e_112111
plain-language theorem explainer
For the Fin-4 multi-index (1,1,2,1,1,1), the folded coupling numerator equals eight times the closed-form kernel entry. Gravity analysts certifying the exact midpoint M2 TT identity cite this as one atomic case among the 4^6 kernel. Proof is a single kernel decide on the concrete integer equality.
Claim. At indices $(a,b,c,d,i,j)=(1,1,2,1,1,1)$ with each coordinate in $\{0,1,2,3\}$, the integer obtained by folding the coupling list equals $8$ times the tabulated closed-form kernel value at that same multi-index.
background
The ambient setting is the Regge exact-midpoint analysis of a 4D TT identity. Two integer kernels on $(\mathrm{Fin},4)^6$ are compared pointwise. One side, the numerator, folds a fixed coupling list and accumulates local contributions at each six-index. The other side is an explicitly matched table with sparse nonzero integers (typical values $4$, $-2$, and so on).
This module is chunk 5 of a decide campaign that discharges all $4^6$ cases of the claimed relation numerator $= 8\cdot$ closed-form. The two upstream definitions supply both sides of the comparison: the fold that builds the numerator, and the match table that defines the closed form.
proof idea
One-line wrapper: decide evaluates both integer expressions at the concrete sextuple $(1,1,2,1,1,1)$ after unfolding the fold and the match table, then confirms the equality in the kernel. No named lemmas are applied.
why it matters
This case is consumed by the master assembly theorem stating that for every $a,b,c,d,i,j\in\mathrm{Fin},4$, the folded numerator equals eight times the closed-form entry. That assembly proceeds by exhaustive fin_cases on the six indices and lands each branch on a chunk decide such as this one. The resulting global identity is a computational certificate inside the Regge midpoint M2 TT analysis for 4D gravity, underwriting discrete curvature bookkeeping used in the Recognition Science gravity stack.
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