e_313133
plain-language theorem explainer
At the six Fin-4 indices (3,1,3,1,3,3), the folded coupling numerator equals eight times the explicit integer kernel. Gravity analysts certifying the 4D Regge exact-midpoint M2TT identity cite this as one concrete cell of the pointwise kernel check. The proof is a single kernel decide on a closed integer equality.
Claim. For indices $(a,b,c,d,i,j)=(3,1,3,1,3,3)$ in $\mathrm{Fin}\,4$, the coupling-list fold numerator equals eight times the explicit six-index integer kernel: $N_2(3,1,3,1,3,3)=8\,Z(3,1,3,1,3,3)$.
background
In the 4D Regge exact-midpoint analysis, the M2TT identity is reduced to a pointwise integer comparison. The numerator $N_2$ (Lean: m2Num) is the fold of contribution terms over a fixed coupling list, evaluated at six indices in $\mathrm{Fin},4$. The explicit kernel $Z$ (Lean: explicitZ) is a pattern-matched $\mathrm{Int}$-valued table on the same six indices, with sample entries such as $Z(0,0,1,1,2,2)=4$ and $Z(0,0,1,2,1,2)=-2$.
This module is chunk 13 of the kernel-decide suite that establishes $N_2=8Z$ cell by cell. The local setting is purely combinatorial: no continuum limit, only finite-index integer arithmetic on the 4D midpoint stencil.
proof idea
One-line wrapper discharged by decide. Substituting the six concrete $\mathrm{Fin},4$ values into the fold definition of the numerator and the pattern table of the explicit kernel yields two closed integer expressions; the kernel reduces their equality automatically. No lemmas beyond the two definitions are invoked.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which asserts $N_2=8Z$ for every sextuple in $(\mathrm{Fin},4)^6$ and is proved by nested fin_cases over all six indices. That global identity is the algebraic backbone of the Regge exact-midpoint M2TT certificate in the 4D gravity analysis stack. Each chunk equality such as this one closes one concrete cell so the assembly can finish without residual goals. Within Recognition Science this sits in the gravity-analysis layer that supports discrete curvature bookkeeping consistent with the forced $D=3$ spatial setting (T8) once the time octave is fixed.
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