e_313320
plain-language theorem explainer
For the six-index tuple (3,1,3,3,2,0) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite it as one atomic cell in the 4D Regge midpoint M2–TT identity. The proof is a single kernel decide on concrete integers.
Claim. For indices $a{=}3,\,b{=}1,\,c{=}3,\,d{=}3,\,i{=}2,\,j{=}0$ in $\mathrm{Fin}\,4$, the summed coupling numerator equals eight times the explicit integer kernel: $N(3,1,3,3,2,0)=8\,Z(3,1,3,3,2,0)$.
background
In the 4D Regge midpoint analysis, two integer-valued kernels on six Fin-4 indices are compared. The numerator $N=m2Num$ is obtained by folding a fixed coupling list and summing each term's contribution at the given indices. The comparison target $Z=explicitZ$ is a sparse closed-form table: most sextuples map to zero, while a short list of patterns (e.g. $(0,0,1,1,2,2)\mapsto 4$, $(0,0,1,2,1,2)\mapsto -2$) records the nonzero entries.
The local module is chunk 13 of a 256-cell decide grid that exhausts all Fin-4 sextuples. The claim under audit is the single cell with indices $(3,1,3,3,2,0)$. The parent assembly theorem will later quantify over every sextuple by fin_cases and invoke each cell.
proof idea
One-line decide. Both sides reduce to concrete Int values: the left by evaluating the fold of couplingZList contributions at $(3,1,3,3,2,0)$, the right by looking up explicitZ and multiplying by 8. Lean’s kernel decision procedure checks integer equality; no lemmas are invoked.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j:\mathrm{Fin},4,; N=8Z$ by fin_cases on each coordinate and discharge of every cell. That identity is the algebraic core of the Regge exact midpoint M2–TT certificate in 4D gravity analysis: it converts a summed coupling expression into an explicit sparse kernel, enabling exact (not approximate) midpoint identities used downstream in the discrete gravity stack.
Within Recognition Science this sits in the gravity domain rather than the T0–T8 forcing chain; it is infrastructure for exact discrete curvature bookkeeping, not a claim about $\varphi$, the eight-tick octave, or $D=3$.
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