e_003101

This module sits in the Regge-exact midpoint analysis for the 4D M2TT identity. The local goal is to certify, index-by-index, that the folded numerator built from the coupling list matches eight times a closed-form kernel table.

Upstream, m2Num is defined by folding couplingZList and summing the contribution of each term at a six-tuple of Fin-4 indices. explicitZ is the matching piecewise integer table on those same indices (sample values include 4, -2, and so on for the listed patterns). The module doc frames the work as "m2Num = 8·explicitZ, chunk 0 (256 kernel decides)": each chunk theorem fixes one concrete six-tuple and discharges the equality by computation.

The ambient setting is discrete gravity bookkeeping for Recognition Science: exact integer identities on a 4-index combinatorial skeleton, not continuum curvature postulates.

One-line computational proof: by decide. Lean evaluates both sides at the concrete Fin-4 sextuple (0,0,3,1,0,1). The left side runs the fold that defines m2Num; the right side multiplies the table lookup explicitZ by 8. Equality of the resulting integers is decided by the kernel, with no lemmas or rewriting beyond evaluation.

This is one cell of the 256-case grid that feeds m2Num_eq_eight_explicitZ in the assemble module. That parent theorem states the universal identity $\forall a,b,c,d,i,j,;\mathrm{m2Num}=8\cdot\mathrm{explicitZ}$ and proves it by nested fin_cases over all six indices, each case landing on a chunk decide such as this one.

In the gravity stack, the identity converts a folded coupling sum into a sparse closed-form kernel, which is the algebraic backbone of the Regge-exact midpoint M2TT certification. It is pure discrete bookkeeping: no continuum limit, no appeal to the forcing chain T0–T8, and no mass-ladder content. Its value is that the assemble theorem can quote a fully discharged computational certificate rather than an axiomatic ansatz.