e_002301
plain-language theorem explainer
At the multi-index (0,0,2,3,0,1), the folded numerator m2Num equals eight times the tabulated kernel value explicitZ. Gravity analysts cite this as one atomic cell of the 4D Regge midpoint TT identity. The proof is a single kernel decide on concrete integers.
Claim. For the multi-index $(a,b,c,d,i,j)=(0,0,2,3,0,1)$ in $(\mathrm{Fin}\,4)^6$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel entry $Z(a,b,c,d,i,j)$.
background
This module sits in the Gravity analysis stack for the exact midpoint Regge identity in the 4D transverse-traceless (TT) sector. The local goal, stated in the module header, is to certify $m_2^{\mathrm{num}}=8\cdot Z$ on chunk 0 by a battery of kernel decides.
The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: start at $0$ and add each contribution term at the six Fin-4 indices. The comparison object $Z$ is an explicit integer table on $(\mathrm{Fin},4)^6$, with sparse nonzero entries such as $4$ on diagonal-type patterns and $-2$ on selected off-diagonal patterns.
The present declaration fixes one concrete six-tuple and asserts numerical equality of the fold against eight times the table lookup.
proof idea
One-line kernel proof: decide evaluates both sides as concrete Int values (the fold of couplingZList on the left, the pattern-match table on the right) and closes the equality by computation. No lemmas are invoked beyond the definitions of m2Num and explicitZ.
why it matters
Parent theorem m2Num_eq_eight_explicitZ assembles the universal statement $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8Z$ by exhaustive fin_cases on all six indices; each case discharges to one of these pointwise decides. This cell is the instance at $(0,0,2,3,0,1)$.
In the broader Recognition gravity stack, the factor-of-eight identity is bookkeeping for the exact midpoint Regge TT kernel in four dimensions, aligning the folded coupling numerator with the closed-form integer table used downstream. It does not itself invoke the forcing chain (T0–T8), RCL, or the $\varphi$-ladder; it is pure discrete kernel certification inside the 4D gravity analysis layer.
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