e_211310
plain-language theorem explainer
For the six-index tuple (2,1,1,3,1,0) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite this as one cell of the 4D Regge midpoint M2TT identity. The proof is a single kernel decide on concrete integers.
Claim. For indices $a{=}2$, $b{=}1$, $c{=}1$, $d{=}3$, $i{=}1$, $j{=}0$ in $\{0,1,2,3\}$, the coupling-fold numerator $N_2(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel $Z(a,b,c,d,i,j)$.
background
In the 4D Regge midpoint analysis, two integer-valued maps on six Fin-4 indices are compared. The numerator $N_2$ (m2Num) is obtained by folding a fixed coupling list and summing local contributions at $(a,b,c,d,i,j)$. The companion map $Z$ (explicitZ) is a sparse closed form: only a handful of index patterns are nonzero (e.g. $4$ or $-2$), and all other tuples evaluate to zero by the default clause.
The module is chunk 9 of a 256-cell kernel certification that $N_2 = 8Z$ pointwise. Each chunk theorem pins one concrete six-tuple so the global identity can be assembled by exhaustive fin_cases rather than a symbolic rewrite of the fold.
proof idea
One-line decide proof. Both sides reduce to concrete Int values: the left by evaluating the fold of coupling contributions at $(2,1,1,3,1,0)$, the right by pattern-matching explicitZ (or hitting its default zero) and multiplying 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 $\forall a,b,c,d,i,j,, N_2=8Z$ by casing on all six Fin-4 indices and dispatching to chunk theorems such as this one. That identity is the algebraic core of the Regge-exact midpoint M2TT certificate in the gravity analysis stack: it converts a summed coupling expression into an explicit sparse kernel, enabling exact midpoint identities without floating-point residual.
Within Recognition Science gravity work this is bookkeeping, not a forcing-chain step (T0–T8), but it is required scaffolding for any claim that the discrete curvature/mass kernel matches the closed form used downstream.
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