{"schema":"postcutoff/event@1","as_of":"2026-10-12T22:30:00+02:00","url":"https://postcutoff.com/e/2026-10-08-erdos-522-littlewood-polynomial-zeros-lean-kawada/","md":"https://postcutoff.com/e/2026-10-08-erdos-522-littlewood-polynomial-zeros-lean-kawada/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"Adam Bicz","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-10-08-erdos-522-littlewood-polynomial-zeros-lean-kawada","date":"2026-10-08","date_precision":"day","short_title":"Erdős Problem #522 (1961) resolved with a Lean-verified proof","deck":"Half the zeros of random ±1 polynomials lie in the unit disk, almost surely (Kawada, MIT CSAIL)","takeaway":"A 1961 question of Erdős about the zeros of polynomials with random ±1 coefficients gets a formally verified answer, accepted against the statement in Google DeepMind's Formal Conjectures. The author says he originated the mathematics; unnamed in-house models from MIT's Kellis Lab helped with the Lean formalization and with checking derivations.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2610.11324: Universal Radial Laws for Random Polynomials: Erdős Problem #522 and Non-Universal Fluctuations","url":"https://arxiv.org/abs/2610.11324","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"Lean formalization and paper repository (chreia/erdos-522)","url":"https://github.com/chreia/erdos-522","type":"code","group":"primary","domain":"github.com"},{"n":3,"title":"Formal Conjectures: Erdos522.erdos_522","url":"https://google-deepmind.github.io/formal-conjectures/theorem/?name=Erdos522.erdos_522","type":"docs","group":"primary","domain":"google-deepmind.github.io"},{"n":4,"title":"Erdős Problem #522 (erdosproblems.com)","url":"https://www.erdosproblems.com/522","type":"docs","group":"primary","domain":"erdosproblems.com"}],"official":4,"filed":"2026-10-12","updated":"2026-10-12","orgs":["MIT CSAIL","Google DeepMind"],"title":"Erdős Problem #522 (1961) resolved with a Lean-verified proof: half the zeros of random ±1 polynomials lie in the unit disk, almost surely (Kawada, MIT CSAIL)","summary":"On 8 Oct 2026 Sebastien Kawada (MIT CSAIL) posted arXiv 2610.11324, which proves that for random Littlewood polynomials (all coefficients ±1, chosen by fair coin flips) the proportion of zeros in the closed unit disk converges to 1/2 almost surely. Erdős asked this in 1961 (Problem #522 on erdosproblems.com); Yakir had proved convergence in probability. The paper goes further: a universal radial law at scale 1/n and non-universal fluctuations. All main results are verified in Lean 4, and the proof \"has been accepted as a solution to its formal statement in Google DeepMind's Formal Conjectures project\". AI disclosure: the Lean formalization and the checking of several derivations \"were aided by internal models developed in the Kellis Lab at MIT CSAIL\"; \"The author originated the mathematical approach\".","key_facts":["Paper: 'Universal Radial Laws for Random Polynomials: Erdős Problem #522 and Non-Universal Fluctuations', arXiv 2610.11324 [math.PR], 8 Oct 2026; Sebastien Kawada, MIT; 82 pages","Problem (paper): 'In 1961 Erdős asked whether ν_n(1)/n → 1/2 almost surely', where ν_n(1) counts the zeros of f_n(z) = ε_0 + ε_1 z + … + ε_n z^n in the closed unit disk and the ε_k are independent random signs","Earlier: Yakir proved the statement in probability (|ν_n(1) − n/2| < n^(9/10) with probability tending to one). Paper: 'In April 2026, Chojecki and Kwon and Zou each posted an informal argument for the almost-sure statement for random signs.'","Radial law: for bounded symmetric, real Gaussian or circular complex Gaussian coefficients, the proportion of zeros in the disk of radius 1 + x/n converges almost surely to ½(1 + coth x − 1/x), uniformly in x; so almost all zeros lie within O(1/n) of the unit circle","Rate for random signs: almost-sure convergence at the unit circle at rate O(n^(−1/4) √(log n)), even with arbitrary dependence between degrees","Fluctuations: the variance of the zero count is (c_G + κ_4/12)·n + O(n^(399/400)), depending on the coefficient law through its fourth cumulant; for random signs the constant is c_G − 1/6","Verification: 'Ours is the first formally verified proof of Erdős Problem #522. Its Lean formalization was completed on September 22, 2026.' The repository lists 173 main theorems depending only on the axioms propext, Classical.choice and Quot.sound, and a single-file proof checkable in the Lean web editor","Formal statement: the theorem proves Erdos522.erdos_522 as written in Google DeepMind's Formal Conjectures project, whose maintainers reviewed the formal proof (acknowledgments)","AI disclosure (acknowledgments): 'The Lean formalization and the checking and refinement of several derivations were aided by internal models developed in the Kellis Lab at MIT CSAIL, including language models, latent representation models, and graph neural networks acting on the dependency structure of the proof. The author originated the mathematical approach and established the results presented here.'"],"key_numbers":[],"tags":["math","probability","random-polynomials","erdos-problems","littlewood-polynomials","lean","formal-conjectures","formal-verification"],"science":{"field":"mathematics","subfield":"probability / random polynomials","problem":"Erdős Problem #522 (1961): for a random polynomial with independent ±1 coefficients, does the proportion of zeros in the closed unit disk tend to 1/2 almost surely?","result":"Yes, with a universal radial law ½(1 + coth x − 1/x) at scale 1/n and a variance formula showing non-universal fluctuations.","open_since":"1961","ai_system":["unnamed internal models of the Kellis Lab (MIT CSAIL)"],"human_role":"Human-led: the author states he originated the approach and established the results; in-house language models and other models aided the Lean formalization and the checking of derivations","verification":{"key":"lean","label":"Lean-verified","text":"Lean 4 with Mathlib, kernel-replayed; accepted as a solution to the Formal Conjectures statement; unrefereed preprint"},"status":"pending","shock":"","short":"Erdős #522: zeros of random ±1 polynomials"},"body_md":"## What happened\n\nA Littlewood polynomial has every coefficient equal to +1 or −1. Pick the signs by coin flips and ask where the zeros fall: they\ncluster near the unit circle, and Erdős asked in 1961 whether exactly half of them, in the limit, lie inside or on it for almost\nevery infinite sequence of flips. Convergence in probability was known. The almost-sure version needs control over all degrees at\nonce, because each new coefficient moves every zero.\n\nKawada's paper proves the almost-sure statement and describes the zeros at finer scales. Magnified by the degree n, their distance\nfrom the circle follows one fixed law for a wide class of coefficient distributions. The fluctuations of the count do depend on\nthe distribution, through its fourth cumulant.\n\nThe whole development is formalized in Lean 4. The theorem matches the statement of Problem #522 that Google DeepMind's Formal\nConjectures project had written down in advance, which removes the usual doubt about whether a formal proof proves the intended\nclaim. The paper says in-house models of the Kellis Lab at MIT helped with the formalization and with checking derivations, and\nthat the mathematical approach is the author's.\n\n## Why it matters\n\nIt is a named Erdős problem closed with a machine-checked proof against an independently written formal statement, days after\nerdosproblems.com froze proof claims and removed its open/solved labels (6 Oct). It also shows a different pattern from the autonomous\nresults of autumn 2026: a human-led proof where AI did the formalization work, using a lab's own models that the paper does not\nname or describe further.","disputed":[{"label":"Verification","text":"Lean 4 with Mathlib, kernel-replayed; accepted as a solution to the Formal Conjectures statement; unrefereed preprint","sources":[]}],"related":[{"id":"2026-10-08-alphaproof-nexus-science-paper","url":"https://postcutoff.com/e/2026-10-08-alphaproof-nexus-science-paper/","date":"2026-10-08","date_precision":"day","short_title":"Google DeepMind's AlphaProof Nexus paper published in Science","deck":"LLM + Lean agent resolved 9 of 353 open Erdős problems and 44 of 492 OEIS conjectures","takeaway":"This is a peer-reviewed, fully formal account of LLM agents solving open research problems at scale and at low cost.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":16,"official":4,"filed":"2026-10-10","updated":"2026-10-10","orgs":["Google DeepMind"]},{"id":"2026-10-06-erdos-problems-site-freezes-proof-claims","url":"https://postcutoff.com/e/2026-10-06-erdos-problems-site-freezes-proof-claims/","date":"2026-10-06","date_precision":"day","short_title":"erdosproblems.com freezes proof claims and drops 'open/solved' labels and solver credits after a wave of unexplained AI proofs","deck":null,"takeaway":"erdosproblems.com was where AI-for-math claims were counted and disputed, from the GPT-5 controversy of October 2025 to the 2026 waves of GPT-6 Astra and Claude results.","category":"research","category_label":"Research","importance":3,"confidence":"high","status":{"key":"confirmed","labels":["Confirmed"]},"sources":4,"official":4,"filed":"2026-10-08","updated":"2026-10-12","orgs":["erdosproblems.com"]},{"id":"2026-09-30-ai-assisted-conjecture-wave-summer-2026","url":"https://postcutoff.com/e/2026-09-30-ai-assisted-conjecture-wave-summer-2026/","date":"2026-09-30","date_precision":"day","short_title":"Summer 2026 flood: dozens of named conjectures settled on arXiv with disclosed AI help","deck":null,"takeaway":"This entry catalogues about 50 of them, with the AI role as the authors state it.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":9,"official":4,"filed":"2026-09-30","updated":"2026-10-12","orgs":["OpenAI","Anthropic","Google DeepMind","various mathematicians"]},{"id":"2026-09-10-astra-leanopenproblems-september-results","url":"https://postcutoff.com/e/2026-09-10-astra-leanopenproblems-september-results/","date":"2026-09-10","date_precision":"day","short_title":"GPT-6 Astra's Epoch AI run adds more Lean-checked results","deck":"Dittert conjecture proved, Ibragimov–Iosifescu and eternal-domination conjectures disproved","takeaway":"Autonomous formal proof search now turns out a steady stream of mid-level resolved conjectures, not one-off headlines.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":6,"official":4,"filed":"2026-09-29","updated":"2026-09-29","orgs":["OpenAI","Epoch AI"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-12","type":"filed","text":"Created from the arXiv PDF and the repository README (found by a manual arXiv scan)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-12","run":null,"sources_read":"The arXiv PDF and the repository README (found by a manual arXiv scan)","updated":"2026-10-12","human_review":null,"version":null},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":161,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":100,"in_training_data":false},{"model_id":"gemini-3-8-flash","name":"Gemini 3.8 Flash","cutoff":"2026-03","days_after":191,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":130,"in_training_data":false}],"short_url":null}