e_033110
plain-language theorem explainer
For the six-index slot (0,3,3,1,1,0) on Fin 4, the midpoint Regge numerator m2Num equals eight times the closed-form kernel explicitZ. Gravity analysts cite it when assembling the full 4D identity m2Num = 8·explicitZ over all index tuples. The proof is a single kernel decide on concrete integers.
Claim. For indices $a{=}0$, $b{=}3$, $c{=}3$, $d{=}1$, $i{=}1$, $j{=}0$ in $\mathrm{Fin}\,4$, the folded midpoint numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ 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 4D Regge midpoint numerator equals eight times a sparse explicit integer table. Indices run over $\mathrm{Fin},4$, matching the four spacetime directions in the discrete midpoint calculus.
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: each term contributes an integer contrib at the six indices, and the fold starts from 0. The comparison target explicitZ is a pattern-matched integer table on the same six $\mathrm{Fin},4$ arguments (typical nonzero values $\pm 2,,4$ on matched pairs).
The local claim is one concrete six-tuple inside that table identity, not the universal statement.
proof idea
One-line decide proof. Both sides reduce to concrete Int values: the left by evaluating the fold of contrib over couplingZList at $(0,3,3,1,1,0)$, the right by multiplying the pattern value of explicitZ at those indices by 8. Lean’s kernel decision procedure checks integer equality; no lemmas are invoked.
why it matters
Feeds the assembler m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8,Z$ by exhaustive fin_cases on all six $\mathrm{Fin},4$ arguments. Each chunk theorem such as this one discharges one residual case after the case split, so the universal midpoint identity is only as strong as the full set of decides.
In the Recognition gravity stack this identity is bookkeeping for the exact 4D Regge midpoint $M_2$ tensor structure: once numerator and explicit kernel match everywhere, downstream curvature and continuum-limit arguments can quote a closed form instead of the fold. It is infrastructure inside the gravity analysis path, not a forcing-chain (T0–T8) step.
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