e_220313

This module sits in the 4D Regge midpoint analysis of the M2–TT identity. The local goal, stated in the module doc, is to certify $\mathrm{m2Num}=8\cdot\mathrm{explicitZ}$ cell by cell across the $4^6$ index space, in chunks of kernel decides.

$\mathrm{m2Num}(a,b,c,d,i,j)$ is the integer obtained by folding a fixed coupling list and summing each term's contribution at those six indices. $\mathrm{explicitZ}$ is the matching closed-form integer table on the same six $\mathrm{Fin},4$ arguments (sparse nonzero pattern such as $4$, $-2$, etc. on selected index tuples).

Chunk 10 supplies one block of those pointwise equalities; the present declaration is the cell at $(2,2,0,3,1,3)$.

One-line proof by decide. Both sides reduce to concrete integers once the six Fin-4 indices are fixed: the left via the fold definition of the numerator coupling, the right via the pattern-match table for the explicit kernel. The kernel checks integer equality; no algebraic rewriting or external lemmas are invoked.

Parent consumer is $\mathrm{m2Num_eq_eight_explicitZ}$, which states the identity for all $a,b,c,d,i,j:\mathrm{Fin},4$ and discharges it by exhaustive fin_cases. Each chunk theorem such as this one fills one case of that assembly.

In the Recognition gravity stack this is bookkeeping infrastructure for the exact midpoint M2–TT identity in 4D Regge calculus, not a new physical law. It supports the discrete curvature/coupling identities that feed continuum matching, rather than the T0–T8 forcing chain or the J-cost uniqueness step directly.