e_021111

In the 4D Regge midpoint analysis, two integer-valued maps on six Fin-4 indices are compared. The summed coupling numerator $N(a,b,c,d,i,j)$ is obtained by folding a fixed coupling list and adding each term's contribution at those indices. The companion map $Z$ is an explicit case table of small integers (entries such as $4$, $-2$, and so on) that is meant to be the closed form of that sum up to a universal factor of eight.

This module is chunk 2 of a 256-way kernel split: each declaration checks one concrete sextuple. The local claim is exactly $N=8Z$ at the fixed indices $(0,2,1,1,1,1)$. Upstream, both $N$ and $Z$ are defined in the kernel certificate module; the assembly theorem later quantifies over all of $(\mathbb{F}_4)^6$.

One-line proof by decide. Both sides reduce to concrete integers once the six Fin-4 arguments are substituted: the left-hand side evaluates the fold that defines the summed numerator, the right-hand side looks up the explicit table and multiplies by eight. No lemmas beyond the two definitions are required; the kernel discharges the integer equality.

Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, N=8Z$ by exhausting all Fin-4 cases. That global identity is the algebraic backbone of the exact midpoint M2 TT relation in the 4D Regge analysis. Within Recognition Science gravity work, such kernel certificates pin the discrete curvature bookkeeping that later couples to continuum limits; they do not themselves invoke the forcing chain (T5--T8) or the Recognition Composition Law, but they clear a necessary computational step before those continuum claims can be stated cleanly.