{ "schema_version": "1.0.0", "source": "vibemath", "problem_id": "vibemathed:bugeaud-problem-10-61", "title": "$(\\xi\\alpha^n)_{n\\ge1}$ is not uniformly distributed modulo one for Pisot $\\alpha$ and $\\xi$ in the Cantor set $C(\\alpha)$", "canonical_statement": "Bugeaud's Problem 10.61, due to Michel Mendès France in 1967: for a Pisot number $\\alpha > 2$ and the Cantor set $C(\\alpha) = \\{(\\alpha-1)\\sum_{k\\ge1}\\varepsilon_k\\alpha^{-k} : \\varepsilon_k \\in \\{0,1\\}\\}$, no $\\xi \\in C(\\alpha)$ has $(\\xi\\alpha^n)_{n\\ge1}$ uniformly distributed modulo one.\n\nThe problem itself remains open. What is proved is a set of criteria for it, and two instances. The criteria: a reduction to symbolic dynamics that is an equivalence; a pressure criterion; and a covering criterion which, for a quadratic setup of norm $b$, applies exactly when $(\\log_2\\alpha - 1)(\\log_2(\\alpha/|b|) - 1) > 1$, a condition that reduces to $\\alpha > 4$ for units. The two instances are both quadratic: at $\\alpha = 2+\\sqrt5$ in the strong form, an explicit interval that every orbit misses at every time, and at $\\alpha = 2+\\sqrt3$ by a confinement-gap certificate.", "plain_summary": "VibeMathed reports this item as partial. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.", "current_status": "partial", "solution_events": [ { "id": "vibemathed:bugeaud-problem-10-61:event", "problem_id": "vibemathed:bugeaud-problem-10-61", "type": "proved", "status": "partial", "occurred_at": "2026-08-31T00:00:00.000Z", "title": "rwst/Pisot-Cantor-61 (paper.pdf)", "summary": "The problem is open, and the repository says so: what is proved are criteria for it and two instances, not the general case.\n\nEvery compared statement that concludes Problem 10.61 does so for a quadratic setup - a real root $\\alpha > 1$ of $X^2 - aX - b$ whose conjugate has modulus below one - and both instances are quadratic. The arbitrary-degree material is conditional ingredients: for the family $X^d - aX^{d-1} - 1$ the real root exceeding $a$ is shown to be Pisot for $a \\ge 3$, with a conjugate-modulus bound and a numerical inequality. No compared statement carries those above degree two, because the covering criterion is proved only for quadratic setups.\n\nThe covering criterion also leaves quadratic $\\alpha$ with route-A exponent at least one undecided, about which nothing is claimed.", "ai_contribution": "ai_discovered", "attempt_ids": [ "vibemathed:bugeaud-problem-10-61:attempt" ], "method_family_ids": [ "vibemathed:bugeaud-problem-10-61:method" ], "source_assertion_ids": [ "vibemathed:bugeaud-problem-10-61:assertion" ] } ], "attempts": [ { "id": "vibemathed:bugeaud-problem-10-61:attempt", "problem_id": "vibemathed:bugeaud-problem-10-61", "model": "Fable 5, Opus 5", "provider": "Anthropic", "attempted_at": "2026-08-31T00:00:00.000Z", "human_collaborators": [ "Ralf Stephan" ], "exposure": "unknown", "prompt_public": null, "outcome": "Directed by a human, the model planned and executed the discovery, the formalization and the write-up. The repository's formalization.yaml records the work as \"(C) 2026 Ralf Stephan, in collaboration with Claude Code\", and describes paper.pdf there as machine-written notes documenting the Lean development rather than a prior paper the formalization followed.", "failed_routes": [], "artifacts": [ { "label": "Registry link", "url": "https://palomar-registry.org/entry?id=PALOMAR-2026-08-31-000013", "kind": "palomar", "license": null }, { "label": "Github repo", "url": "https://github.com/rwst/Pisot-Cantor-61", "kind": "code", "license": null }, { "label": "Preprint", "url": "https://www.researchgate.net/publication/414009170_Criteria_for_the_non-equidistribution_of_xan_on_the_Cantor_set_Ca", "kind": "paper", "license": null } ], "sources": [ { "label": "rwst/Pisot-Cantor-61 (paper.pdf)", "url": "https://github.com/rwst/Pisot-Cantor-61/blob/main/paper.pdf", "kind": "primary-mathematical-source", "license": null }, { "label": "VibeMathed: $(\\xi\\alpha^n)_{n\\ge1}$ is not uniformly distributed modulo one for Pisot $\\alpha$ and $\\xi$ in the Cantor set $C(\\alpha)$", "url": "https://vibemathed.com/problem/bugeaud-problem-10-61", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" } ], "cost_usd": null, "independence": "unknown" } ], "verifications": [ { "id": "vibemathed:bugeaud-problem-10-61:verification", "problem_id": "vibemathed:bugeaud-problem-10-61", "solution_event_id": "vibemathed:bugeaud-problem-10-61:event", "level": "lean_checked_statement_unaudited", "mathematical_correctness": "supported", "statement_fidelity": "unaudited", "peer_review": "none", "verifier": "VibeMathed (source-reported label)", "verified_at": null, "note": "Lean-checked, statement unaudited. Registered in the Palomar registry as PALOMAR-2026-08-31-000013: status registered, trust high, pinned to commit d61132ff, mirrored to PalomarArchive.\n\nSeventeen statements are compared with leanprover/comparator, which checks three things per theorem - that the statement in Solution is definitionally the same statement as in the trusted Challenge module, compared constant by constant; that the proof uses no axiom outside a permitted list; and that the resulting environment is re-accepted by the Lean kernel. All seventeen registered theorems sit in the axiom-free lane, permitting only propext, Quot.sound and Classical.choice.\n\nThe repository maintains a second lane permitting one cited literature input, LY.entropyRate_floor. Two theorems consume it and neither is among the registered seventeen; the $\\alpha = 2+\\sqrt3$ instance is registered in its axiom-free form.\n\nNot Lean-verified, on the anchoring half. A Palomar listing is a strong precondition rather than the anchoring itself: it makes the audit cheap, but nobody without a stake has checked that the formal statements say what Problem 10.61 says, and Palomar states plainly that a listing is not a certificate of novelty or relevance.", "sources": [ { "label": "VibeMathed: $(\\xi\\alpha^n)_{n\\ge1}$ is not uniformly distributed modulo one for Pisot $\\alpha$ and $\\xi$ in the Cantor set $C(\\alpha)$", "url": "https://vibemathed.com/problem/bugeaud-problem-10-61", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" }, { "label": "rwst/Pisot-Cantor-61 (paper.pdf)", "url": "https://github.com/rwst/Pisot-Cantor-61/blob/main/paper.pdf", "kind": "primary-mathematical-source", "license": null } ] } ], "sources": [ { "label": "rwst/Pisot-Cantor-61 (paper.pdf)", "url": "https://github.com/rwst/Pisot-Cantor-61/blob/main/paper.pdf", "kind": "primary-mathematical-source", "license": null }, { "label": "VibeMathed: $(\\xi\\alpha^n)_{n\\ge1}$ is not uniformly distributed modulo one for Pisot $\\alpha$ and $\\xi$ in the Cantor set $C(\\alpha)$", "url": "https://vibemathed.com/problem/bugeaud-problem-10-61", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" } ], "recommended_mode": "verify", "recommended_exposure": "result_only", "opportunity_signals": [ { "id": "vibemathed:bugeaud-problem-10-61:signal:recent_partial_result", "problem_id": "vibemathed:bugeaud-problem-10-61", "kind": "recent_partial_result", "reason": "A source-reported partial result may support bounded expansion.", "generated_at": "2026-09-13T16:28:40.178Z" } ], "uncertainties": [ "VibeMath has not independently verified the mathematical claim.", "AI-attempt independence and training-data exposure are unknown unless explicitly documented.", "VibeMath has not independently audited the mathematical statement, proof, or novelty claim." ], "generated_at": "2026-09-13T16:28:40.178Z" }