e_022111
plain-language theorem explainer
For the six-index slot (0,2,2,1,1,1) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite these micro-identities when assembling the global Regge midpoint M2 TT identity in 4D. The proof is a single kernel decide on concrete integers.
Claim. With indices in $\{0,1,2,3\}$, the folded coupling numerator at $(0,2,2,1,1,1)$ equals eight times the explicit integer kernel at the same indices: $N(0,2,2,1,1,1)=8\,Z(0,2,2,1,1,1)$.
background
This module is one chunk of a 256-case kernel certification that the 4D Regge midpoint M2 TT numerator equals eight times a sparse explicit integer table. Indices run over $\mathrm{Fin},4$, i.e. the four coordinate directions of a 4-simplex edge labeling.
The numerator $N=\mathrm{m2Num}$ is defined by folding a fixed coupling list: start at $0$ and add a contribution for each coupling triple at the six free indices. The table $Z=\mathrm{explicitZ}$ is a pattern-matched integer function on the same six indices (typical nonzero values $\pm 2,,4$ on a handful of permutations; zero elsewhere).
The local claim is the single-point instance of $N=8Z$ at $(0,2,2,1,1,1)$. Sibling theorems cover the other index tuples in the same chunk.
proof idea
One-line proof by decide. Both sides reduce to concrete Int values: the left by evaluating the fold of contributions at fixed indices, the right by pattern-matching explicitZ (or defaulting to zero) and multiplying by eight. Lean’s kernel decision procedure checks integer equality; no lemmas are invoked beyond definitional reduction.
why it matters
Feeds the assembler m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, N=8Z$ by exhaustive fin_cases on all six $\mathrm{Fin},4$ arguments and dispatch into these pointwise theorems. That global identity is the certified algebraic core of the Regge-exact midpoint M2 TT relation in 4D gravity analysis inside the monolith.
In the broader Recognition Science gravity stack, such exact discrete identities pin the curvature/coupling bookkeeping before continuum or phenomenological limits are taken. This declaration is pure scaffolding glue: one of 256 decides, not a conceptual step of the T0–T8 forcing chain, but required so the assembled theorem has zero sorry.
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