e_110221

In the 4D Regge midpoint analysis, two integer-valued kernels on six indices in $\mathrm{Fin},4$ are compared. The numerator $N=\mathrm{m2Num}$ is defined by folding a fixed coupling list: each term contributes an integer via a local contribution map, and the fold starts at 0. The comparison target $Z=\mathrm{explicitZ}$ is a closed-form pattern-matched integer table on the same six indices (sample clauses include values $4$, $-2$, and so on).

The local module is chunk 5 of a decide battery whose sole job is to certify $N=8Z$ at many concrete multi-indices. The ambient claim is the pointwise identity over all $4^6$ tuples, later assembled by exhaustive finite case split.

One-line computational proof: decide evaluates both sides at the fixed indices $(1,1,0,2,2,1)$ and checks integer equality. No lemmas are invoked beyond the reducible definitions of the folded numerator and the explicit kernel table.

Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, N(a,b,c,d,i,j)=8Z(a,b,c,d,i,j)$ and discharges the universal quantifier by six nested fin_cases sweeps. Each chunk theorem such as this one closes one concrete cell of that grid.

In the Recognition gravity stack this identity is bookkeeping for the exact midpoint TT kernel in 4D Regge calculus: once $N=8Z$ is certified everywhere, downstream curvature and mass-ladder comparisons can quote a single closed kernel instead of a folded sum. It is pure finite verification, not a continuum GR derivation.