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

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.

Exact FrontierDelta

proved→proved

Scope and record

Occurred: Oct 29, 2026

Delta type: REGISTRY REVISION

Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Improved integrality of Donaldson–Thomas invariants of loop quivers (GKS Conjecture 1.3 for twist knots) · DT improved integrality

Confidence: Not scored

Registry verification: unreviewed · announcement · partial

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Attribution

VibeMathed
registry · event recorded by

Claude Fable 5
model · ai model contributor · Anthropic

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Named evidence

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

specialist registry · passed

No independent human review, and the repository says so itself. Evidence in the repo: two structurally independent implementations (plethystic CoHA engine vs. direct cyclic-word enumeration) agreeing on all computed invariants; Theorem 1 checked numerically for $m\le10$ , $n\le120$ and the Kazandzidis inputs to $n=300$ with sharpness; and a two-round adversarial AI referee report, which is author-side and does not count as independent verification here. The Lean was read here on 24 August 2026 rather than taken on trust. Dtformal.lean carries no sorry, no admit, no native_decide and no axiom declarations whatsoever, on Lean 4.33.0. The classical Kazandzidis congruences enter as explicit hypotheses (`KazOdd`, `Kaz2`) rather than as axioms, and those definitions are faithful to the real congruences - `KazOdd` is $p^{\kappa+v_p(NK(N-K)\binom{N}{K})}\mid\binom{pN}{pK}-\binom{N}{K}$ , and `Kaz2` carries the $(-1)^{K(N-K)}$ sign the $p=2$ case needs - so the conditional theorems are substantive rather than vacuous. `sharp_two`, `sharp_three` and `orbit_sum_zero` take no such hypothesis, matching the claim that sharpness and the combinatorial core are unconditional. Lean was not compiled here, and the output of AxiomCheck.lean is not committed to the repository.

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

witness · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

Source ↗

Garoufalidis, Kucharski, Sułkowski - Knots, BPS states, and algebraic curves (Conjecture 1.3)

specialist registry · pending

problem-record link supplied by the VibeMathed registry.

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Basor, Conrey, Morrison - Knots and ones (the earlier m=3 / figure-eight case)

independent reproduction · pending

independent link supplied by the VibeMathed registry.

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Lineage and corrections

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

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

Basor, Conrey, Morrison - Knots and ones (the earlier m=3 / figure-eight case) evidence for this event

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

VibeMathed record: Improved integrality of Donaldson–Thomas invariants of loop quivers (GKS Conjecture 1.3 for twist knots) evidence for this event

Garoufalidis, Kucharski, Sułkowski - Knots, BPS states, and algebraic curves (Conjecture 1.3) evidence for this event

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

This event attributed to Claude Fable 5

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