e_221110
plain-language theorem explainer
For the six-index cell (2,2,1,1,1,0) on Fin 4, the folded coupling numerator equals eight times the tabulated explicit kernel entry. Gravity analysts assembling the exact 4D Regge midpoint M2–TT identity cite this as one decided cell in a chunked kernel exhaust. The proof is a single decide on two closed integer expressions.
Claim. Let $N$ be the fold of coupling contributions over the coupling list and $Z$ the explicit six-index integer kernel on $(\mathrm{Fin}\,4)^6$. Then $N(2,2,1,1,1,0)=8\,Z(2,2,1,1,1,0)$.
background
In the 4D Regge exact-midpoint M2–TT analysis, two integer kernels on six indices in $\mathrm{Fin},4$ are compared. The explicit kernel $Z$ is a pattern table: selected index sextuples map to fixed integers such as $4$ or $-2$, with the remaining patterns implicit. The numerator $N$ is defined by folding a coupling list, accumulating a contribution at each tuple for the given indices.
This module is chunk 10 of a kernel-decide sweep whose local claim is $N=8Z$ on each concrete cell. Upstream, $N$ and $Z$ are pure definitions in the kernel-cert module; no analytic hypothesis is attached to either side at a fixed index.
proof idea
One-line wrapper: decide. After unfolding the fold that defines the numerator and the match table that defines the explicit kernel at the concrete indices $(2,2,1,1,1,0)$, both sides reduce to closed Int values. The decision procedure checks literal equality; no lemma rewrite or induction is required.
why it matters
Parent theorem is the universal assembly m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, N(a,b,c,d,i,j)=8Z(a,b,c,d,i,j)$ and discharges the quantifiers by fin_cases on all six indices. Each cell theorem such as this one supplies the goal that remains after those case splits land on a fixed sextuple.
In the Recognition gravity stack this identity is bookkeeping for the exact midpoint M2–TT kernel in 4D Regge calculus: it converts a folded coupling sum into an eightfold multiple of a sparse explicit table, so later curvature or deficit identities can quote a closed form rather than a fold. It does not itself touch the T0–T8 forcing chain, $\phi$, or the eight-tick octave; those enter only if the assembled kernel is fed into a continuum or recognition-limit argument elsewhere.
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