e_213330

In the 4D Regge midpoint analysis, the numerator of the discrete M2TT identity is built by folding a fixed coupling list: each term contributes an integer depending on six Fin-4 indices, and the fold sum is the numerator $N(a,b,c,d,i,j)$. Parallel to that sum sits an explicit integer table $Z$ on the same six indices, given by a finite pattern-match (e.g. diagonal-type slots return $\pm 4$ or $-2$).

The local module is chunk 9 of a 256-cell kernel certification that $N=8Z$ holds at every multi-index. The ambient goal is an exact algebraic identity for the midpoint Regge curvature/mass term in four dimensions, reduced to integer arithmetic on $\mathbb{F}_4$ labels rather than symbolic continuum limits.

Upstream, $N$ is the fold of contrib over couplingZList, and $Z$ is the closed-form case table; both live in the kernel certificate module imported here.

One-line computational proof: decide evaluates both sides at the concrete sextuple $(2,1,3,3,3,0)$ and checks integer equality. No lemmas are invoked beyond the definitions of the folded numerator and the explicit kernel table; the kernel reduces the Fin-4 arithmetic and the fold to a ground fact.

Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, N=8Z$ and discharges the claim by exhaustive fin_cases over all six Fin-4 arguments, each leaf a chunk identity of this form. Without the pointwise cells, the universal numerator identity does not close.

In the gravity stack this is bookkeeping for the exact discrete midpoint identity underlying Regge-side curvature bookkeeping in 4D, not a continuum GR derivation. It sits downstream of the kernel certificate definitions and upstream of the assembled equality used by later midpoint/TT analyses. No forcing-chain (T0–T8) step is claimed here; the link is purely to the discrete gravity numerator algebra.