e_033000
plain-language theorem explainer
For the single multi-index (0,3,3,0,0,0), the folded numerator coupling m2Num equals eight times the tabulated kernel value explicitZ. Gravity analysts cite it as one atomic case in the chunked verification that the 4D Regge midpoint M2 TT numerator matches 8Z everywhere. The proof is a pure kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(0,3,3,0,0,0)$ in $\mathrm{Fin}\,4$, the folded numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel $Z(a,b,c,d,i,j)$.
background
In the 4D Regge midpoint analysis, two integer-valued maps on six $\mathrm{Fin},4$ indices are compared. The numerator $m_2^{\mathrm{num}}$ is the fold of a fixed coupling list: each term contributes via a local contrib and the accumulator starts at 0. The comparison target explicitZ is a sparse pattern of small integers (entries such as $4$, $-2$, and defaults) encoding the closed-form kernel.
The module is chunk 3 of a 256-way split of the full $4^6$ table. The local claim is the pointwise identity $m_2^{\mathrm{num}}=8\cdot Z$ at one multi-index, so that a later assembler can recombine chunks without re-running the whole decide tree at once.
proof idea
One-line computational proof: decide evaluates both sides at the concrete six-tuple $(0,3,3,0,0,0)$. The left side reduces by unfolding the fold over couplingZList; the right side multiplies the looked-up explicitZ entry by 8. Equality of the resulting integers is discharged by the kernel.
why it matters
Feeds the universal statement m2Num_eq_eight_explicitZ, which asserts $\forall a,b,c,d,i,j,, m_2^{\mathrm{num}}=8,Z$ by exhaustive fin_cases on all six indices. That identity is the algebraic core of the Regge exact midpoint M2 TT certificate in 4D gravity: it converts a folded coupling sum into an explicit sparse kernel, enabling exact (not approximate) checks of the midpoint curvature identities. Within Recognition Science gravity analysis this is bookkeeping infrastructure, not a forcing-chain step, but it closes a concrete computational gap in the 4D Regge sector.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.