e_313213
plain-language theorem explainer
At multi-index (3,1,3,2,1,3) the folded Regge coupling numerator equals eight times the explicit integer kernel value. Gravity analysts proving the exact midpoint M2TT identity in four dimensions cite this as one of 256 kernel point checks. The proof is a single kernel decision on two concrete integers.
Claim. For indices $a=3$, $b=1$, $c=3$, $d=2$, $i=1$, $j=3$ in $\mathrm{Fin}\,4$, the folded coupling numerator equals eight times the explicit table entry: $m_2^{\mathrm{num}}(3,1,3,2,1,3)=8\,Z(3,1,3,2,1,3)$.
background
In the 4D Regge midpoint analysis two integer maps on six indices in $\mathrm{Fin},4$ are compared. The numerator is a fold of a fixed coupling list: each term adds a contribution at the given multi-index. The explicit factor is a closed case table of small integers (typical entries $4$, $-2$, and so on).
This module is chunk 13 of a 256-way partition of that kernel. Each chunk discharges a block of concrete six-tuples for the pointwise identity numerator $=8\cdot$ explicit table. The parent assembly result then universalizes over all of $(\mathrm{Fin},4)^6$ by exhaustive case split on the six indices.
proof idea
With all six $\mathrm{Fin},4$ arguments fixed, both sides reduce to concrete integers: the numerator via the fold over the coupling list, the right-hand side via the explicit case table. The proof is a one-line decide, which asks the kernel to check that integer equality. No algebraic rewriting or intermediate lemmas are needed beyond the two upstream definitions.
why it matters
This point check is consumed by the assembly theorem asserting the same identity for every six-tuple in $(\mathrm{Fin},4)^6$. That universal form is the algebraic backbone of the exact midpoint M2TT identity in the gravity analysis stack. Within Recognition Science the check sits in the gravity phenomenology layer: it underwrites discrete exactness claims for the Regge sector rather than a step of the T0–T8 forcing chain. Closing all 256 chunks discharges the numerator-versus-table comparison that the assembly theorem packages.
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