e_111111
plain-language theorem explainer
At multi-index (1,1,1,1,1,1) the folded coupling numerator equals eight times the explicit integer kernel Z. Gravity analysts checking the 4D Regge exact-midpoint M₂ TT identity cite this as one atomic Fin-4 case. Proof is a single kernel decide on the integer equality.
Claim. For indices $a=b=c=d=i=j=1$ in $\mathrm{Fin}\,4$, the folded coupling numerator satisfies $m_2^{\mathrm{num}}(1,1,1,1,1,1)=8\,Z_{\mathrm{expl}}(1,1,1,1,1,1)$.
background
In the 4D Regge exact-midpoint analysis, the M₂ TT numerator identity is reduced to an integer comparison on six-tuples drawn from Fin 4. The left-hand side folds a fixed coupling list: each term contributes an integer at the chosen indices, and the fold sums them. The right-hand side is an explicit piecewise integer table on the same domain.
This module is chunk 5 of that comparison (256 kernel decides). Local goal: verify numerator = 8 · explicit table on the chunk, case by case. Upstream, the table and the fold are pure definitions; no analytic lemmas are required before the decide steps.
proof idea
One-line computational discharge: by decide. Both sides are closed integer terms at the constant indices (1,1,1,1,1,1). The kernel evaluates the fold that defines the numerator and the pattern match that defines the explicit table, then confirms equality. No named algebraic lemmas are applied.
why it matters
Supplies one atomic case to the universal assembly theorem that states the numerator equals eight times the explicit kernel for every Fin-4 six-tuple. That assembly runs by exhaustive fin_cases and is the certified numerator half of the Regge exact-midpoint M₂ TT identity in four dimensions. In the Recognition gravity stack this pins discrete curvature bookkeeping that must later match continuum limits tied to forced spatial dimension D = 3 and the eight-tick octave at the continuum interface. Closes one decide cell inside chunk 5.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.