e_003003

This module is chunk 0 of a 256-case kernel certification that the numerator mass-squared coupling equals eight times an explicit integer table on all 4D multi-indices. The ambient setting is the exact midpoint analysis of the 4D Regge calculus identity used in the gravity sector.

The numerator coupling $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: sum the local contributions of each term at the six Fin-4 indices. The explicit kernel $Z$ is a closed integer table on $(\mathbb{F}_4)^6$ (sample values include $4$, $-2$, and so on for paired index patterns). The claim is the equality of these two integers at one concrete multi-index.

Upstream, both $m_2^{\mathrm{num}}$ and $Z$ live in the kernel-cert module; this chunk only discharges individual decide goals so the assembler can recombine them.

One-line computational proof: decide evaluates both sides at the fixed indices $(0,0,3,0,0,3)$ and checks integer equality. No lemmas are invoked beyond the definitions of the folded numerator and the explicit kernel table. The same pattern is repeated for every multi-index in the chunk.

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 Fin-4 arguments. Each pointwise decide (this one included) is a leaf of that case split.

In the Recognition gravity stack, the identity certifies that the midpoint Regge numerator matches the closed-form kernel used downstream in curvature and mass-coupling bookkeeping. It is pure discrete algebra on Fin 4, not a continuum GR derivation, and does not itself invoke the forcing chain (T5–T8) or the Recognition Composition Law; those enter only at higher layers that consume the certified kernel.