e_120213
plain-language theorem explainer
At the six Fin-4 indices (1,2,0,2,1,3), the folded Regge coupling numerator equals eight times the explicit integer kernel value. Gravity analysts assembling the 4D midpoint M2TT identity cite this as one concrete point of the 256-case kernel table. The proof is a single kernel decide on closed integer arithmetic.
Claim. For indices $a=1$, $b=2$, $c=0$, $d=2$, $i=1$, $j=3$ in $\mathrm{Fin}\,4$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8\cdot Z(a,b,c,d,i,j)$, where $Z$ is the explicit integer kernel on six Fin-4 arguments.
background
This module is chunk 6 of a 256-point kernel certification that the folded numerator m2Num agrees with eight times an explicit integer table explicitZ on every sextuple in $(\mathrm{Fin},4)^6$. The local setting is the 4D Regge midpoint analysis for the M2TT identity in the Gravity domain.
Upstream, m2Num a b c d i j is the fold of contrib · a b c d i j over couplingZList, returning an Int. The companion explicitZ is a total pattern-matched table Fin 4 → ⋯ → Fin 4 → Int with sparse nonzero entries (e.g. value 4 on (0,0,1,1,2,2), value −2 on several mixed pairs). The identity under check is pointwise equality of the fold against eight times that table.
The present declaration fixes one concrete sextuple from that grid: (1,2,0,2,1,3).
proof idea
One-line proof by decide. Both sides reduce to concrete integers once the six Fin-4 arguments are substituted: the left via the fold definition of m2Num, the right via the pattern match of explicitZ scaled by 8. Lean’s kernel decision procedure discharges the resulting closed Int equality with no lemmas or case splits.
why it matters
Parent theorem m2Num_eq_eight_explicitZ states the universal claim $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8Z$ and proves it by exhaustive fin_cases on all six arguments. Each leaf of that case tree is one of these chunk theorems; this declaration is the leaf for (1,2,0,2,1,3).
In the Recognition gravity stack the certified numerator–kernel match is the algebraic backbone of the exact 4D Regge midpoint M2TT identity. Without the 256 point checks, the assemble theorem cannot close, and downstream curvature or continuum-limit arguments that quote the identity remain conditional on an unchecked table. The chunk split keeps each decide goal tiny while the assemble theorem recombines them into the full $\forall$ statement.
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