e_013010
plain-language theorem explainer
At the six-index point (0,1,3,0,1,0), the Regge coupling numerator m2Num equals eight times the explicit integer table explicitZ. Gravity analysts certifying the 4D midpoint M2TT identity cite this as one atomic kernel case among the Fin-4 grid. The proof is a single decide on concrete integers after unfolding the two definitions.
Claim. For the multi-index $(a,b,c,d,i,j)=(0,1,3,0,1,0)$ on $\mathrm{Fin}\,4$, the folded coupling numerator equals eight times the explicit closed-form integer: $N(0,1,3,0,1,0)=8\,Z(0,1,3,0,1,0)$.
background
In the 4D Regge midpoint M2TT analysis, the numerator m2Num is the integer obtained by folding a fixed coupling contribution list over six indices in Fin 4. The companion explicitZ is a pattern-matched integer table on the same six indices (typical values 4, -2, and so on), intended as the closed form of that sum divided by eight.
This module is chunk 1 of a decide-split certification that m2Num = 8 · explicitZ holds pointwise on the full $4^6$ grid. Both m2Num and explicitZ are pure definitions imported from the kernel certificate module; no analytic continuum input enters here.
proof idea
One-line computational proof. After the definitions of the folded numerator and the explicit table are in scope, decide evaluates both sides at the concrete sextuple $(0,1,3,0,1,0)$ and checks integer equality. No intermediate lemmas are invoked.
why it matters
The atom is consumed by the universal assembler m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j:\mathrm{Fin},4,; N=8Z$ and discharges the grid by nested fin_cases. That identity is the algebraic backbone of the exact midpoint M2TT relation in the 4D Regge gravity analysis: once every kernel cell matches, the discrete curvature bookkeeping is certified before any continuum comparison. It sits inside the gravity domain of the Recognition mirror and does not itself touch the forcing chain (T0–T8), the J-cost, or the phi ladder.
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