and $Z$ is the closed table used as the target closed form.

One-line kernel proof: decide evaluates both sides at the concrete six-tuple $(1,1,0,3,1,2)$ and checks integer equality. No lemmas are invoked beyond the definitions of the numerator fold and the explicit table; the Fin-4 values are closed under reduction to Int.

Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, m_2^{\mathrm{num}}=8Z$ by exhaustive fin_cases on all six indices. Each chunk lemma such as this one discharges one residual goal in that case split.

In the Recognition gravity stack this identity is bookkeeping for the exact midpoint mass-squared kernel in 4D Regge calculus: once the numerator matches eight times the explicit table everywhere, later curvature and continuum-matching arguments can quote a single closed form rather than a fold. It does not itself touch the T0–T8 forcing chain, $\phi$-ladder masses, or $\alpha$; it is infrastructure inside the gravity analysis layer.