e_022113
plain-language theorem explainer
For the six-index cell (0,2,2,1,1,3), the folded coupling numerator equals eight times the explicit integer table value. Gravity analysts certifying the 4D Regge midpoint M2TT numerator identity cite this atomic equality. The proof is a single kernel decide on concrete integers.
Claim. For indices $(0,2,2,1,1,3)\in(\mathrm{Fin}\,4)^6$, the folded coupling numerator $m_2^{\mathrm{num}}(0,2,2,1,1,3)$ equals $8$ times the explicit closed-form integer $Z(0,2,2,1,1,3)$.
background
In the 4D Regge midpoint M2TT analysis, the numerator $m_2^{\mathrm{num}}$ is an integer-valued six-index array on $\mathrm{Fin},4$. It is defined by folding a fixed coupling list and summing a contribution function at each tuple $(a,b,c,d,i,j)$.
The companion table $Z$ (explicitZ) is a sparse pattern-matched map $(\mathrm{Fin},4)^6\to\mathbb{Z}$ giving the closed-form value of that same numerator cell. Typical nonzero entries are $\pm 2$ or $4$ on paired index patterns.
This module is chunk 2 of the 256 kernel decides that establish the pointwise identity $m_2^{\mathrm{num}}=8\cdot Z$ on every cell.
proof idea
One-line wrapper: by decide. Both sides are closed integer terms once the six $\mathrm{Fin},4$ arguments are fixed to $0,2,2,1,1,3$, so the kernel evaluates the fold defining the numerator and the pattern match defining $Z$, then checks equality.
why it matters
The parent theorem m2Num_eq_eight_explicitZ states $\forall(a,b,c,d,i,j),,m_2^{\mathrm{num}}=8\cdot Z$ and proves it by exhaustive fin_cases on the six indices, dispatching each cell to a chunk equality such as this one. That global numerator identity is part of the certified 4D Regge exact-midpoint M2TT kernel used in the gravity analysis layer of Recognition Science. It does not itself touch the forcing chain (T0–T8) or the RCL; it is infrastructure for the discrete curvature side.
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