e_301130
plain-language theorem explainer
For the six-index tuple (3,0,1,1,3,0) on Fin 4, the folded Regge coupling numerator equals eight times the tabulated explicit kernel integer. Gravity analysts closing the 4D midpoint M2TT identity cite it as one atomic case among the 256 kernel decides. The proof is a single computational decide on both integer sides.
Claim. For indices $(a,b,c,d,i,j)=(3,0,1,1,3,0)$ with each index in $\mathrm{Fin}\,4$, the folded coupling numerator equals $8$ times the explicit integer kernel value at those indices.
background
This module is one chunk of the case-split that establishes the numerator identity behind the exact Regge midpoint M2TT relation in 4D. The local setting is stated as: m2Num equals eight times explicitZ, proved by 256 kernel decides split across chunks.
The numerator m2Num is the fold of a fixed coupling list: it sums a contribution function over that list at a six-tuple of Fin-4 indices, yielding an integer. The comparison value explicitZ is a fully expanded integer table on the same six Fin-4 arguments (pattern-matched constants such as 4, -2, and so on).
The identity claimed here is only the single cell with indices (3,0,1,1,3,0). Sibling theorems in the same chunk cover the neighboring cells.
proof idea
One-line computational proof: decide evaluates both the folded numerator and eight times the tabulated kernel entry at the concrete Fin-4 sextuple and checks integer equality. No algebraic lemmas are invoked; the kernel certificate definitions supply enough closed-form data for the decision procedure.
why it matters
The parent theorem is the universal statement that m2Num equals eight times explicitZ for every six-tuple in Fin 4. That assembler runs exhaustive fin_cases on all six indices and discharges each residual goal by a chunk theorem of this form.
In the Recognition gravity stack this cell-level equality is bookkeeping for the exact midpoint M2TT identity in 4D Regge analysis: once every kernel cell matches, the numerator side collapses to the explicit eight-fold table and the continuum or discrete gravity identities that depend on that table can proceed without residual case debt. It does not itself invoke the T0–T8 forcing chain; it is infrastructure inside the gravity analysis layer.
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