e_200210
plain-language theorem explainer
For the Fin-4 multi-index (2,0,0,2,1,0), the folded coupling numerator equals eight times the explicit integer kernel value. Analysts certifying the 4D Regge midpoint M2TT identity cite this cell among the 256 kernel decides. The proof is a single computational decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(2,0,0,2,1,0)$ in $\mathrm{Fin}\,4$, the folded coupling numerator equals eight times the explicit kernel integer: $N(2,0,0,2,1,0)=8\,Z(2,0,0,2,1,0)$.
background
In the 4D Regge midpoint analysis, the M2TT identity is checked pointwise by comparing a folded integer numerator against a closed-form kernel table on six Fin-4 indices. The numerator accumulates contributions by folding a fixed coupling list; the explicit kernel is a sparse case table of small integers (values such as $4$ and $-2$ on selected patterns).
This module is chunk 8 of the 256 kernel decides that discharge the pointwise claim numerator $=8\cdot$ explicit kernel. The local setting is pure integer arithmetic on finite indices; no continuum or metric reconstruction is performed here.
Both the fold definition and the explicit case table are supplied by the KernelCert import. Downstream assembly will quantify over all six indices by exhaustive case split.
proof idea
One-line computational wrapper: decide evaluates both sides at the concrete six-tuple $(2,0,0,2,1,0)$ and checks integer equality. No intermediate lemmas are applied beyond unfolding the fold definition of the numerator and the case table of the explicit kernel.
why it matters
This cell is one of the 256 pointwise facts consumed by the assembly theorem that states the equality for every six-tuple in $\mathrm{Fin},4$, proved there by nested fin_cases. That assembly is the algebraic backbone of the Regge exact midpoint M2TT identity certificate in 4D gravity analysis.
Within the Recognition gravity stack, such kernel identities pin discrete curvature bookkeeping before any continuum or phenomenological claim. The decide closes this cell completely; no sorry or hypothesis interface remains on this index.
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