e_112233
plain-language theorem explainer
Pointwise identity: the folded Regge coupling numerator at multi-index (1,1,2,2,3,3) equals eight times the explicit integer table at those same indices. Gravity analysts cite it when assembling the full 4D midpoint M2 TT numerator identity over (Fin 4)^6. Proof is a single kernel decide on two concrete integers.
Claim. For indices $a=b=1$, $c=d=2$, $i=j=3$ in $\mathrm{Fin}\,4$, 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 5 of a 256-case kernel certification that the 4D Regge midpoint M2 TT numerator matches an explicit integer table. The ambient setting is discrete gravity analysis: couplings on oriented 4-simplices with indices in $\mathrm{Fin},4$.
The numerator $N$ (Lean m2Num) is defined by folding a fixed coupling list, accumulating a contribution at each multi-index $(a,b,c,d,i,j)$. The comparison object $Z$ (Lean explicitZ) is a pattern-matched integer table on the same six indices, with sparse nonzero entries such as $4$ or $-2$ on selected diagonal and off-diagonal patterns.
The local claim is one concrete sextuple in that table-versus-fold comparison. Sibling chunks cover the other index patterns; together they feed the exhaustive identity.
proof idea
One-line proof by decide. Both sides reduce to concrete Int values: the left-hand side by evaluating the fold of couplingZList with the contribution function at $(1,1,2,2,3,3)$, the right-hand side by looking up explicitZ at those indices and multiplying by 8. The kernel checks integer equality; no algebraic lemmas are invoked.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j:\mathrm{Fin},4,; N=8Z$ and discharges the universal quantifier by exhaustive fin_cases on all six indices. Each chunk such as this one supplies a decided pointwise fact the assembler (or the kernel path behind the cases) relies on.
In the Recognition gravity stack this closes a numerator identity used in the exact midpoint Regge analysis in 4D, consistent with the forced spatial dimension $D=3$ plus time from the T8 landmark, here appearing as $\mathrm{Fin},4$ index structure. It is bookkeeping, not a new physical law: it certifies that the folded coupling expression collapses to the sparse explicit table, so later curvature or mass-ladder arguments can quote the closed form.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.