Improved integrality of Donaldson–Thomas invariants of loop quivers (GKS Conjecture 1.3 for twist knots)

For the $m$-loop quiver, the numerical Donaldson–Thomas invariants $\mathrm{DT}^{(m)}_n$ (Kontsevich–Soibelman/Reineke) satisfy $v_p(\mathrm{DT}^{(m)}_n)\ge v_p(n)$ for every prime $p\ge5$, with exact defects at $p=2,3$: $v_3\ge v_3(n)-[m\equiv2\ (3)]$ and $v_2\ge v_2(n)-[m\equiv2,3\ (4)]$, and these bounds are attained. Hence the optimal integer with $n\mid\gamma(m)\mathrm{DT}^{(m)}_n$ for all $n$ is exactly $\gamma(m)=2^{\varepsilon_2(m)}3^{\varepsilon_3(m)}$. Via the identification of twist-knot extremal BPS invariants with loop-quiver DT invariants, this proves the Improved Integrality Conjecture (Garoufalidis–Kucharski–Sułkowski 2015, Conj. 1.3, an observation they credit to Kontsevich) for all twist knots with optimal constants, reproducing all twelve $\gamma^\pm$ values GKS tabulated empirically. Supporting new results: a derivative theorem for Gaussian binomials at roots of unity, the first $q$-supercongruence for DT invariants ($\Phi_p(q)^2\mid R_n$), an exact necklace formula for the quantized invariants, and a self-contained proof of the signed $p=2$ Kazandzidis supercongruence.

The sharp valuation bounds and optimal $\gamma(m)$ are proved for ALL loop quivers $m\ge2$, hence for the extremal BPS invariants of all twist knots (both rows, matching every twist-knot entry of GKS Table 1). Scope limits: the $m=3$/figure-eight divisibility $2n_r/r\in\mathbb Z$ was previously proved by Basor–Conrey–Morrison (arXiv:1703.00990), whose per-$r$ $2$-adic characterization for $m=3$ is finer than the uniform bound; the torus-knot case of GKS Conj. 1.3 (multi-vertex quivers, $\gamma$ growing with the knot) remains open and is not claimed; the general-knot conjecture remains open. The Lean formalization covers the reduction to the classical Kazandzidis congruences, not those congruences themselves.