e_111110
plain-language theorem explainer
At the single index sextuple (1,1,1,1,1,0) the folded Regge midpoint numerator equals eight times the closed-form integer table. Gravity analysts cite it as one of the 256 kernel point checks that assemble the global identity. The proof is a pure `decide` on concrete Fin-4 integers.
Claim. For indices $a=b=c=d=i=1$ and $j=0$ in $\mathrm{Fin}\,4$, the folded coupling numerator equals eight times the explicit integer table: $N(1,1,1,1,1,0)=8\,Z(1,1,1,1,1,0)$.
background
This module is chunk 5 of a 256-way kernel split that certifies the 4D Regge exact-midpoint identity $N=8Z$ pointwise on $(\mathrm{Fin},4)^6$.
The numerator $N=\mathrm{m2Num}$ is defined by folding a fixed coupling list: start at $0$ and add each contribution $\mathrm{contrib}(t;a,b,c,d,i,j)$. The closed form $Z=\mathrm{explicitZ}$ is an explicit integer-valued pattern match on the six $\mathrm{Fin},4$ arguments (typical nonzero values $\pm 2,\pm 4$).
Both definitions live in the kernel certificate module imported here. The present theorem fixes one concrete sextuple and checks the scalar equality at that point.
proof idea
One-line computational proof: by decide. After the six indices are substituted, both sides reduce to concrete integers (the fold of contrib versus eight times the matched explicitZ clause), and Lean’s decision procedure discharges the equality.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, N=8Z$ by exhaustive fin_cases on all six arguments and invokes one kernel lemma per cell. Without the full 256-chunk cover the universal identity does not close.
In the Recognition gravity stack this identity is the algebraic backbone of the exact midpoint Regge analysis in 4D (spatial dimension $D=3$ plus time, consistent with the T8 forcing landmark). It is pure integer bookkeeping, not a dynamical claim about curvature or mass ladders.
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