e_203100
plain-language theorem explainer
For the six-index tuple (2,0,3,1,0,0) 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 identity in 4D. The proof is a single kernel decide on concrete integers.
Claim. For indices $a{=}2$, $b{=}0$, $c{=}3$, $d{=}1$, $i{=}0$, $j{=}0$ in $\mathrm{Fin}\,4$, the folded coupling numerator equals eight times the explicit integer kernel: $N(2,0,3,1,0,0)=8\,Z(2,0,3,1,0,0)$.
background
In the 4D Regge midpoint analysis, two integer-valued kernels on six Fin-4 indices are compared. The numerator $N=m2Num$ is defined by folding a fixed coupling list and summing local contributions at each multi-index. The comparison target $Z=explicitZ$ is a sparse closed-form table of small integers (entries such as $4$, $-2$, and so on on selected patterns).
The module is one of several decide-chunks that discharge the pointwise identity $N=8Z$ on a 256-cell partition of the six-index domain. The local setting is purely combinatorial: no continuum limit or curvature hypothesis is invoked; only integer equality of the two kernels at a fixed tuple.
Upstream, $m2Num$ and $explicitZ$ are the sole dependencies. The former accumulates contributions; the latter supplies the explicit right-hand side used throughout the identity certificates.
proof idea
One-line proof by decide. Both sides evaluate to concrete integers once the six Fin-4 arguments are substituted into the fold definition of $m2Num$ and the pattern-match table of $explicitZ$, so the kernel closes the equality without further lemmas.
why it matters
Feeds the assembly theorem $m2Num_eq_eight_explicitZ$, which states $\forall a,b,c,d,i,j:\mathrm{Fin},4,; m2Num=8\cdot explicitZ$ by exhaustive fin_cases over the six indices. Each chunk such as this one supplies one concrete cell so the global forall can be built without a monolithic decide.
In the Recognition gravity stack this identity is bookkeeping for the exact midpoint M2TT kernel in 4D Regge calculus: matching the folded numerator to eight times the explicit table certifies the discrete curvature/coupling algebra used downstream. It does not itself invoke the forcing chain (T0–T8), phi, or the eight-tick octave; it is a pure integer certificate inside the gravity analysis layer.
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