e_320103

In the 4D Regge midpoint analysis, two integer-valued maps on six Fin-4 indices appear. The numerator $N(a,b,c,d,i,j)$ is obtained by folding a fixed coupling list and summing a local contribution at each term. The companion map $Z$ is an explicit pattern-matched integer table (typical entries $4$, $-2$, and so on) that is meant to be the closed form of that fold, up to a universal factor of eight.

This module is chunk 14 of a 256-cell kernel certification: each cell pins one concrete six-tuple. The local claim is exactly $N=8Z$ at $(3,2,0,1,0,3)$. The surrounding assemble theorem then quantifies over all of $(\mathrm{Fin},4)^6$ by exhaustive case split.

One-line proof by decide. Both sides reduce to concrete integers: the left-hand side evaluates the fold that defines the numerator at the fixed indices, the right-hand side multiplies the table value of the explicit kernel by eight. Lean’s kernel decision procedure checks the resulting integer equality; no lemmas beyond the two definitions are required.

Feeds the assemble theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, N=8Z$ and is proved by nested fin_cases over the six indices, discharging each cell by a chunk identity of this form. That global identity is the algebraic backbone of the exact midpoint M2TT relation in the 4D Regge gravity analysis. Within Recognition Science gravity work it certifies that the folded coupling numerator collapses to a sparse explicit kernel, clearing a computational step toward the continuum or continuum-limit identities used downstream. It does not itself invoke the forcing chain (T5–T8) or the Recognition Composition Law; it is pure discrete kernel bookkeeping.