e_312200

This module is one chunk of a 256-case kernel certification that the 4D Regge midpoint M2TT numerator equals eight times a sparse explicit integer table. Indices run over $\mathrm{Fin},4$, i.e. coordinate labels ${0,1,2,3}$ in a 4D discrete gravity setting.

The numerator $N=\mathrm{m2Num}$ is defined by folding a fixed coupling list: start at 0 and add a contribution for each coupling term at the six indices. The comparison target $Z=\mathrm{explicitZ}$ is a pattern-matched integer table (typical nonzero entries are $\pm 2$ or $4$) that packages the closed form of that sum.

Chunk 13 discharges one concrete six-tuple among the full $4^6$ grid. Sibling lemmas cover neighboring tuples; the assemble theorem glues them into a universal identity.

One-line proof by decide. Both sides reduce to concrete Int values: the left by evaluating the fold of contributions at fixed indices $(3,1,2,2,0,0)$, the right by looking up explicitZ (or defaulting to 0 if unmatched) and multiplying by 8. Lean’s kernel decision procedure checks integer equality; no algebraic rewriting is required.

Feeds the assemble theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, N=8Z$ by exhaustive fin_cases over $\mathrm{Fin},4$. That identity is the certified algebraic core of the Regge-exact midpoint M2TT analysis in the Gravity domain: it replaces a folded coupling sum by a sparse closed form, enabling exact discrete curvature bookkeeping rather than floating-point checks.

Within Recognition Science gravity work, such kernel certificates underwrite exact discrete identities used when matching continuum limits and mass/ladder structure. This declaration is pure casework scaffolding: it does not itself invoke $\phi$, the eight-tick octave, or $D=3$, but it is a necessary brick in the exact 4D gravity numerator pipeline those landmarks sit on.