e_110332
plain-language theorem explainer
For the six-index tuple (1,1,0,3,3,2) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite this as one cell of the 4D Regge midpoint M2 TT identity. The proof is a single kernel decide on concrete integers.
Claim. For indices $a=1,b=1,c=0,d=3,i=3,j=2$ in $\mathrm{Fin}\,4$, 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 5 of a 256-cell kernel certification that the Regge midpoint mass-squared numerator in 4D equals eight times a sparse explicit integer table. Indices run over $\mathrm{Fin},4$, matching the four spacetime directions of the discrete TT sector.
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at 0 and add each local contribution contrib at the six indices. The comparison target explicitZ is a pattern-matched integer table (typical nonzero entries are $\pm 2,\pm 4$) that packages the closed-form kernel after combinatorial reduction.
The local claim is one concrete six-tuple evaluation inside that table identity, not the universal quantifier.
proof idea
One-line decide. Both sides reduce to concrete Int values: the left by evaluating the fold of contrib over couplingZList at $(1,1,0,3,3,2)$, the right by looking up explicitZ at the same indices and multiplying by 8. No lemmas are invoked beyond kernel computation.
why it matters
Feeds the assembler m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, m_2^{\mathrm{num}}=8,Z$ by exhaustive fin_cases on all six $\mathrm{Fin},4$ arguments. Each chunk theorem such as this one discharges one branch of that case split.
In the gravity analysis stack this identity is the algebraic core of the exact midpoint Regge M2 TT relation in 4D: once every cell matches, the folded numerator may be replaced by the sparse explicit kernel everywhere downstream. It is bookkeeping for the discrete curvature/mass-squared sector, not a forcing-chain (T0–T8) step, but it locks a computational lemma the continuum-limit gravity arguments rely on.
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