e_120301

In the 4D Regge midpoint analysis, two integer-valued kernels on six Fin-4 indices are compared. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at 0 and add each contribution term at the given indices. The comparison target $Z_{\mathrm{expl}}$ is an explicit case table of small integers (entries such as $4$, $-2$, and so on) on the same six-index domain.

The local module is chunk 6 of a 256-cell kernel certificate: each cell asserts $m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$ at one concrete multi-index. The factor 8 is the global normalization relating the folded sum to the closed form. Upstream, only the two kernel definitions are needed; no analytic lemma is imported beyond those defs.

One-line proof by decide. Both sides reduce to concrete Int values once the six Fin-4 arguments are literals: the fold that defines the numerator evaluates fully, the explicit table lookup returns a fixed integer, and the kernel checks equality with the factor eight. No rewrite lemmas or induction are used.

This cell is consumed by m2Num_eq_eight_explicitZ, which states the identity for every six-tuple in Fin 4 by exhaustive fin_cases and dispatches each branch to a chunk theorem of this form. Closing all 256 cells upgrades the pointwise kernel certificate into a uniform algebraic identity used in the Regge midpoint M2–TT analysis in four dimensions.

Within Recognition gravity work, that identity is bookkeeping infrastructure for curvature/edge coupling numerics on the discrete complex, not a forcing-chain landmark (T5–T8) by itself. It removes a residual computational gap between the folded coupling definition and the explicit integer table so later gravity lemmas can quote a single forall rather than case-bash.