e_002010

The module is a chunk of the 4D Regge exact-midpoint $M_2$ TT-identity certification. The numerator $N(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: start at $0$ and add each local contribution at the six $\mathrm{Fin},4$ indices. The companion table $Z$ is an explicit integer-valued kernel on the same six indices, with sparse nonzero entries such as $Z(0,0,1,1,2,2)=4$ and $Z(0,0,1,2,1,2)=-2$.

The local claim is one cell of the global relation $N=8Z$. Module documentation frames the file as chunk 0 of that identity, discharged by 256 kernel decides. Upstream, both $N$ and $Z$ live in the kernel-certificate module that supplies the fold and the pattern-matched table.

One-line computational proof: by decide. Both sides are closed integer terms once the six concrete $\mathrm{Fin},4$ indices are substituted into the fold definition of the numerator and the pattern match of the explicit kernel, so the decision procedure evaluates the equality in $\mathbb{Z}$ and closes the goal. No lemmas are invoked beyond the two upstream definitions.

The parent theorem is the full assembly statement: for every sextuple in $(\mathrm{Fin},4)^6$, the folded numerator equals eight times the explicit kernel. That proof introduces the six indices and runs fin_cases on each, so each concrete cell such as this one is a required leaf. In the Recognition gravity stack this cell is scaffolding for the exact midpoint $M_2$ TT identity in 4D Regge calculus, tying the discrete curvature coupling to a closed integer kernel rather than a residual symbolic sum. It does not itself invoke the forcing chain (T0–T8) or the Recognition Composition Law; it is pure discrete-gravity bookkeeping that later geometric claims rely on.