e_200022

This module is chunk 8 of a 256-cell kernel certification that the midpoint mass-squared numerator equals eight times an explicit integer kernel on all of $(\mathbb{F}_4)^6$. The ambient setting is the 4D Regge exact-midpoint M2TT identity in the Gravity analysis stack.

The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is the fold of a fixed coupling list: each term contributes an integer weight at the six Fin-4 indices, and the fold sums those contributions from zero. The explicit kernel $Z$ is a total function $(\mathrm{Fin},4)^6\to\mathbb{Z}$ given by a finite pattern match (e.g. $Z(0,0,1,1,2,2)=4$, $Z(0,0,1,2,1,2)=-2$, and so on).

The claim checked here is the single cell with indices $(2,0,0,0,2,2)$. Sibling theorems in the same chunk cover the other cells of this block.

One-line kernel proof: decide evaluates both sides as concrete integers. The left side reduces by unfolding the fold definition of the numerator against the coupling list; the right side multiplies the matched explicit-kernel entry by eight. No lemmas are invoked beyond decidable equality on $\mathbb{Z}$.

Parent theorem m2Num_eq_eight_explicitZ assembles all 256 cells into the universal statement $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8Z$. That assembler runs nested fin_cases on the six indices and dispatches each branch to a chunk theorem of this form.

In the Recognition gravity stack this identity is bookkeeping for the exact midpoint mass-squared tensor structure in 4D Regge calculus: once every cell matches, the closed-form kernel $Z$ may replace the folded coupling sum in downstream curvature and continuum-limit arguments. It does not itself invoke the forcing chain (T5–T8) or the Recognition Composition Law; it is pure finite combinatorial certification supporting those later geometric claims.