e_303003

This module is one chunk of a 256-case kernel certification that the Regge midpoint numerator equals eight times an explicit integer table on $(\mathrm{Fin},4)^6$. The local slogan is $N=8Z$, proved by exhaustive decide.

The numerator $N(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: start at $0$ and add each term's contribution at those six indices. The table $Z$ is a pattern-matched integer function on the same six $\mathrm{Fin},4$ arguments (typical nonzero values $\pm 2,,4$ on selected index patterns).

The ambient setting is 4D discrete gravity analysis: certifying an exact midpoint identity between an $M_2$ numerator built from couplings and a closed-form kernel, case by case before assembly.

One-line proof by decide. Both sides reduce to concrete Int values for the fixed indices $(3,0,3,0,0,3)$: the left via the fold definition of the numerator, the right via the pattern match of the explicit kernel (scaled by 8). No lemmas are invoked beyond kernel evaluation of equality on integers.

Feeds the assembly theorem that $\forall a,b,c,d,i,j,, N=8Z$ on $(\mathrm{Fin},4)^6$, proved by nested fin_cases that dispatch to these pointwise chunk identities. Without the full set of 256 decides, the global Regge exact midpoint $M_2$–TT numerator identity stays uncertified.

In the Recognition gravity stack this is pure discrete-kernel bookkeeping: it locks the algebraic factor of eight between the folded coupling numerator and the explicit $Z$ table used downstream in 4D midpoint analysis. It does not itself invoke the forcing chain (T5–T8) or the RCL; it is infrastructure for the exact discrete gravity identity those layers later consume.