Problem detail · source-aware
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Gill introduced the probabilistic automatic complexity $A_P(w)$ of a string: the least number of states of a probabilistic finite automaton for which $w$ is the unique most probably accepted string of its length. He asked whether $A_P$ is unbounded, no string with $A_P>3$ being known. The paper proves $A_P(w)\le 3$ for every string over every finite alphabet, with an explicit three-state witness.
The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement fidelity, correctness, priority, or novelty.
The author states the construction was found in conversation with, and verified with the assistance of, Claude Fable 5, and that the proof was formalized in Lean with assistance from Harmonic's Aristotle.
Provider: Anthropic · Prompt public: unknown · Independence: unknown
No independent review. A Lean formalization is reported but no artifact was located to check, so this does not carry the Lean-verified tier. Preprint, not refereed.
Correctness: unknown · statement fidelity: unaudited · peer review: none
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
Source-reported tools: construction.
Independent: unknown · difference confidence: 0
VibeMath has not independently audited the mathematical statement, proof, or novelty claim.