e_123101

In the 4D Regge exact-midpoint analysis, the TT-sector numerator is assembled from a finite coupling list. The folded numerator $N(a,b,c,d,i,j)$ is the integer obtained by folding that list and summing each term's contribution at six $\mathbb{F}_4$ indices. The explicit kernel $Z$ is a closed-form integer table on the same six indices (pattern-matched cases such as $(0,0,1,1,2,2)\mapsto 4$ and sign-flipped off-diagonal entries).

This module is chunk 6 of the 256-cell decide grid that checks $N=8Z$ at every multi-index. The local claim is the single cell with indices $(1,2,3,1,0,1)$. Upstream, $N$ and $Z$ are defined in the kernel certificate module; downstream they are reassembled into the universal identity.

One-line proof by decide. Both sides are closed integer expressions once the six Fin 4 arguments are literals, so the kernel reduces the equality to true with no further lemmas.

Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j., N=8Z$ by exhausting all $\mathbb{F}_4$ cases. That universal identity is the algebraic backbone of the 4D Regge midpoint M2–TT numerator certification in the Gravity analysis stack. It is bookkeeping rather than a new physical law: it locks the folded coupling sum to the explicit kernel so later curvature and continuum-limit arguments can quote a single clean factor of eight. No Recognition forcing-chain step (T5–T8) is discharged here; the result is infrastructure inside the discrete gravity certificate.