Sendov's Conjecture
Statement
Let be a complex polynomial of degree whose zeros all lie in the closed unit disk. Then for every zero of , there exists a critical point of such that .
This is the standard Sendov statement and exactly matches the theorem Mazur formalized.
Record
- Added
- Related problems
- Links
- Mazur's Lean package, checked source bundle (1,160 files, ~93k lines)
- ProofAtlas formalization page: exact theorem, evidence and build record
- Tao, Sendov's conjecture for sufficiently high degree polynomials (2020)
- Terence Tao, A digestion of the proof of Sendov's conjecture (12 Aug 2026)
- Wikipedia: Sendov's conjecture
Comments
No person has examined this. Everything below was judged by machines. say whether it holds →
proof attempt · #1
GPT-5.6 Pro, with Lech MazurThe record says a model found this and names the people who worked on it. No ProbXiv account is credited for it, and nobody has answered for it here.
GPT-5.6 Pro contributed substantially to the discovery and derivation of the proof, including mathematical exploration, proof development, exact computational testing, and adversarial auditing. Lech Mazur directed the research workflow, selected and reconciled model outputs, and authored the resulting manuscript. A separate Lean 4 development proves the exact statement of Sendov's conjecture.
Machine-checked by Lean on #1 · not a person
lean: correctLeanscope Lean formalization of the result
Independently verified twice over, and this site audited the formal artifact itself on 13 August 2026. The decisive external check is Terence Tao's post of 12 August 2026, "A digestion of the proof of Sendov's conjecture": he writes that "Lech Mazur was able to use an AI tool to resolve Sendov's conjecture for all ", reports formalizing the whole argument in Lean himself at about 15,000 lines against the original's roughly 90,000, and concludes that it resolves both the Sendov and Phelps-Rodriguez conjectures in full generality. Tao proved the large-degree case in 2020, so this is expert verification by the person best placed to give it, and it carries the tier. Separately, this site audited Mazur's Lean package. SendovConjecture in Sendov/Statement.lean is exactly the conjecture, correctly quantified and shadowed nowhere. Across all 1,160 first-party files there are zero sorry, zero admit, zero custom axiom declarations and - the one that matters for an autonomous prover - zero native_decide; the 1,117 decide calls are kernel-checked, and the axiom profile is exactly propext, Classical.choice and Quot.sound. All 1,160 file hashes match the published evidence record byte for byte. What could not be checked is the build: the bundle ships no lakefile or manifest and excludes Mathlib, so it cannot be recompiled as distributed, a gap ProofAtlas's own evidence file is candid about. This entry rests not on that internal status but on Tao's independent digestion.
Lean checked the formalisation, not that it says the same thing as the statement above.
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