e_201131

This module sits in the 4D Regge exact-midpoint analysis: one proves that a summed numerator built from coupling contributions coincides with eight times a closed-form integer table on every sextuple of Fin-4 indices. The module doc frames the work as chunk 8 of that identity, discharged by 256 kernel decisions.

The numerator is defined by folding a fixed coupling list: start at 0 and add each contribution at the six indices. The explicit kernel is a pattern-matched integer table on Fin-4 sextuples (sample clauses give values such as 4, -2, and so on). Both objects live in the kernel-certificate module imported here.

The local claim is only the single cell with indices (2,0,1,1,3,1). Sibling theorems cover the other cells in the same chunk.

One-line proof by decide. Both sides reduce to concrete integers: the left-hand side evaluates the fold of contributions at the fixed indices; the right-hand side is eight times the matching clause of the explicit table. Lean’s decision procedure checks integer equality; no lemmas beyond the two definitions are invoked.

The parent theorem is the full assembly identity: for every $a,b,c,d,i,j$ in Fin 4, the numerator equals eight times the explicit kernel. That proof introduces the six indices and runs fin_cases on each, so each concrete cell theorem (including this one) is the leaf that closes one branch.

In the Recognition gravity stack this identity is bookkeeping for the 4D Regge midpoint M2–TT comparison: once every cell matches, the summed coupling form may be replaced by the closed table. It does not itself invoke the forcing chain (T5–T8) or the Recognition Composition Law; it is pure discrete kernel certification underneath the continuum gravity claims.