e_120231
plain-language theorem explainer
For the six-index slot (1,2,0,2,3,1) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite it when assembling the exact midpoint M2 TT identity in 4D Regge calculus. The proof is a single kernel decide on concrete integers.
Claim. With indices in $\{0,1,2,3\}$, the summed coupling numerator at $(a,b,c,d,i,j)=(1,2,0,2,3,1)$ equals $8$ times the explicit integer kernel at those same indices: $N(1,2,0,2,3,1)=8\,Z(1,2,0,2,3,1)$.
background
This module is one chunk of a 256-case kernel certification that the Regge midpoint mass-squared numerator equals eight times a tabulated integer kernel on four-dimensional simplicial data. Indices run over $\mathrm{Fin},4$, labeling discrete directions in the 4D complex.
The numerator $N=\texttt{m2Num}$ is defined by folding a fixed coupling list: it accumulates every contribution $\texttt{contrib}$ at the six indices $(a,b,c,d,i,j)$. The comparison target $Z=\texttt{explicitZ}$ is a pure pattern-match table $\mathrm{Fin},4^6\to\mathbb{Z}$ (examples: $(0,0,1,1,2,2)\mapsto 4$, $(0,0,1,2,1,2)\mapsto -2$).
Chunk 6 discharges one concrete sextuple among the siblings $e_120000,\ldots$; the parent assembly theorem then glues all 256 decides into a universal identity.
proof idea
One-line decide proof. Both sides reduce to concrete integers: the left by evaluating the fold of contrib over couplingZList at $(1,2,0,2,3,1)$, the right by looking up explicitZ and multiplying by 8. Lean’s kernel decision procedure checks integer equality; no lemmas beyond the two definitions are invoked.
why it matters
Feeds directly into m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j:\mathrm{Fin},4,; N=8Z$ by exhaustive fin_cases. That universal identity is the algebraic core of the exact midpoint M2 TT certificate in 4D Regge gravity analysis.
In the Recognition Science gravity stack, such kernel equalities underwrite discrete curvature and mass-squared bookkeeping on the simplicial complex before continuum limits or phenomenological matching. The chunking (256 decides) keeps each certificate tiny and kernel-checkable while the assemble theorem restores the clean forall form used downstream.
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