e_000013

In the 4D Regge midpoint analysis, two integer-valued kernels on six $\mathrm{Fin},4$ indices are compared. The numerator $N=\mathrm{m2Num}$ is defined by folding a fixed coupling list: each term contributes an integer via a local contribution map, and the fold starts at $0$. The comparison target $Z=\mathrm{explicitZ}$ is a closed-form piecewise integer table on the same index space (sample values include $4$, $-2$, and so on for distinguished index patterns).

The local module is chunk 0 of a 256-cell decide grid whose sole job is to certify $N=8Z$ at every multi-index. Upstream, both $N$ and $Z$ are imported from the kernel certificate module; no analytic expansion is performed here, only evaluation at one concrete point.

One-line proof by decide. Lean reduces both sides at the concrete six-tuple $(0,0,0,0,1,3)$: the fold that defines the numerator and the pattern-match that defines the explicit kernel are fully computational on Fin 4 and Int, so the equality is discharged by the kernel without lemmas or case splits in this file.

This cell is consumed by the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, N=8Z$ and proves it by exhaustive fin_cases over all six indices. Each chunk theorem such as this one supplies one of the 256 ground instances the assembler relies on. In the broader gravity stack, the identity $N=8Z$ is the algebraic certificate that the midpoint M2 TT coupling matches its explicit closed form, a step toward exact discrete curvature identities in the Recognition Science gravity analysis. It does not itself invoke the forcing chain (T0–T8) or the J-cost; it is pure index algebra supporting the Regge side.