e_032113

In the 4D Regge exact-midpoint M2/TT analysis, two integer-valued kernels on six Fin-4 indices are compared. The numerator $N=\mathrm{m2Num}$ is obtained by folding a fixed coupling list: each term contributes an integer via a local contrib map, and the fold starts at 0. The comparison target $Z=\mathrm{explicitZ}$ is a sparse explicit table $\mathrm{Fin},4^6\to\mathbb{Z}$ with a handful of nonzero entries (e.g. $4$, $-2$) and zero elsewhere.

The module is chunk 3 of a 256-cell kernel certification: every concrete six-tuple is discharged separately so the assembler can recombine them. Local setting (module doc): prove $N=8\cdot Z$ cellwise by kernel decides.

One-line computational proof: decide. Both sides reduce to concrete integers once the six Fin-4 arguments are substituted. The left side evaluates the fold that defines the numerator; the right side looks up the explicit table and multiplies by 8. Lean’s decision procedure checks integer equality; no lemmas beyond the two definitions are required.

Feeds the assembler theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, N(a,b,c,d,i,j)=8,Z(a,b,c,d,i,j)$ by exhaustive fin_cases on all six indices. Each chunk cell such as this one supplies one discharged goal in that case split.

In the broader gravity stack this identity is bookkeeping for the exact midpoint M2/TT kernel in 4D Regge calculus: it certifies that the folded coupling numerator is exactly eight times a sparse closed form, so later curvature or deficit identities can quote the closed form instead of the fold. It does not itself invoke the RS forcing chain (T5–T8) or the Recognition Composition Law; it is pure discrete kernel algebra supporting the gravity side of the monolith.