e_031122
plain-language theorem explainer
Pointwise identity: the folded Regge midpoint coupling numerator at index sextuple (0,3,1,1,2,2) equals eight times the explicit integer table entry. Gravity analysts cite it as one of 256 kernel cells feeding the global m2Num = 8·explicitZ assembly. Proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(0,3,1,1,2,2)$ in $(\mathrm{Fin}\,4)^6$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit closed-form integer $Z(a,b,c,d,i,j)$.
background
In the 4D Regge exact-midpoint analysis, the numerator coupling $m_2^{\mathrm{num}}$ is defined by folding a fixed list of local contributions over a sextuple of $\mathrm{Fin},4$ indices (edge/vertex labels on the simplex). The companion table $Z$ is an explicit integer-valued pattern match on the same sextuple, recording the closed form that the fold is claimed to match.
This module is chunk 3 of a 256-cell kernel certification: each cell fixes one concrete sextuple and asserts $m_2^{\mathrm{num}}=8Z$ there. The factor 8 is the combinatorial normalization tying the summed contributions to the tabulated values. Upstream, $m_2^{\mathrm{num}}$ is the fold of contrib over couplingZList; $Z$ is the case table (e.g. $(0,0,1,1,2,2)\mapsto 4$, off-diagonal pairs $\mapsto -2$).
proof idea
One-line kernel proof: decide. Both sides reduce to concrete Int values (left via the fold definition of the numerator, right via the pattern match for $Z$ at $(0,3,1,1,2,2)$), so the equality is a decidable integer comparison with no further lemmas.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall(a,b,c,d,i,j),, m_2^{\mathrm{num}}=8Z$ by exhaustive fin_cases on all six $\mathrm{Fin},4$ arguments. Each chunk cell such as this one discharges one branch of that case tree. The identity is infrastructure for the Regge midpoint $M_2$ TT analysis in the gravity sector: it certifies that the folded coupling numerator agrees with the explicit closed form used downstream in curvature and mass-squared identities. It does not itself invoke the T0–T8 forcing chain; it is a pure combinatorial certificate inside the 4D discrete gravity layer.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.