e_220111
plain-language theorem explainer
For the six Fin-4 indices (2,2,0,1,1,1), the folded coupling numerator equals eight times the explicit integer table entry. Gravity analysts certifying the Regge-exact midpoint M2 TT identity cite this as one of 256 kernel-decided cells. The proof is a single kernel `decide` on concrete Int equality.
Claim. With six arguments in $\mathrm{Fin}\,4$, the folded coupling numerator at $(2,2,0,1,1,1)$ equals $8$ times the explicit closed-form integer table at the same multi-index: $m_2^{\mathrm{num}}(2,2,0,1,1,1)=8\,Z(2,2,0,1,1,1)$.
background
The Regge-exact midpoint stack compares two integer maps on six $\mathrm{Fin},4$ indices. The numerator folds a fixed coupling list, summing each contribution at the given multi-index. Its companion is a sparse pattern-matched table of small integers (typical nonzero entries $4$, $-2$, and similar).
This module is chunk 10 of an exhaustive 256-cell certification that the numerator equals eight times the table pointwise. Both maps are defined in the kernel-certificate module: the numerator is the fold; the table is the closed form.
The setting is discrete 4D curvature bookkeeping for the M2 TT identity in the Gravity analysis layer.
proof idea
One-line kernel discharge: by decide. With all six arguments concrete $\mathrm{Fin},4$ literals, both sides reduce to closed Int values, so decidable equality finishes the goal. No lemmas are invoked.
why it matters
Supplies one cell to the assembler theorem that states the numerator equals eight times the explicit table for every six-tuple in $\mathrm{Fin},4$. That assembler runs nested fin_cases over all indices and lands on chunk theorems such as this one. The global equality is algebraic scaffolding for the Regge-exact midpoint M2 TT identity in 4D, the discrete curvature identity this Gravity analysis stack is building. The cell itself is pure bookkeeping, not a new physical law; it closes one of the 256 kernel obligations in the chunked certification.
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