e_313021
plain-language theorem explainer
Pointwise identity: the folded Regge midpoint numerator at multi-index (3,1,3,0,2,1) equals eight times the explicit integer table at those indices. Gravity analysts cite it as one kernel cell in the 4D midpoint mass-term certification. The proof is a single kernel decide on concrete Int values.
Claim. For indices $(a,b,c,d,i,j)=(3,1,3,0,2,1)$ in $(\mathrm{Fin}\,4)^6$, the folded coupling numerator equals eight times the explicit closed-form integer: $m_2^{\mathrm{num}}(3,1,3,0,2,1)=8\,Z_{\mathrm{expl}}(3,1,3,0,2,1)$.
background
This module is one chunk of the 4D Regge exact-midpoint certification that the folded numerator of the mass-term kernel agrees with an explicit integer table, scaled by eight. The local slogan is $m_2^{\mathrm{num}}=8\cdot Z_{\mathrm{expl}}$, discharged by 256 kernel decides per chunk.
The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: start at $0$ and add each contribution $\mathrm{contrib}(t;a,b,c,d,i,j)$. The comparison object $Z_{\mathrm{expl}}$ is a total function $(\mathrm{Fin},4)^6\to\mathbb{Z}$ given by a finite case table (e.g. $(0,0,1,1,2,2)\mapsto 4$, mixed pairs $\mapsto -2$, and so on).
Upstream, both definitions live in the kernel-certificate module; this chunk only evaluates them at one concrete sextuple.
proof idea
One-line computational proof: decide evaluates both sides at the fixed indices $(3,1,3,0,2,1)$ and checks integer equality. No lemmas are invoked beyond the reducible definitions of the folded numerator and the explicit table.
why it matters
Feeds the assembly theorem $m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$ for every $(a,b,c,d,i,j)\in(\mathrm{Fin},4)^6$, proved by exhaustive fin_cases on the six indices. That global identity is the algebraic backbone of the 4D Regge exact-midpoint mass-term certificate in the Gravity analysis stack.
In the Recognition framework this sits on the gravity side of the discrete geometric calculus (eight-tick / $D=3$ spatial structure upstream in the forcing chain), certifying that the midpoint discretization of the quadratic form matches its closed integer kernel rather than a floating approximation. Closing all chunks removes scaffolding from the Regge midpoint identity path.
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