e_033130
plain-language theorem explainer
For the six-index slot (0,3,3,1,3,0) on Fin 4, the folded Regge midpoint numerator m2Num equals eight times the explicit kernel value explicitZ. Gravity analysts cite it when assembling the full 4D m2Num = 8·Z identity from 256 kernel cases. The proof is a single kernel decide on two concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(0,3,3,1,3,0)$ in $\mathrm{Fin}\,4$, the midpoint numerator $m_2(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel $Z(a,b,c,d,i,j)$.
background
This module is chunk 3 of a 256-case kernel certification that the Regge exact-midpoint numerator equals eight times a closed-form integer table in 4D. Indices run over $\mathrm{Fin},4$, labeling discrete edge/face slots in the midpoint scheme.
Upstream, $m_2$ is defined by folding a fixed coupling list: start at $0$ and add each contribution contrib t a b c d i j. The companion explicitZ is a pattern-matched integer table on the same six indices (sample entries include $4$, $-2$, and so on for particular tuples).
The local claim is one concrete cell of the identity $m_2=8Z$, discharged by computation rather than by a symbolic expansion of the fold.
proof idea
One-line proof: decide. Both sides reduce to concrete Int values (the fold for m2Num at this tuple, and the pattern match for explicitZ), so the kernel checks integer equality and closes the goal. No lemmas beyond the two definitions are invoked.
why it matters
Parent theorem m2Num_eq_eight_explicitZ states $\forall a,b,c,d,i,j,, m_2=8Z$ and proves it by exhaustive fin_cases on all six $\mathrm{Fin},4$ indices. This declaration is one of the 256 atomic cells that assembly stitches together (chunk 3 in the module split).
In the Recognition gravity stack, the identity certifies that the discrete midpoint curvature numerator matches the explicit kernel used in 4D Regge-type analysis. It is bookkeeping infrastructure for the exact midpoint TT identity, not a new physical law; it locks the algebraic bridge between the folded coupling definition and the tabulated $Z$ so downstream curvature and mass-ladder arguments can quote a single closed form.
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