e_310222

This module is one chunk of the 4D Regge exact-midpoint analysis: it certifies pointwise that the folded numerator equals eight times a closed-form integer table. The ambient setting is discrete gravity on a 4-index simplex kernel, with all indices ranging over $\mathrm{Fin},4$.

The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list and summing a local contribution at each term. The comparison target $Z(a,b,c,d,i,j)$ is an explicit integer-valued pattern match on the six indices (typical nonzero values $\pm 2,,4$). The module header states the goal as $m_2^{\mathrm{num}}=8\cdot Z$ via 256 kernel decides, of which this declaration is one slot.

One-line proof by decide. Both sides reduce to concrete Int values once the six Fin 4 arguments are fixed: the left side evaluates the fold of couplingZList with contrib, the right side multiplies the pattern-matched explicitZ entry by 8. No lemmas are invoked beyond kernel evaluation of those two definitions.

Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8,Z$ by exhaustive fin_cases on all six indices. Each chunk theorem such as this one discharges one concrete case so the global identity is a pure case split rather than a symbolic fold argument.

In the Recognition gravity stack this identity is bookkeeping for the exact midpoint Regge kernel in 4D: once numerator and closed form agree everywhere, downstream curvature and mass-ladder comparisons can quote a single compact $Z$ table instead of the folded sum. It does not itself touch T5–T8 or the RCL; it is infrastructure inside the gravity analysis layer.