e_310103
plain-language theorem explainer
For the Fin-4 index sextuple (3,1,0,1,0,3), the folded midpoint mass-squared numerator equals eight times the explicit kernel integer. Gravity analysts assembling the 4D Regge midpoint M2TT identity cite this as one of 256 pointwise checks. The proof is a single kernel decide on concrete integers.
Claim. For indices $a=3$, $b=1$, $c=0$, $d=1$, $i=0$, $j=3$ in $\mathrm{Fin}\,4$, the folded coupling numerator $N_2(a,b,c,d,i,j)$ equals $8$ times the explicit kernel value $Z(a,b,c,d,i,j)$.
background
In the 4D Regge exact-midpoint gravity analysis, the mass-squared numerator is an integer obtained by folding a fixed coupling list and summing a contribution at six indices in $\mathrm{Fin},4$. The explicit kernel is a closed-form integer table on the same six-index domain; the identity under check is that the folded numerator is exactly eight times that table entry.
This module is chunk 13 of the 256 kernel decides that certify the pointwise equality. Upstream, the numerator is the fold of couplingZList via contrib, and the kernel is the pattern-matched explicitZ table. Downstream assembly exhausts all sextuples by fin_cases and invokes these chunks.
proof idea
One-line proof by decide. Both sides are closed integer terms once the six indices are fixed to (3,1,0,1,0,3); the kernel evaluates the fold and the table entry and accepts the equality. No lemmas beyond the two definitions are required.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states that for every sextuple in $(\mathrm{Fin},4)^6$ the folded numerator equals eight times the explicit kernel. That universal identity is the algebraic core of the Regge exact-midpoint M2TT 4D certificate in the gravity stack. Each chunk such as this one closes one concrete branch of the 256-way case split; without it the assembly cannot finish.
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