{"schema":"postcutoff/event@1","as_of":"2026-10-30T13:42:00+01:00","url":"https://postcutoff.com/e/2026-10-06-openai-anderson-localization-2d-delocalization-3d/","md":"https://postcutoff.com/e/2026-10-06-openai-anderson-localization-2d-delocalization-3d/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":"2026-10-06-openai-anderson-localization-2d-delocalization-3d","date":"2026-10-06","date_precision":"day","short_title":"OpenAI claims Anderson localization in 2D and extended states in 3D","deck":null,"takeaway":"Two manuscripts in OpenAI's 6 Oct 2026 math release (family 261, dated 23 Sept) claim the two central conjectures about electrons in a random lattice. In two dimensions, the Anderson model has pure-point spectrum, so all states are localized, at every disorder strength. In three or more dimensions, weak disorder leaves purely absolutely continuous spectrum, meaning extended states, on an energy interval. These are Barry Simon's Problems 1 and 2 of 2000, for uniform disorder. Specialists Jakšić and Last shortened the last step of the 3D proof in an appendix, and a third party has posted a Lean formalization of the 3D result. OpenAI's own Lean covers only the 2D spectrum's location.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"OpenAI: Pure-Point Spectrum for the Two-Dimensional Anderson Model at Every Positive Disorder (PDF)","url":"https://github.com/openai/math/blob/main/preprints/Pure-Point-Spectrum-for-the-Two-Dimensional-Anderson-Model-at-Every-Positive-Disorder-September-23-2026/paper.pdf","type":"paper","group":"primary","domain":"github.com"},{"n":2,"title":"OpenAI: Absolutely Continuous Spectrum for Weak-Disorder Anderson Models in Dimensions at Least Three (PDF)","url":"https://github.com/openai/math/blob/main/preprints/Absolutely-Continuous-Spectrum-for-Weak-Disorder-Anderson-Models-in-Dimensions-at-Least-Three-September-23-2026/paper.pdf","type":"paper","group":"primary","domain":"github.com"},{"n":3,"title":"Jakšić, Last: Extended States on the Bethe Lattice Revisited (arXiv 2610.11611, Appendix B on the OpenAI preprint)","url":"https://arxiv.org/abs/2610.11611","type":"paper","group":"primary","domain":"arxiv.org"},{"n":4,"title":"cadamcat/anderson-ac-lean4: Lean 4 formalization of the 3D theorem (Yao Xu)","url":"https://github.com/cadamcat/anderson-ac-lean4","type":"code","group":"primary","domain":"github.com"},{"n":5,"title":"OpenAI Lean scope note 261 (2D spectrum only)","url":"https://github.com/openai/math/blob/main/lean/docs/261.md","type":"code","group":"primary","domain":"github.com"},{"n":6,"title":"oai-math-revised: community notes on the 3D manuscript","url":"https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Absolutely-Continuous-Spectrum-for-Weak-Disorder-Anderson-Models-in-Dimensions-at-Least-Three-September-23-2026/AGENT_NOTES.md","type":"discussion","group":"reactions","domain":"github.com"}],"official":5,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"],"title":"OpenAI's release claims both halves of the Anderson localization picture: localization in 2D at every disorder and extended states in 3D and up at weak disorder (Simon's Problems 1 and 2)","summary":"Family 261: \"Pure-Point Spectrum for the Two-Dimensional Anderson Model at Every Positive Disorder\" and \"Absolutely Continuous Spectrum for Weak-Disorder Anderson Models in Dimensions at Least Three\" (23 Sept 2026, released 6 Oct), both for independent uniform site potentials. Known before: localization at strong disorder or band edges in any dimension, localization at all disorder in one dimension, and extended states on tree (Bethe) lattices, but no extended states on ℤ^d and no full 2D localization. Vojkan Jakšić and Yoram Last (arXiv 2610.11611, appendix of 7 Oct) give a shorter passage from the 3D paper's boundary-positivity statement to purity. Yao Xu's GitHub project anderson-ac-lean4 (14–15 Oct) states a sorry-free Lean proof of the 3D theorem. OpenAI's Lean scope note covers only the 2D spectrum as a set, \"not… the pure-point spectral type\".","key_facts":["2D (abstract, verbatim): 'For each fixed positive disorder strength, we prove that the nearest-neighbor Anderson operator on the square lattice with independent uniform site potentials almost surely has pure-point spectral type throughout its spectrum. This resolves the pure-point assertion of the two-dimensional Anderson localization conjecture for the uniform single-site law.'","3D and up (verbatim): 'For each fixed dimension d ≥ 3 and each fixed sufficiently small positive disorder strength, we prove that the Anderson operator on ℤ^d with independent uniform site potentials almost surely has purely absolutely continuous spectrum with nonzero weight on an open energy interval… This settles the purely absolutely continuous energy-range question in Simon's Problem 1 for this model.'","Context: Anderson's 1958 model of electrons in a disordered crystal; physics (scaling theory, 1979) predicts localization at any disorder in 2D and a mobility edge in 3D. Rigorous results covered strong disorder and band edges (Fröhlich–Spencer, Aizenman–Molchanov), all disorder in 1D, and weak disorder on the Bethe lattice (Klein), but not the two central cases, which Barry Simon listed as Problems 1 and 2 in 2000","Expert engagement: Jakšić (Politecnico di Milano) and Last (Hebrew University), 'Extended States on the Bethe Lattice Revisited' (arXiv 2610.11611, 8 Oct). Appendix B, 'written on October 7, 2026… one day after OpenAI publicly released the preprint [10], claiming to prove purely absolutely continuous spectrum', shows that their structural argument gives purity directly from the preprint's boundary-positivity estimate (its Eq. 177)","Third-party Lean: cadamcat/anderson-ac-lean4 (Yao Xu, 'Developed with AI assistance'): 'The final theorem uses only the axioms propext, Classical.choice and Quot.sound. No proof in the repository uses sorry or an added axiom.' The statement file imports only Mathlib, with a non-vacuity check; not independently reviewed by us","OpenAI's Lean scope note 261 (verbatim): the formalization 'identifies its spectrum as the real interval [−4−h, 4+h]… It does not assert the pure-point spectral type claimed in the accompanying paper.' The challenge PlanarAndersonSpectrum was accepted in oai-math-revised's Comparator run"],"key_numbers":[],"tags":["physics","mathematical-physics","anderson-localization","random-schrodinger-operators","spectral-theory","lean","openai-math-release"],"science":{"field":"physics","subfield":"mathematical physics (random Schrödinger operators, Anderson localization)","problem":"Simon's Problem 2 (localization at all disorder in 2D) and Problem 1 (absolutely continuous spectrum / extended states at weak disorder in d ≥ 3) for the Anderson model","result":"Claimed for uniform site potentials: pure-point spectrum throughout the spectrum in 2D at every disorder; purely absolutely continuous spectrum on an open energy interval in every d ≥ 3 at small disorder","open_since":"1958","ai_system":["OpenAI internal model (math release)"],"human_role":"Autonomous AI output released by OpenAI; an expert appendix shortens one step; a third-party AI-assisted Lean formalization of the 3D result","verification":{"key":"lean-unreviewed","label":"Lean proof, not yet reviewed","text":"Unrefereed manuscripts; 3D theorem formalized in a third-party Lean project (statement fidelity unreviewed); 2D pure-point claim unformalized"},"status":"pending","shock":"Both open halves of the Anderson picture, localization in the plane and extended states in space, are claimed in one family."},"body_md":"## What happened\n\nIn 1958 Philip Anderson showed that disorder can trap electrons: in a sufficiently random crystal, wave functions stay localized\ninstead of spreading. Physics predicts that in two dimensions any amount of disorder localizes every state, while in three dimensions\nweak disorder leaves a band of extended, conducting states. Mathematicians proved localization at strong disorder, at band edges and in\none dimension, and extended states on tree-like lattices. The two central cases on the ordinary lattice stayed open. Barry Simon put\nthem first and second on his 2000 list of open problems for Schrödinger operators.\n\nOpenAI's release claims both, for the standard model with uniformly distributed random potentials. Two days after the release, Vojkan\nJakšić and Yoram Last, who had just written a new proof of the tree-lattice result, added an appendix showing that their method shortens\nthe 3D paper's final step from a boundary estimate to pure absolute continuity. They describe the preprint as \"claiming to prove\" the\nresult and do not assess the rest of it. A separate developer, Yao Xu, has published a Lean 4 formalization of the 3D theorem that he\nsays uses no unproved steps. OpenAI's own formal proof for the family covers only the location of the 2D spectrum.\n\n## Why it matters\n\nAnderson localization underlies the physics of disordered metals and insulators, and these two statements are its best-known open\nmathematical problems. If correct, the release settles both for the uniform model. The 3D claim has more support (an expert appendix\nand a third-party formal proof). The 2D localization claim rests on the manuscript alone, and we found no public assessment by\nspecialists as of 29 Oct.","disputed":[{"label":"Verification","text":"Unrefereed manuscripts; 3D theorem formalized in a third-party Lean project (statement fidelity unreviewed); 2D pure-point claim unformalized","sources":[]}],"related":[{"id":"2026-10-24-oai-math-revised-community-hub","url":"https://postcutoff.com/e/2026-10-24-oai-math-revised-community-hub/","date":"2026-10-24","date_precision":"day","short_title":"oai-math-revised: mathematicians launch a community hub to revise OpenAI's AI-written math manuscripts (guest post on Tao's blog)","deck":null,"takeaway":"This is the first organized attempt by research mathematicians to digest and credit the OpenAI catalogue.","category":"research","category_label":"Research","importance":3,"confidence":"high","status":{"key":"confirmed","labels":["Confirmed"]},"sources":6,"official":5,"filed":"2026-10-25","updated":"2026-10-27","orgs":["OpenAI","mathcommunity"]},{"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":79,"official":13,"filed":"2026-10-07","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2026-10-06-openai-heisenberg-ferromagnet-bose-einstein-condensation","url":"https://postcutoff.com/e/2026-10-06-openai-heisenberg-ferromagnet-bose-einstein-condensation/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI claims quantum ferromagnet order and interacting Bose condensation","deck":null,"takeaway":"Two families in OpenAI's 6 Oct 2026 math release claim long-open theorems of quantum statistical mechanics. Family 271: the quantum Heisenberg ferromagnet in dimension 3 or more has a spontaneously magnetized equilibrium state at low temperature, plus Bloch's T^3/2 law with its exact coefficient. Family 267: a dilute gas of hard-sphere bosons Bose-condenses in its ground state and at a fixed positive temperature. The magnetization and ground-state condensation statements have Lean proofs that passed an independent Comparator run. OpenAI published a reasoning summary for 271, and Scientific American listed both among the release's most important claims. No expert verification yet.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":8,"official":7,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-29","type":"filed","text":"Created (families 251–378; both abstracts, Lean scope note, Jakšić–Last paper and appendix read, anderson-ac-lean4 README, oai-math-revised notes)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-29","run":null,"sources_read":"Families 251–378; both abstracts, Lean scope note, Jakšić–Last paper and appendix read, anderson-ac-lean4 README, oai-math-revised notes","updated":"2026-10-29","human_review":null,"version":{"date":"2026-10-29"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":159,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":98,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":189,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":128,"in_training_data":false}],"short_url":"https://postcutoff.com/s/openai-anderson-localization-2d"}