e_112331

In the 4D Regge midpoint analysis, the mass-squared numerator is assembled by folding a fixed coupling list: $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ sums a contribution functional over that list. The companion table $Z$ is an explicit integer-valued function on six $\mathrm{Fin},4$ arguments, with sparse nonzero pattern (e.g. $4$ on matched pairs, $-2$ on certain crossed pairs).

The local module is chunk 5 of a 256-cell kernel certification that $m_2^{\mathrm{num}}=8Z$ pointwise. The ambient goal is an exact algebraic identity for the midpoint M2TT kernel in four dimensions, reduced to finitely many integer checks because every index lives in a four-element type.

Upstream, both $m_2^{\mathrm{num}}$ and $Z$ are pure definitions in the kernel certificate module; no analytic hypotheses remain once the fold and the table are fixed.

One-line kernel proof: decide evaluates both sides at the concrete six-tuple $(1,1,2,3,3,1)$ and checks integer equality. No lemmas are invoked beyond the definitions of the fold numerator and the explicit table; the checker closes the ground term.

Feeds the assembly theorem $m_2^{\mathrm{num}}=8Z$ for all six indices, which is proved by exhaustive fin_cases and therefore depends on each cell such as this one. That global identity is the algebraic backbone of the exact midpoint M2TT kernel certificate in the gravity analysis stack.

Within Recognition Science gravity work, such kernel identities pin discrete curvature/mass bookkeeping before continuum or phenomenological layers. The chunking into 256 decides is pure engineering of the finite check; the physics content is the factor-of-eight match between the folded coupling sum and the closed table $Z$.