e_000012
plain-language theorem explainer
For the six-index slot (0,0,0,0,1,2) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite this as one atomic cell in the 256-way case split that certifies the Regge midpoint M2 identity. The proof is a single kernel decide on concrete integers.
Claim. With indices in $\{0,1,2,3\}$, the integer numerator $m_2(0,0,0,0,1,2)$ obtained by folding the coupling list equals $8$ times the explicit kernel value $Z(0,0,0,0,1,2)$.
background
This module is chunk 0 of a 256-cell kernel certification that $m_2 = 8\cdot Z$ pointwise on $(\mathrm{Fin},4)^6$. The ambient setting is the exact midpoint analysis of the 4D Regge M2/TT identity used in the gravity sector.
The numerator $m_2(a,b,c,d,i,j)$ is defined by folding couplingZList and summing the local contribution of each coupling term at those six indices. The comparison target explicitZ is a total function $(\mathrm{Fin},4)^6\to\mathbb{Z}$ given by a finite pattern of integer cases (typical nonzero values $\pm 2,,4$).
The present cell fixes the concrete tuple $(0,0,0,0,1,2)$. Sibling theorems cover the other index combinations in the same chunk; the assembler later glues all cells by exhaustive fin_cases.
proof idea
One-line kernel proof: by decide. Both sides reduce to concrete integers once the six Fin-4 arguments are closed, so the equality is discharged by native decision on $\mathbb{Z}$ arithmetic with no further lemmas.
why it matters
Feeds the parent theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j:\mathrm{Fin},4,; m_2(a,b,c,d,i,j)=8,Z(a,b,c,d,i,j)$ and is proved by six nested fin_cases that invoke each cell (including this one).
That global identity is the algebraic backbone of the Regge exact-midpoint M2/TT certification in 4D. In the Recognition gravity stack it supplies a machine-checked numerator identity on the discrete curvature kernel, rather than a floating-point or schematic check. It does not itself touch the T0–T8 forcing chain, phi-ladder masses, or the alpha band; those sit upstream of the continuum limit this discrete identity supports.
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