The Lukic Conjecture

Let $\mu$ be a probability measure on the unit circle with Verblunsky coefficients $\alpha$ . Lukic conjectured that a weighted entropy condition with finitely many critical points is equivalent to a decomposition of $\alpha$ into components localized at those points. A counterexample with two critical points of multiplicity three refutes it: the sequence satisfies the decomposition conditions while the corresponding weighted entropy is $-\infty$ .

Result: Disproved
Status: Resolved
AI contribution: AI-discovered
Method: Construction
Field: Spectral theory
Posed by: Milivoje Lukić
Year posed: —
Years open: —
Solved: 2026-07-29
Model: GPT-5.6
Vendor: OpenAI
Collaborators: Jun Yan
Verification: Unreviewed
Publication: Preprint
Significance: 20 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What the AI did

The counterexample - two phase modes with common power-decay exponent $3/20$ - was found by GPT-5.6, as the abstract states directly; the author developed and verified the construction.

Verification

Single-author arXiv preprint with the counterexample stated explicitly; not yet peer-reviewed.

Source

arXiv:2607.26419 - A Mixed-Resonance Counterexample to the Lukic Conjecture

The Lukic Conjecture

What the AI did

Verification

Source

Discussion