e_033210
plain-language theorem explainer
For the six-index slot (0,3,3,2,1,0) on Fin 4, the folded midpoint numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite it when assembling the exact Regge midpoint M2 identity in 4D. The proof is a single kernel decide on that concrete sextuple.
Claim. For indices $(a,b,c,d,i,j)=(0,3,3,2,1,0)$ with each index in $\{0,1,2,3\}$, the folded coupling numerator $N_2(a,b,c,d,i,j)$ equals $8$ times the explicit kernel integer $Z(a,b,c,d,i,j)$.
background
This module is one chunk of the 4D Regge exact-midpoint $M_2$ identity certification: the claim that the folded numerator equals eight times an explicit integer kernel on every sextuple of Fin-4 indices. The module header states the local goal as $m2Num = 8\cdot explicitZ$, chunk 3, discharged by 256 kernel decides.
Upstream, $m2Num$ is defined by folding a coupling list and summing a contribution at each term for fixed indices $(a,b,c,d,i,j)$. The companion $explicitZ$ is a total pattern-matched integer table on the same six Fin-4 arguments (sample clauses give values such as $4$, $-2$, etc.). The present declaration pins one concrete cell of that table equality.
proof idea
One-line decide on the ground sextuple $(0,3,3,2,1,0)$. Both sides reduce to concrete integers: the left via the fold definition of the numerator, the right via the pattern table for the explicit kernel, so the kernel closes the equality with no further lemmas.
why it matters
Feeds the assembly theorem $m2Num_eq_eight_explicitZ$, which states the identity for all Fin-4 sextuples by exhaustive fin_cases and invokes each chunk cell such as this one. That global equality is the algebraic backbone of the Regge exact-midpoint $M_2$ TT identity in 4D gravity analysis inside the monolith. It is bookkeeping infrastructure rather than a T0–T8 forcing step: it certifies a discrete curvature/coupling identity used downstream in the gravity sector, not a derivation of $\phi$, the eight-tick octave, or $D=3$.
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