e_223231

In the 4D Regge midpoint analysis, two integer-valued maps on six $\mathrm{Fin},4$ indices are compared. The numerator $\mathrm{m2Num}$ is the fold of a fixed coupling list: it sums a local contribution over every term in that list. The companion $\mathrm{explicitZ}$ is a sparse pattern-matched table of small integers (entries such as $4$, $-2$, and so on) that is meant to be the closed form of that fold, up to a universal factor of eight.

The present module is chunk 10 of a 256-cell kernel certificate: each cell pins one concrete sextuple. The module header states the goal as $\mathrm{m2Num}=8\cdot\mathrm{explicitZ}$ decided cellwise. Upstream, both maps live in the kernel-certificate module that supplies the shared definitions used by every chunk.

One-line tactic proof: decide. Lean reduces both sides of the integer equality for the fixed indices $(2,2,3,2,3,1)$ and checks they match. No lemmas are invoked beyond the definitions of $\mathrm{m2Num}$ (fold of contributions) and $\mathrm{explicitZ}$ (pattern table).

The parent theorem $\mathrm{m2Num_eq_eight_explicitZ}$ asserts the identity for every sextuple in $(\mathrm{Fin},4)^6$. Its proof is a six-fold fin_cases cascade that lands on the individual cell theorems; this declaration is the cell for $(2,2,3,2,3,1)$. Together the cells close the algebraic certificate that the folded Regge midpoint numerator is exactly eight times the explicit table, a computational step inside the 4D gravity analysis chain. It does not itself touch the T0–T8 forcing landmarks; it is pure integer bookkeeping supporting the Regge side of the gravity stack.