{ "schema_version": "1.0.0", "source": "vibemath", "problem_id": "vibemathed:smale-s-mean-value-conjecture-k-1", "title": "Smale’s Mean Value Conjecture ($K=1$)", "canonical_statement": "Smale conjectured that in his mean value theorem for complex polynomials, the universal constant $4$ could be replaced by $1$. Equivalently, for every complex polynomial $p$ of degree at least $2$ and every $z\\in\\mathbb C$, there should exist a critical point $c$ of $p$ such that\n$$\n\\frac{|p(z)-p(c)|}{|z-c|}\\le |p'(z)|.\n$$\nThe conjecture is false: there exists a polynomial $p$ with $p(0)=0$ and $p'(0)=1$ such that\n$$\n\\left|\\frac{p(c)}{c}\\right|>1\n$$\nfor every critical point $c$.", "plain_summary": "VibeMathed reports this item as candidate. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.", "current_status": "candidate", "solution_events": [ { "id": "vibemathed:smale-s-mean-value-conjecture-k-1:event", "problem_id": "vibemathed:smale-s-mean-value-conjecture-k-1", "type": "disproved", "status": "candidate", "occurred_at": "2026-09-03T00:00:00.000Z", "title": "Github", "summary": "The formal proof constructs a complex polynomial $p$ such that\n$$\np(0)=0,\\qquad p'(0)=1,\n$$\nand for every critical point $c$ of $p$,\n$$\n\\left|\\frac{p(c)}{c}\\right|>1.\n$$\nThus at $z=0$ there is no critical point satisfying\n$$\n\\frac{|p(0)-p(c)|}{|c|}\\le |p'(0)|=1,\n$$\nwhich disproves Smale's conjectured universal constant $K=1$.\n\nThe counterexample has very large unspecified degree and violates the bound only by a small margin, so it is consistent with Smale's original $K=4$ theorem, the known low-degree positive cases, and previous asymptotic improvements toward $1$.", "ai_contribution": "ai_discovered", "attempt_ids": [ "vibemathed:smale-s-mean-value-conjecture-k-1:attempt" ], "method_family_ids": [ "vibemathed:smale-s-mean-value-conjecture-k-1:method" ], "source_assertion_ids": [ "vibemathed:smale-s-mean-value-conjecture-k-1:assertion" ] } ], "attempts": [ { "id": "vibemathed:smale-s-mean-value-conjecture-k-1:attempt", "problem_id": "vibemathed:smale-s-mean-value-conjecture-k-1", "model": "GPT-6 Astra (pre-release)", "provider": "OpenAI", "attempted_at": "2026-09-03T00:00:00.000Z", "human_collaborators": [ "Tom Adamczewski" ], "exposure": "unknown", "prompt_public": null, "outcome": "A pre-release GPT-6 Astra autonomously attempted the Formal Conjectures statement in Epoch AI's LeanOpenProblems evaluation. No human saw or steered the proof search. Astra found a new counterexample construction and wrote the Lean proof. The proof builds a holomorphic model on a nonconvex compact set, approximates it polynomially, and adds a high-power perturbation that forces all critical points into a region where the $K=1$ inequality fails. Claude was later used to write repository documentation from the completed run; it was not the mathematical solver.", "failed_routes": [], "artifacts": [ { "label": "X Announcement", "url": "https://x.com/tmkadamcz/status/2095785941193404457", "kind": "discussion", "license": null }, { "label": "Challenge.lean: the compared statement, copied from Formal Conjectures", "url": "https://github.com/tadamcz/mean-value-problem/blob/main/Challenge.lean", "kind": "lean-statement", "license": null }, { "label": "Solution.lean and the Lean development", "url": "https://github.com/tadamcz/mean-value-problem/blob/main/Solution.lean", "kind": "lean-proof", "license": null }, { "label": "Formal Conjectures source statement (Google DeepMind)", "url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Wikipedia/MeanValueProblem.lean", "kind": "lean-statement", "license": null }, { "label": "Epoch AI LeanOpenProblems, the evaluation the run was part of", "url": "https://github.com/epoch-research/LeanOpenProblems", "kind": "other", "license": null } ], "sources": [ { "label": "Github", "url": "https://github.com/tadamcz/mean-value-problem", "kind": "primary-mathematical-source", "license": null }, { "label": "VibeMathed: Smale’s Mean Value Conjecture ($K=1$)", "url": "https://vibemathed.com/problem/smale-s-mean-value-conjecture-k-1", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" } ], "cost_usd": 265, "independence": "unknown" } ], "verifications": [ { "id": "vibemathed:smale-s-mean-value-conjecture-k-1:verification", "problem_id": "vibemathed:smale-s-mean-value-conjecture-k-1", "solution_event_id": "vibemathed:smale-s-mean-value-conjecture-k-1:event", "level": "lean_verified_statement_audited", "mathematical_correctness": "supported", "statement_fidelity": "audited", "peer_review": "none", "verifier": "VibeMathed (source-reported label)", "verified_at": null, "note": "Lean-verified on this site's ladder: kernel-checked, and the statement was written independently of the prover. Checked here on 5 September 2026 from a clone of the repository at f99ab98: 2,266 lines of Lean across the development, zero `sorry` outside the Challenge.lean stub, zero `axiom` declarations, no `native_decide`, `unsafe` or `implemented_by`; the mathlib revision is pinned; CI runs Comparator against the trusted statement allowing only propext, Quot.sound and Classical.choice. The compared statement is byte-identical to Formal Conjectures' `mean_value_problem` at commit 9cbe1d3c, which is Smale's K = 1 form exactly: the quantified K is unused there, and the division convention in Lean (x/0 = 0) only makes the conjecture easier to satisfy, so it cannot help a disproof. Candidate rather than resolved because, two days after the run, no mathematician outside it has read the proof; the witness is a nonconstructive limiting perturbation of large unspecified degree and violates the bound by a small margin, which is consistent with everything known. The proof account was machine-generated and is unaudited.", "sources": [ { "label": "VibeMathed: Smale’s Mean Value Conjecture ($K=1$)", "url": "https://vibemathed.com/problem/smale-s-mean-value-conjecture-k-1", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" }, { "label": "Github", "url": "https://github.com/tadamcz/mean-value-problem", "kind": "primary-mathematical-source", "license": null } ] } ], "sources": [ { "label": "Github", "url": "https://github.com/tadamcz/mean-value-problem", "kind": "primary-mathematical-source", "license": null }, { "label": "VibeMathed: Smale’s Mean Value Conjecture ($K=1$)", "url": "https://vibemathed.com/problem/smale-s-mean-value-conjecture-k-1", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" } ], "recommended_mode": "verify", "recommended_exposure": "result_only", "opportunity_signals": [ { "id": "vibemathed:smale-s-mean-value-conjecture-k-1:signal:verify_now", "problem_id": "vibemathed:smale-s-mean-value-conjecture-k-1", "kind": "verify_now", "reason": "The source labels this result as a candidate; verification should precede reuse.", "generated_at": "2026-09-13T16:28:40.178Z" } ], "uncertainties": [ "The source status is candidate and must not be represented as solved.", "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" }