e_323231

This module is chunk 14 of a 256-case kernel certification that the 4D Regge midpoint M2 numerator equals eight times an explicit integer table $Z$. Indices run over $\mathrm{Fin},4$, labeling discrete edge/face slots in the midpoint scheme.

The numerator m2Num is defined by folding a fixed coupling list: each term contributes an integer depending on the six indices, and the fold starts at 0. The companion explicitZ is a sparse pattern match on those same six indices, returning small integers such as $4$, $-2$, or (by default) $0$.

The local goal is purely algebraic identity checking: no continuum limit or variational argument appears here. Upstream, both sides are closed definitions in the kernel certificate module; this chunk only evaluates one concrete sextuple.

One-line proof by decide. Both sides reduce to concrete Int values: the left via the fold definition of the numerator, the right via the pattern match for explicit $Z$ scaled by 8. Lean’s kernel decision procedure discharges the integer equality with no lemmas or rewriting.

Feeds the assembler m2Num_eq_eight_explicitZ, which states the identity for every sextuple in $(\mathrm{Fin},4)^6$ by exhaustive fin_cases and invokes each chunk theorem such as this one. That global equality is the certified algebraic core of the 4D Regge midpoint M2–TT identity used in the gravity analysis stack.

Within Recognition Science gravity work, such kernel identities lock discrete curvature bookkeeping before continuum or phenomenological claims. This declaration is one of 256 parallel cases; alone it is trivial, but the full cover is what the assembler needs. No forcing-chain (T0–T8) step is touched directly; the link is infrastructure for discrete gravity identities rather than the phi-ladder or RCL.