e_032111

In the 4D Regge midpoint analysis, two integer-valued kernels on six Fin-4 indices are compared cellwise. The numerator $m_2^{\mathrm{num}}$ is the fold of a fixed coupling list: it sums a local contribution over that list and returns an Int. The comparison target $Z_{\mathrm{explicit}}$ is a closed pattern-match table on the same six indices (sample entries include $4$, $-2$, and so on).

The module is chunk 3 of a 256-cell partition of the Fin-4^6 grid. Each cell asserts the scalar identity $m_2^{\mathrm{num}}=8,Z_{\mathrm{explicit}}$ at one concrete multi-index. The factor eight is the global normalization that makes the folded sum match the explicit table.

Upstream, both kernels live in the KernelCert module; this chunk only evaluates them at fixed arguments.

One-line proof by decide. Both sides reduce to concrete integers once the six Fin-4 arguments are substituted: the fold that defines the numerator and the pattern match that defines the explicit kernel become ground Int expressions, and the kernel discharges the equality.

Parent theorem m2Num_eq_eight_explicitZ assembles the full universal statement $\forall a,b,c,d,i,j,, m_2^{\mathrm{num}}=8,Z_{\mathrm{explicit}}$ by exhaustive fin_cases on all six indices. Each chunk cell such as this one supplies one branch of that case split.

In the Recognition gravity stack this identity is bookkeeping for the exact midpoint M2–TT relation in 4D Regge calculus: it certifies that the folded coupling numerator is exactly eight times the closed-form kernel, so later curvature and mass-ladder arguments can quote the compact explicit table rather than the fold. It does not itself invoke T5–T8 or the RCL; it is pure discrete kernel algebra supporting those layers.