e_012122
plain-language theorem explainer
Kernel-checked equality of the folded Regge coupling numerator with eight times the explicit integer table at the single index sextuple (0,1,2,1,2,2). Gravity analysts assembling the 4D midpoint M2 TT identity cite it as one of 256 atomic cases. The proof is a one-line decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(0,1,2,1,2,2)$ in $(\mathrm{Fin}\,4)^6$, the folded coupling numerator equals eight times the explicit integer table: $m_2^{\mathrm{num}}(0,1,2,1,2,2)=8\,Z_{\mathrm{expl}}(0,1,2,1,2,2)$.
background
This module is chunk 1 of a 256-case kernel certification that the 4D Regge midpoint numerator agrees with an explicit integer table, scaled by eight. The ambient setting is the exact midpoint M2 TT identity in four dimensions.
The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: start from 0 and add each term's contribution at those six $\mathrm{Fin},4$ indices. The table $Z_{\mathrm{expl}}$ is a total function on the same six indices into $\mathbb{Z}$, specified by pattern-matching (e.g. $(0,0,1,1,2,2)\mapsto 4$, $(0,0,1,2,1,2)\mapsto -2$, and so on).
The local claim is only the instance at $(0,1,2,1,2,2)$. Sibling theorems cover the other sextuples in the same chunk.
proof idea
One-line computational proof: decide. Both sides reduce to concrete integers once the fold for the numerator and the pattern match for the explicit table are evaluated at the fixed indices $(0,1,2,1,2,2)$; the kernel checks equality of those integers. No lemmas beyond the two definitions are invoked.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j:\mathrm{Fin},4,; m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$ and discharges the universal quantifier by exhaustive fin_cases on all six indices. Each atomic equality such as this one is a certificate cell in that case split.
In the Recognition gravity stack, the identity is part of the exact midpoint analysis of the 4D Regge M2 TT sector: replacing the folded coupling sum by a closed integer table (times eight) is what makes later curvature and mass-ladder comparisons algebraic rather than combinatorial. It does not itself touch the T0–T8 forcing chain, but it is infrastructure for the gravity side of the RS ledger.
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