e_213320
plain-language theorem explainer
For the six-index tuple (2,1,3,3,2,0) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite this as one of the 256 kernel decides that assemble the exact midpoint M2TT identity. The proof is a single decide on concrete integers.
Claim. For indices $a{=}2$, $b{=}1$, $c{=}3$, $d{=}3$, $i{=}2$, $j{=}0$ in $\mathrm{Fin}\,4$, the folded coupling numerator equals eight times the explicit integer kernel: $N(2,1,3,3,2,0)=8\,Z(2,1,3,3,2,0)$.
background
In the Regge exact-midpoint analysis for 4D gravity, two integer-valued kernels on six Fin-4 indices are compared. The numerator $N=m2Num$ is obtained by folding a fixed coupling list and summing local contributions at $(a,b,c,d,i,j)$. The comparison target $Z=explicitZ$ is a sparse closed-form table of small integers (entries such as $4$, $-2$, and defaults) on the same index domain.
This module is chunk 9 of a 256-way case split: each chunk theorem asserts $N=8Z$ at one concrete six-tuple. The local setting is purely combinatorial certification of the midpoint M2TT identity, not a continuum limit argument.
Upstream, $m2Num$ and $explicitZ$ are defined in the kernel certificate module; the present fact only evaluates them at one point.
proof idea
One-line kernel decide. Both sides reduce to concrete Int values once the six Fin-4 arguments are substituted, so decide closes the equality with no lemmas beyond the definitions of $m2Num$ and $explicitZ$.
why it matters
Feeds the assembly theorem $m2Num_eq_eight_explicitZ$, which states $\forall a,b,c,d,i,j,, N=8Z$ by exhaustive fin_cases over Fin 4. That global identity is the algebraic backbone of the Regge exact-midpoint M2TT certificate in the gravity analysis stack.
Within Recognition Science gravity work, such kernel equalities underwrite discrete curvature bookkeeping before continuum or phenomenological layers. This chunk does not itself touch T0–T8, the RCL, or the phi ladder; it is pure index arithmetic supporting the 4D midpoint identity.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.