e_310010
plain-language theorem explainer
Pointwise identity: the folded coupling numerator at multi-index (3,1,0,0,1,0) equals eight times the explicit integer table entry. Gravity analysts cite it as one of 256 kernel cells in the 4D Regge midpoint M2 TT numerator check. Proof is a single kernel decide on two concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(3,1,0,0,1,0)$ in $(\mathrm{Fin}\,4)^6$, the summed coupling numerator equals eight times the closed-form table value: $N(3,1,0,0,1,0)=8\,Z(3,1,0,0,1,0)$.
background
In the 4D Regge exact-midpoint analysis, the numerator of the M2 TT identity is assembled by folding a fixed coupling list: $N(a,b,c,d,i,j)$ sums integer contributions over that list. Beside it sits an explicit integer table $Z$ on the same six $\mathrm{Fin},4$ indices, given by a finite pattern-match (e.g. selected pairs evaluate to $4$, $-2$, and so on).
The module is chunk 13 of a 256-cell kernel: each cell asserts $N=8Z$ at one concrete multi-index. The factor eight is the global normalization relating the folded sum to the closed form. Upstream, $N$ is the fold of contrib over couplingZList; $Z$ is the pattern-matched table used as the target closed form.
proof idea
One-line kernel proof: decide evaluates both sides at the fixed indices $(3,1,0,0,1,0)$ and checks integer equality. No lemmas are invoked; both $N$ and $Z$ reduce to concrete Int values on this cell.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, N=8Z$ by exhaustive fin_cases over $(\mathrm{Fin},4)^6$ and discharge of each cell. That global identity is the numerator half of the 4D Regge exact-midpoint M2 TT certificate in the Gravity analysis stack. Without the pointwise cells, the assembly cannot close. Landmark link is structural (exact discrete gravity identity), not a T0–T8 forcing step.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.