e_032020
plain-language theorem explainer
Pointwise identity: the folded Regge coupling numerator at multi-index (0,3,2,0,2,0) equals eight times the explicit integer table at those same indices. Gravity analysts cite it as one kernel cell in the 4D midpoint M2TT identity. The proof is a single decidability check on concrete integers.
Claim. For the multi-index $(0,3,2,0,2,0)\in(\mathbb{F}_4)^6$, the folded coupling numerator equals eight times the explicit closed-form integer: $N(0,3,2,0,2,0)=8\,Z(0,3,2,0,2,0)$.
background
In the 4D Regge midpoint analysis, two integer-valued kernels on six indices in $\mathbb{F}_4$ are compared. The numerator $N=m2Num$ is defined by folding a fixed coupling list: start at $0$ and add each contribution term at the given indices. The comparison object $Z=explicitZ$ is an explicit case table of small integers (entries such as $4$, $-2$, and so on) on the same six-index domain.
The local module is chunk 3 of a 256-cell kernel certification that $N=8Z$ holds everywhere. Each cell is an equality at one concrete sextuple; the present cell is indices $(0,3,2,0,2,0)$. Upstream, only the definitions of $N$ and $Z$ are required: no analytic lemma, just the fold and the table.
proof idea
One-line computational proof: decide evaluates both sides at the fixed indices $(0,3,2,0,2,0)$ and checks integer equality. The left side reduces by unfolding the fold over couplingZList; the right side is eight times a single table lookup in explicitZ. No lemmas beyond kernel evaluation are invoked.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j\in\mathbb{F}_4$, $N=8Z$, by exhaustive fin_cases on all six indices. Each chunk theorem such as this one discharges one concrete branch of that case split.
In the gravity stack this identity is bookkeeping for the exact midpoint M2TT kernel in 4D Regge calculus: once $N=8Z$ is certified on the full finite domain, later analytic steps may replace the folded sum by the closed table. It does not itself invoke the RS forcing chain (T5–T8) or the Recognition Composition Law; it is pure discrete kernel algebra supporting the continuum gravity side.
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