e_123301
plain-language theorem explainer
For the concrete Fin-4 index tuple (1,2,3,3,0,1), the folded Regge midpoint numerator equals eight times the tabulated explicit kernel integer. Gravity analysts building the 4D midpoint M2TT identity cite this as one of 256 kernel cases. Proof is a single kernel decide on two closed integer expressions.
Claim. For indices $a=1$, $b=2$, $c=3$, $d=3$, $i=0$, $j=1$ in $\mathrm{Fin}\,4$, the folded coupling numerator equals eight times the explicit kernel entry: $N(1,2,3,3,0,1)=8\,Z(1,2,3,3,0,1)$.
background
In the 4D Regge midpoint analysis, six indices in $\mathrm{Fin},4$ label a discrete coupling slot. The numerator $N(a,b,c,d,i,j)$ is defined by folding a contribution function over a fixed coupling list and summing integer terms. The companion table $Z(a,b,c,d,i,j)$ is an explicit pattern-match on those six indices, returning a small integer (for example $4$, $-2$, and so on on the listed patterns).
This module is chunk 6 of a 256-case kernel split whose sole job is to check $N=8Z$ at every concrete index tuple. The local setting is pure integer arithmetic on closed terms: once the six Fin-4 values are fixed, both sides reduce to numerals.
proof idea
One-line proof by decide. With all six indices concrete, m2Num unfolds to a finite integer fold and explicitZ to a single table entry; the kernel evaluates both sides and closes the equality. No lemmas are invoked beyond the definitions of the two functions.
why it matters
The parent theorem m2Num_eq_eight_explicitZ states the identity for all six Fin-4 arguments and discharges the universal claim by exhaustive fin_cases. Each concrete case such as this one supplies a decided goal in that assembly. Without the full 256-case cover, the exact midpoint M2TT numerator identity in 4D would remain open at the discrete level. The result is bookkeeping inside the Gravity analysis stack, not a new physical law; it certifies that the folded coupling numerator matches the closed-form kernel table used downstream.
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