e_032122
plain-language theorem explainer
For the six-index slot (0,3,2,1,2,2) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite it when assembling the exact midpoint M2 identity in 4D Regge calculus. The proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(0,3,2,1,2,2)$ in $\mathrm{Fin}\,4$, the folded coupling numerator equals eight times the explicit kernel integer: $N(0,3,2,1,2,2)=8\,Z(0,3,2,1,2,2)$.
background
This module is one chunk of the 4D Regge exact-midpoint M2 identity certification. The local goal, stated in the module header, is to prove $N=8Z$ on a block of the $4^6$ index space by kernel decides (256 cases per chunk).
The numerator $N=\mathrm{m2Num}$ is the fold of a fixed coupling list: it sums a contribution function over that list at six $\mathrm{Fin},4$ indices. The closed form $Z=\mathrm{explicitZ}$ is an integer-valued pattern match on those same six indices (sample values include $4$, $-2$, and so on).
The parent assembly theorem will quantify over all six indices and discharge each concrete slot by a named equality of this form.
proof idea
One-line proof by decide. Both sides reduce to concrete integers once the six indices are fixed: the left-hand side evaluates the fold that defines the numerator, the right-hand side multiplies the pattern-matched kernel value by eight. No lemmas beyond the two definitions are invoked.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, N=8Z$ and proves it by exhaustive fin_cases on each coordinate, routing each cell to a chunk equality such as this one.
In the gravity analysis stack this identity is the algebraic core of the exact midpoint M2/TT certification in 4D Regge calculus: once $N=8Z$ is global, midpoint curvature numerators match the explicit kernel everywhere on the discrete index torus. It is bookkeeping rather than a new physical law, but without the chunk equalities the assembly cannot close.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.