e_313030
plain-language theorem explainer
For the six-index slot (3,1,3,0,3,0) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite it as one of the 256 kernel decides that assemble the global identity m2Num = 8·explicitZ. The proof is a single decide on concrete integers.
Claim. For indices $a=3,b=1,c=3,d=0,i=3,j=0$ 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 midpoint analysis, two integer-valued kernels on six $\mathrm{Fin},4$ indices are compared. The numerator $N_2$ (Lean m2Num) is obtained by folding a fixed coupling list and summing a local contribution at each term. The comparison target $Z$ (Lean explicitZ) is a sparse closed-form table: most sextuples map to $0$, with a short list of nonzero patterns such as $(0,0,1,1,2,2)\mapsto 4$ and $(0,0,1,2,1,2)\mapsto -2$.
The module is chunk 13 of a 256-way case split that exhausts all sextuples. The local claim is the instance of $N_2=8Z$ at the concrete point $(3,1,3,0,3,0)$. Upstream, both sides are pure definitions; no analytic lemma is required beyond evaluating those definitions on Fin 4.
proof idea
One-line kernel decide. Both sides reduce to concrete Int values once the six Fin 4 arguments are substituted into the fold for $N_2$ and the pattern match for $Z$. Lean’s decide closes the resulting integer equality; there is no algebraic rewriting or named lemma application beyond definitional evaluation.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, N_2=8Z$ by nested fin_cases over all six indices. Each chunk theorem such as this one discharges one leaf of that case tree. In the broader Gravity/Regge midpoint development, the identity certifies that the folded coupling numerator is exactly eight times the explicit kernel, a bookkeeping step toward exact midpoint identities in the 4D discrete gravity analysis. It is pure finite enumeration, not a forcing-chain (T0–T8) step, but it hardens the algebraic skeleton those continuum claims sit on.
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