e_222311

In the 4D Regge exact-midpoint analysis, two integer kernels on six Fin-4 indices are compared. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at 0 and add each contribution term evaluated at $(a,b,c,d,i,j)$. The companion $Z_{\mathrm{explicit}}$ 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 8.

This module is chunk 10 of a 256-case decide grid that discharges one sextuple at a time. The local claim is only the equality at the concrete point $(2,2,2,3,1,1)$; neighboring chunks handle the other Fin-4 combinations.

One-line computational proof: by decide. Lean reduces both sides of $m_2^{\mathrm{num}}(2,2,2,3,1,1)=8\cdot Z_{\mathrm{explicit}}(2,2,2,3,1,1)$ to concrete integers from the fold definition and the pattern table, then checks equality in $\mathbb{Z}$. No lemmas are invoked beyond the two definitions.

The parent theorem is $m_2^{\mathrm{num}}=8,Z_{\mathrm{explicit}}$ for every $(a,b,c,d,i,j)\in(\mathrm{Fin},4)^6$, proved by exhaustive fin_cases that routes each sextuple to a chunk equality such as this one. Without the atomic decides, the assembly cannot close. In the Recognition gravity stack this identity is the algebraic certificate that the midpoint M2–TT kernel matches its explicit 4D form, a prerequisite for later continuum and continuum-limit arguments in the Regge sector. It does not itself touch the forcing chain (T0–T8) or the RCL; it is pure discrete kernel bookkeeping.