e_211133
plain-language theorem explainer
Pointwise identity: the midpoint mass-squared numerator at multi-index (2,1,1,1,3,3) equals eight times the explicit integer kernel value there. Gravity analysts cite it when assembling the full 4D Regge midpoint M2TT numerator identity. The proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(2,1,1,1,3,3)$ in $\{0,1,2,3\}^6$, the folded coupling 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
This module is chunk 9 of a 256-case kernel certification that the Regge exact-midpoint mass-squared numerator in 4D equals eight times an explicit integer table. Indices run over $\mathrm{Fin},4$, i.e. coordinate labels ${0,1,2,3}$.
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at $0$ and add a local contribution at each coupling triple for the six indices. The comparison target $\mathrm{explicitZ}$ is a closed integer table on the same six indices (sample entries include $4$ on diagonal-type pairs such as $(0,0,1,1,2,2)$ and $-2$ on mixed pairs such as $(0,0,1,2,1,2)$).
The local claim is one concrete cell of that table equality, here at $(2,1,1,1,3,3)$.
proof idea
Both sides are closed integer expressions once the six $\mathrm{Fin},4$ arguments are fixed. The proof is the single tactic decide, which evaluates the fold defining the numerator and the matching clause of the explicit kernel and checks integer equality. No lemmas beyond the two definitions are required.
why it matters
The parent assembly theorem m2Num_eq_eight_explicitZ states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8,Z$ and discharges the universal quantifier by six nested fin_cases sweeps. Each sweep leaf is one of these chunk theorems; this declaration is the leaf for $(2,1,1,1,3,3)$.
In the Recognition gravity stack, that global identity is the certified algebraic core of the 4D Regge exact-midpoint M2TT relation used downstream in curvature and mass-squared bookkeeping. It does not itself invoke the forcing chain (T5–T8) or the Recognition Composition Law; it is pure discrete kernel arithmetic supporting the gravity analysis layer.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.