e_200032

In the 4D Regge midpoint analysis the M2-TT identity asserts that a folded numerator built from coupling contributions coincides with eight times a sparse explicit integer kernel. The numerator is defined by folding a contribution map over a fixed coupling list at a six-tuple of indices in $\mathrm{Fin},4$. The comparison table is a pattern-matched integer function on the same domain, nonzero only on a short list of index patterns (for example the value $4$ on $(0,0,1,1,2,2)$ and $-2$ on several related permutations).

This module is chunk 8 of the 256-case kernel-decide campaign that discharges the pointwise identity. The local setting is pure integer arithmetic on a finite index set; no continuum limit or physical units enter the certificate.

One-line proof by decide. Both sides reduce by definition: the left-hand side evaluates the fold that defines the numerator at $(2,0,0,0,3,2)$; the right-hand side evaluates the pattern match that defines the explicit table at the same indices, then multiplies by eight. The kernel checks the resulting integer equality. No intermediate lemmas are invoked.

This cell feeds the assembly theorem that states the identity for every six-tuple in $\mathrm{Fin},4$ by exhaustive case split. That global equality is the algebraic content of the Regge exact midpoint M2-TT identity in four dimensions: the folded coupling numerator is identically eight times the explicit sparse kernel. Within the Recognition Science gravity stack the certificate supports discrete curvature bookkeeping used downstream for continuum limits and effective Newtonian potentials. It does not itself invoke the forcing chain (T0-T8), the Recognition Composition Law, or the phi-ladder mass formula; those enter at higher layers.