e_213121
plain-language theorem explainer
Pointwise identity: the folded Regge coupling numerator at multi-index (2,1,3,1,2,1) equals eight times the explicit integer table at that same index. Gravity analysts cite it when discharging one Fin-4 cell of the 4D midpoint M2TT numerator identity. The proof is a single kernel decide on concrete integers.
Claim. For indices $a=2,b=1,c=3,d=1,i=2,j=1$ in $\mathrm{Fin}\,4$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer value $Z(a,b,c,d,i,j)$.
background
In the 4D Regge midpoint M2TT analysis, two integer-valued maps on six $\mathrm{Fin},4$ indices are compared. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at $0$ and add a local contribution at each coupling term. The comparison target is an explicit table $Z$ on the same six indices, given by a finite pattern-match with small integer outputs (entries such as $4$, $-2$, and so on).
The ambient module is chunk 9 of a 256-cell kernel certification that $m_2^{\mathrm{num}}=8\cdot Z$ holds at every multi-index. Each chunk theorem pins one concrete cell; the present cell is $(2,1,3,1,2,1)$.
proof idea
Both sides evaluate to concrete integers once the six $\mathrm{Fin},4$ arguments are fixed. The proof is the single tactic decide, which runs the kernel decision procedure on that closed integer equality. No algebraic rewriting or intermediate lemmas are required beyond the definitions of the folded numerator and the explicit table.
why it matters
The parent assembly theorem states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}(a,b,c,d,i,j)=8,Z(a,b,c,d,i,j)$ and discharges the universal claim by exhaustive $\mathrm{Fin},4$ case split. This declaration supplies one of those cells (chunk 9). In the broader gravity stack it is bookkeeping for the exact midpoint M2TT identity in four dimensions: once every cell matches, the folded numerator may be replaced by the closed table factor of eight, simplifying later curvature and coupling identities in the Regge analysis.
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