e_232012
plain-language theorem explainer
Pointwise identity: the folded Regge coupling numerator at index sextuple (2,3,2,0,1,2) equals eight times the explicit integer table at those indices. Gravity analysts cite it as one kernel cell in the 4D midpoint M2 TT identity. The proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(2,3,2,0,1,2)$ in $(\mathrm{Fin}\,4)^6$, the folded coupling numerator equals eight times the explicit table value: $N_2(2,3,2,0,1,2)=8\,Z(2,3,2,0,1,2)$.
background
In the 4D Regge midpoint analysis, the numerator $N_2$ (Lean m2Num) is the fold of a fixed coupling list: sum the local contribution of each coupling term at a sextuple of $\mathrm{Fin},4$ indices. The companion table $Z$ (Lean explicitZ) is a sparse closed-form integer function on the same domain, with nonzero values only on a short list of index patterns (e.g. $4$ or $-2$ on selected pairs).
This module is chunk 11 of the kernel that discharges $N_2=8Z$ by exhaustive evaluation. The ambient claim is the exact midpoint M2 TT identity in four dimensions; the kernel cert module supplies both the fold definition and the explicit table used here.
proof idea
One-line computational proof: decide evaluates both sides at the concrete sextuple $(2,3,2,0,1,2)$ and checks integer equality. No lemmas are invoked beyond the reducible definitions of the fold and the table.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, N_2(a,b,c,d,i,j)=8Z(a,b,c,d,i,j)$ and proves it by six nested fin_cases over $\mathrm{Fin},4$. Each chunk theorem such as this one is a single leaf of that case tree (256 kernel decides in this family).
In the Recognition gravity stack, the identity certifies that the midpoint Regge numerator collapses to a sparse explicit integer kernel, a prerequisite for closed-form TT-sector identities in 4D. It is pure algebraic bookkeeping, not a dynamical law, but without the pointwise cells the assembly theorem cannot close.
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