e_010022
plain-language theorem explainer
For the six-index slot (0,1,0,0,2,2) on Fin 4, the folded numerator m2Num equals eight times the explicit kernel value explicitZ. Gravity analysts cite these micro-identities when assembling the global Regge midpoint M2TT identity in 4D. The proof is a single kernel decide on concrete integers.
Claim. With indices in $\mathrm{Fin}\,4$, the folded coupling numerator at $(0,1,0,0,2,2)$ equals eight times the tabulated explicit kernel entry at the same indices: $N(0,1,0,0,2,2)=8\,Z(0,1,0,0,2,2)$.
background
This module is one chunk of the 4D Regge-exact midpoint M2TT identity certification. The local goal, stated in the module header, is to prove the pointwise relation numerator = 8 · explicit kernel on a block of the 4^6 index space by kernel decides.
The numerator $N(a,b,c,d,i,j)$ is defined by folding a fixed coupling list and summing a contribution at each term. The explicit kernel $Z$ is a total function on six Fin-4 indices given by a finite pattern table of small integers (entries such as 4, -2, and defaults).
Upstream, both $N$ and $Z$ live in the KernelCert module. Downstream assembly will range over all Fin-4 sextuples by fin_cases and invoke one such equality per cell.
proof idea
One-line closed proof: by decide. Lean evaluates both sides at the concrete sextuple (0,1,0,0,2,2) in the integer kernel and checks definitional equality. No lemmas are invoked beyond the evaluability of the fold defining the numerator and the pattern match defining the explicit kernel.
why it matters
Feeds the parent theorem m2Num_eq_eight_explicitZ, which asserts the identity for every sextuple in Fin 4 by exhaustive fin_cases and one chunk lemma per cell. That global equality is the algebraic backbone of the Regge-exact midpoint M2TT identity in 4D gravity analysis inside the monolith.
In the Recognition Science gravity stack this is bookkeeping, not a new physical law: it certifies that the folded coupling numerator matches the closed-form kernel used in continuum or continuum-limit comparisons. It does not itself invoke the forcing chain (T0–T8), RCL, or the phi ladder; those enter only if later gravity theorems specialize constants.
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