e_312232
plain-language theorem explainer
For the single multi-index (3,1,2,2,3,2) in (Fin 4)^6, the folded Regge midpoint numerator m2Num equals eight times the explicit integer kernel explicitZ. Gravity analysts cite it as one of the 256 decided cases that assemble the exact 4D TT midpoint identity. The proof is a one-line kernel decide.
Claim. Let $a,b,c,d,i,j\in\{0,1,2,3\}$. Writing $N$ for the folded coupling numerator obtained by summing the list of elementary contributions at those six indices, and $Z$ for the explicit integer kernel table at the same indices, one has $N(3,1,2,2,3,2)=8\,Z(3,1,2,2,3,2)$.
background
In the Regge-exact midpoint analysis for 4D gravity, two integer-valued maps on six indices in $\mathrm{Fin},4$ are compared. The numerator $N=\mathrm{m2Num}$ is defined by folding a fixed coupling list: start from $0$ and add each elementary contribution at the given multi-index. The comparison table $Z=\mathrm{explicitZ}$ is a sparse case-split returning small integers (typically $\pm 2,\pm 4$, or $0$ off the listed patterns).
The local module is chunk 13 of a 256-case kernel: the claim is that $N=8Z$ holds pointwise. Upstream, both $N$ and $Z$ are pure definitions in the kernel certificate module; no analytic hypothesis is imposed beyond the finite discrete domain.
The ambient goal is an exact algebraic identity for the midpoint mass-squared TT sector in four dimensions, reduced to exhaustive evaluation on $(\mathrm{Fin},4)^6$.
proof idea
One-line wrapper: by decide. Lean reduces both sides of the equality on the concrete six-tuple $(3,1,2,2,3,2)$ by unfolding the fold that defines the numerator and the case table that defines the explicit kernel, then closes the resulting integer equality by the decision procedure. No lemmas beyond the two definitions are invoked.
why it matters
This case is one brick in the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,,N=8Z$ and is proved by nested fin_cases over the six indices. That global identity is the algebraic core of the Regge-exact midpoint M2 TT certificate in 4D: it converts the folded coupling expression into an explicit sparse kernel that can be inspected and reused downstream.
Within Recognition Science gravity work, such exact discrete identities underwrite continuum limits and curvature identifications without residual numerical error on the finite index set. The chunking into 256 decides keeps each certificate tiny and machine-checkable.
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