e_213210
plain-language theorem explainer
Pointwise identity: the folded Regge coupling numerator at index sextuple (2,1,3,2,1,0) equals eight times the explicit integer table entry. Gravity analysts cite it when assembling the full 4D midpoint M2TT numerator identity over all Fin-4 indices. The proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(2,1,3,2,1,0)$ in $\{0,1,2,3\}^6$, the folded coupling numerator $N(a,b,c,d,i,j)$ equals $8$ times the explicit closed-form integer $Z(a,b,c,d,i,j)$.
background
This module is chunk 9 of a 256-way kernel split proving that the 4D Regge midpoint M2TT numerator equals eight times an explicit integer table. The ambient setting is discrete gravity analysis: edge couplings on a 4-simplex skeleton, with all free indices ranging over $\mathrm{Fin},4\simeq{0,1,2,3}$.
The numerator $N$ is defined by folding a fixed coupling list: start at $0$ and add a local contribution for each coupling term at the six indices. The table $Z$ is a pattern-matched integer function on the same six indices (typical values $\pm 2,\pm 4$, and zero off the listed patterns). The claim is the pointwise relation $N=8Z$ at one concrete sextuple.
Upstream, both $N$ and $Z$ live in the kernel certificate module; this chunk only discharges one decide instance.
proof idea
One-line computational proof: decide. Lean reduces both sides at the concrete Fin-4 sextuple $(2,1,3,2,1,0)$ to closed integers (the fold for $N$, the match for $Z$) and checks equality by the kernel decision procedure. No lemmas are invoked beyond the definitions of the numerator fold and the explicit table.
why it matters
Parent theorem m2Num_eq_eight_explicitZ assembles the universal statement $\forall a,b,c,d,i,j,, N=8Z$ by exhaustive fin_cases on all six indices; each case lands on a chunk identity of this form. Without the pointwise decides, the assembly cannot close.
In the Recognition gravity stack this certifies the exact algebraic numerator identity behind the 4D Regge midpoint M2TT analysis, so continuum or continuum-limit arguments can quote a fully reduced integer table rather than an opaque fold. It is bookkeeping inside the gravity analysis layer, not a forcing-chain (T0–T8) step, but it hardens the discrete curvature side that later couples to RS mass and coupling claims.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.