e_313331
plain-language theorem explainer
For the six-index slot (3,1,3,3,3,1) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite these pointwise identities when assembling the global Regge midpoint M2TT relation. The proof is a single kernel decide on concrete integers.
Claim. For indices $a{=}3,\,b{=}1,\,c{=}3,\,d{=}3,\,i{=}3,\,j{=}1$ in $\mathrm{Fin}\,4$, the folded coupling numerator equals eight times the explicit integer kernel: $N(3,1,3,3,3,1)=8\,Z(3,1,3,3,3,1)$.
background
This module is chunk 13 of a 256-case kernel certification that the Regge-exact midpoint numerator equals eight times a tabulated integer kernel on every six-tuple in $(\mathrm{Fin},4)^6$.
The numerator $N(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: start at $0$ and add each contribution term evaluated at the six indices. The companion kernel $Z$ is an explicit pattern-matched integer table on the same six $\mathrm{Fin},4$ arguments (sample entries include $4$, $-2$, and so on).
The local goal is purely algebraic bookkeeping: check $N=8Z$ at one concrete point so a later assembler can quantify over all indices by case split.
proof idea
One-line decide proof. Both sides reduce to concrete integers once the six Fin 4 arguments are substituted: the fold defining the numerator evaluates fully, and the pattern match for the explicit kernel returns a fixed Int. Lean’s kernel decision procedure discharges the resulting integer equality.
why it matters
Feeds the assembler theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, N=8Z$ by exhaustive fin_cases on all six indices. Each chunk lemma such as this one supplies one of the 256 concrete equalities that make that universal statement go through.
In the gravity analysis stack this identity is the algebraic core of the Regge-exact midpoint M2TT relation in 4D: once the numerator is identified with eight times the explicit kernel everywhere, downstream curvature and mass-window arguments can quote a closed form rather than a fold. It is bookkeeping, not a new physical law, but without the pointwise certificates the global identity remains unproved in Lean.
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