{"schema":"postcutoff/event@1","as_of":"2026-10-23T13:24:00+02:00","url":"https://postcutoff.com/e/2026-10-22-simon-l2-conjecture-disproved-chatgpt/","md":"https://postcutoff.com/e/2026-10-22-simon-l2-conjecture-disproved-chatgpt/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"human editor","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-10-22-simon-l2-conjecture-disproved-chatgpt","date":"2026-10-22","date_precision":"day","short_title":"Barry Simon's L² conjecture (Problem 8 of his 2000 list) disproved","deck":"Becker and Yang build a counterexample largely found with ChatGPT","takeaway":"One of Barry Simon's fifteen problems for Schrödinger operators, open since 2000, is false: a small, smooth potential satisfying his L² condition can leave no continuous spectrum at all. The authors say ChatGPT produced the key mechanism and, 'very much to our surprise', extended it to the full spectrum.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2610.31637: Becker, Yang, Lenses make many points: pure point spectrum disproves Simon's L² conjecture","url":"https://arxiv.org/abs/2610.31637","type":"paper","group":"primary","domain":"arxiv.org"}],"official":1,"filed":"2026-10-23","updated":"2026-10-23","orgs":["OpenAI"],"title":"Barry Simon's L² conjecture (Problem 8 of his 2000 list) disproved: Becker and Yang build a counterexample largely found with ChatGPT","summary":"On 22 Oct 2026 Simon Becker and Mengxuan Yang posted arXiv 2610.31637, a counterexample to Barry Simon's multidimensional L² conjecture (Problem 8 in \"Schrödinger operators in the twenty-first century\", 2000). For every d ≥ 2 they construct a smooth, bounded, nonnegative \"lens-type\" potential with arbitrarily small sup norm and weighted L² integral such that −Δ + V has only pure point spectrum on (0, ∞), so not even singular continuous spectrum survives. The construction was developed in conversations with ChatGPT and then rewritten by the authors. Unrefereed preprint.","key_facts":["Conjecture (Simon 2000, Problem 8; Simon 2019, Conjecture 20.2): if ∫|x|^{1−d}|V(x)|² dx < ∞ then −Δ + V has absolutely continuous spectrum of infinite multiplicity at almost every positive energy. The 1D analogue is the Deift–Killip theorem (1999)","Theorem 1.1: for d ≥ 2, 0 ≤ σ < 1/3 and δ > 0 there is a smooth bounded nonnegative V with ‖V‖∞ + I_{d,0}(V) + I_{d,σ}(V) < δ such that σ(H_V) = σ_ess(H_V) = [0, ∞) and the continuous spectral projection is zero; the positive eigenvalues are dense","Construction: potentials on widely separated annuli, each a weak angular 'lens' that focuses waves into narrow channels plus bounded periodic 'reflecting barriers' whose spectral gaps cover a prescribed energy window; pure point spectrum via Combes–Hislop spectral averaging; 3D by cylindrical extension, all d by tensor sums","AI disclosure (verbatim): 'Our interest in this conjecture was renewed when it remained among the few of Simon's problems left untouched by OpenAI … We developed the first construction of the potential in discussions with ChatGPT, obtaining a proof of absence of absolutely continuous spectrum in a bounded interval … When prompted with this question, ChatGPT showed that a rather intricate adjustment of the parameters of the potential … could yield point spectrum in a bounded interval through a mechanism very similar to the one used in the final proof and already including the spectral-averaging argument. Very much to our surprise, ChatGPT was then able to extend this result to the full spectrum by pushing the parameters of the construction further. We thoroughly rewrote the resulting argument and manuscript'","ChatGPT model version not stated. Simon Becker earlier used GPT-6.1 Sol for counterexamples to Fried's conjecture (October 2026)"],"key_numbers":[],"tags":["math","spectral-theory","schrodinger-operators","counterexample","chatgpt","openai","simon-problems"],"science":{"field":"mathematics","subfield":"spectral theory / Schrödinger operators","problem":"Barry Simon's multidimensional L² conjecture (Problem 8, 'Schrödinger operators in the twenty-first century')","result":"False in every dimension d ≥ 2: smooth, bounded, nonnegative, arbitrarily small potentials satisfying the (even weighted) L² condition for which −Δ + V has only pure point spectrum on (0, ∞).","open_since":"2000","ai_system":["ChatGPT"],"human_role":"AI-assisted: authors chose the problem and suspected falsity; ChatGPT developed the construction and the extension to the full spectrum; authors rewrote the argument","verification":{"key":"expert","label":"Expert-checked","text":"Unrefereed arXiv preprint (2610.31637); checked by its authors"},"status":"pending","shock":"The conjecture was expected to be the multidimensional Deift–Killip theorem; instead even singular continuous spectrum can be destroyed."},"body_md":"## What happened\n\nIn 2000 Barry Simon published fifteen open problems for Schrödinger operators. Problem 8 asked whether the condition ∫|x|^{1−d}|V|² < ∞, which by the Deift–Killip theorem guarantees absolutely continuous spectrum in one dimension, does the same in higher dimensions. Simon restated it as Conjecture 20.2 in 2019.\n\nSimon Becker and Mengxuan Yang ([arXiv 2610.31637](https://arxiv.org/abs/2610.31637), 22 Oct) show it is false in every dimension d ≥ 2. Their potential is a sum of annular blocks: a weak \"lens\" bends waves into narrow channels that end at bounded periodic barriers whose spectral gaps cover a given energy window. Spectral averaging then gives pure point spectrum, dense in [0, ∞), so the operator has no continuous spectrum at all, even under the stronger weighted condition with |x|^σ, σ < 1/3.\n\nThe authors write that they returned to the problem because it \"remained among the few of Simon's problems left untouched by OpenAI\". ChatGPT produced the first construction, then the mechanism used in the final proof, and finally the extension to all energies. The authors rewrote the argument and take full responsibility for it.\n\nNo expert reactions were found on 23 Oct (X and blogs not searched).\n\n## Why it matters\n\nSimon's 2000 list is one of the standard problem lists of mathematical physics. This is a clean negative answer to one of its problems, with the AI credited for the decisive steps, not just for checking. It also shows researchers choosing targets by what OpenAI's 6 Oct release did *not* cover.","disputed":[{"label":"Verification","text":"Unrefereed arXiv preprint (2610.31637); checked by its authors","sources":[]}],"related":[{"id":"2026-10-23-ai-assisted-results-october-23-listing","url":"https://postcutoff.com/e/2026-10-23-ai-assisted-results-october-23-listing/","date":"2026-10-23","date_precision":"day","short_title":"AI-assisted results in the 23 Oct 2026 arXiv listing","deck":"DeepMind's Gemini 4 Argon loop on Lehmer's problem, Demmel's Claude Code-built MRRR SVD solver, a Claude–GPT agent harness proof with Lean, GPT-6 Astra on k-Hessian and Steklov problems","takeaway":"Friday's arXiv announcement brought a disproof of Barry Simon's L² conjecture and the Critical 2D KPZ process (both with ChatGPT), an expert diagnosis of OpenAI's withdrawn Hodge papers and a hand-written version of OpenAI's Barnette proof. This page lists the next tier.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":23,"official":23,"filed":"2026-10-23","updated":"2026-10-23","orgs":["Google DeepMind","OpenAI","Anthropic","Harmonic","various mathematicians"]},{"id":"2026-10-06-openai-math-release-722-manuscripts","url":"https://postcutoff.com/e/2026-10-06-openai-math-release-722-manuscripts/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI releases 722 AI-written math manuscripts claiming hundreds of open problems","deck":"Including quasi-Riemann, Unique Games, Hodge for CM abelian varieties and free group factors","takeaway":"If even a fraction of these results hold up, this is the largest single jump in mathematical knowledge on record, produced by an AI system in about six weeks.","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":84,"official":13,"filed":"2026-10-07","updated":"2026-10-20","orgs":["OpenAI"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-23","type":"filed","text":"Created (PDF read)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-23","run":null,"sources_read":"PDF read","updated":"2026-10-23","human_review":null,"version":null},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":175,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":114,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":205,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":144,"in_training_data":false}],"short_url":"https://postcutoff.com/s/barry-simon-l2-conjecture"}