{"schema":"postcutoff/event@1","as_of":"2026-10-30T13:42:00+01:00","url":"https://postcutoff.com/e/2026-10-06-openai-l-equals-rl-equals-bpl-lean-claim/","md":"https://postcutoff.com/e/2026-10-06-openai-l-equals-rl-equals-bpl-lean-claim/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-l-equals-rl-equals-bpl-lean-claim","date":"2026-10-06","date_precision":"day","short_title":"OpenAI claims L = RL = BPL with a Lean proof","deck":null,"takeaway":"OpenAI's 6 Oct 2026 math release (family 103, manuscript dated 23 Sept) claims L = RL = BPL. That would mean randomness does not help logarithmic-space computation, a central open problem of complexity theory. The best unconditional bound was Saks and Zhou's O(log^{3/2} n) space, slightly improved by Hoza. A Lean formalization added on 7 Oct proves the equality in an explicit Turing-machine model. The community comparator run of 26 Oct accepted it. A re-check found the proof's definitions identical to the statement file's and the proof built on standard axioms only. It notes that it does not show 'that the machine model is the standard one'. No complexity theorist has assessed it publicly as of 29 Oct.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"OpenAI: Exact Derandomization of Logarithmic Space: L = RL = BPL (23 Sept 2026, paper)","url":"https://github.com/openai/math/blob/main/preprints/Exact-Derandomization-of-Logarithmic-Space-L-equals-RL-equals-BPL-September-23-2026/paper.pdf","type":"paper","group":"primary","domain":"github.com"},{"n":2,"title":"openai/math Lean scope note for family 103","url":"https://github.com/openai/math/blob/main/lean/docs/103.md","type":"code","group":"primary","domain":"github.com"},{"n":3,"title":"oai-math-revised commit: definition bodies match at the checked commit","url":"https://github.com/mathcommunity/oai-math-revised/commit/0794a6fa7575b065267cb34fb9fbae3c171bc2c4","type":"code","group":"primary","domain":"github.com"},{"n":4,"title":"oai-math-recheck (ZHAO Han): family 103 re-check","url":"https://github.com/Beltran12138/oai-math-recheck/tree/f3507cf8b0b5761e792f8b0adbecbc0f1dff5966/families/103","type":"code","group":"primary","domain":"github.com"},{"n":5,"title":"oai-math-revised: comparator run of 26 Oct (results.tsv)","url":"https://github.com/mathcommunity/oai-math-revised/blob/main/checks/comparator-2026-10-26/results.tsv","type":"code","group":"primary","domain":"github.com"},{"n":6,"title":"oai-math-revised: NOTES for L = RL = BPL (re-check record, 28 Oct)","url":"https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Exact-Derandomization-of-Logarithmic-Space-L-equals-RL-equals-BPL-September-23-2026/NOTES.md","type":"discussion","group":"reactions","domain":"github.com"},{"n":7,"title":"HN comment on 'RL=L' rumours (7 Oct)","url":"https://news.ycombinator.com/item?id=49987581","type":"discussion","group":"reactions","domain":"news.ycombinator.com"}],"official":5,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"],"title":"OpenAI's release claims L = RL = BPL, full derandomization of logarithmic space, with a Lean proof; a community re-check confirms the statement's definitions but not that the machine model is the standard one","summary":"\"Exact Derandomization of Logarithmic Space: L = RL = BPL\" (OpenAI, 23 Sept 2026, released 6 Oct): \"We prove L = RL = BPL, resolving the derandomization problem for polynomial-time randomized logarithmic space\", with an \"effective compiler\" from randomized to deterministic logspace machines and a deterministic O(log n + q)-space approximation of acceptance probabilities. The Lean formalization (scope note lean/docs/103.md) \"proves L = RL = BPL for binary languages in an explicit Turing-machine model with logarithmic workspace\". Community checks: accepted by the oai-math-revised comparator run (26 Oct); ZHAO Han's re-check found the challenge and solution definitions byte-identical, with only propext, Classical.choice and Quot.sound as axioms. Open point: whether the formal model matches the standard classes.","key_facts":["Theorem 1.1 (verbatim): 'L = RL = BPL.' Theorem 1.2: for a polynomial-time O(log n)-space randomized machine, a deterministic algorithm on input (x, 1^q) approximates the acceptance probability within 2^−q using O(log(n + 2) + q) work space and (n + 2)^{O(1)} 2^{O(q)} time. About 358k characters, with 13 sections ending in 'The effective compiler'","Model (paper): finite control, endmarked read-only input, finitely many work tapes, fresh fair bits; 'Space counts traversed writable cells in bits, including every simultaneously live counter, numerical register and suspended call'. RL: acceptance probability 0 on no-inputs and ≥ 1/2 on yes-inputs; BPL: ≤ 1/3 and ≥ 2/3; polynomial running time on every random tape","Prior state (as recounted in the paper): Nisan's generator, the Nisan–Zuckerman simulation, Saks–Zhou's O(log^{3/2} n) deterministic space for general bounded-error logspace (JCSS 1999), and Hoza's slight improvement of that bound (APPROX/RANDOM 2021). Full derandomization (BPL = L) remained open","Lean: added to openai/math in the 7 Oct update (commit dated 8 Oct, 05:03 UTC). The scope note says the target asserts equality of the language classes; 'The paper's explicit compiler and numerical running-time bounds are not separately stated in this Comparator target'","Comparator run (oai-math-revised, 26 Oct): LogspaceEquality 'accepted', with the caveat that the configuration listed 20 definitions 'whose bodies the comparator does not compare, only their types'. OpenAI's 26 Oct update emptied that list, so later runs compare the bodies too","Re-check (ZHAO Han, @Beltran12138, oai-math-recheck family 103; recorded 28 Oct, prepared with Claude Code): the 55 OAI constants reachable from the statement, bodies included, dump byte-identical from the challenge and from OpenAI's solution, on arm64 and x86_64. A negative control that changes only the body of BPL (error 1/3 → 1/2) is rejected. 39/39 modules build against Mathlib d13f23b. 'Not shown: that the machine model is the standard one for L, RL and BPL. The re-check gives a reading of the definitions, not a proof of that.'","Reactions: none found from complexity theorists as of 29 Oct (web, HN). An HN commenter (7 Oct) recalled 'rumblings' two weeks earlier that 'RL=L' and other results 'were about to be proven'. The 6–7 Oct viral list of claims ranked 'BPL = L' #161 among '500 most important open problems' (an LLM-ranked list)"],"key_numbers":[],"tags":["computer-science","complexity-theory","derandomization","space-bounded-computation","bpl","lean","openai-math-release","formal-verification"],"science":{"field":"computer-science","subfield":"computational complexity (space-bounded derandomization)","problem":"Is BPL = L? Can bounded-error randomized logarithmic-space computation be simulated deterministically in logarithmic space (also RL = L)?","result":"Claimed proof of L = RL = BPL, with a Lean formalization in an explicit Turing-machine model whose statement definitions have been community-checked against the proof's","open_since":"","ai_system":["OpenAI internal model (math release)"],"human_role":"Autonomous AI output released by OpenAI; community volunteers (with Claude Code) re-checked the Lean statement","verification":{"key":"peer-review","label":"Peer-reviewed","text":"Lean 4 proof accepted by an independent comparator run (standard axioms only); definitions identical between statement and proof; faithfulness of the formal machine model to the standard classes not independently established; manuscript unrefereed"},"status":"pending","shock":"One of complexity theory's flagship derandomization questions arrives not as a paper for referees but as a kernel-checked Lean theorem, so the open question shifts from 'is the proof right?' to 'are the definitions right?'"},"body_md":"## What happened\n\nBPL is the class of problems that a randomized machine with logarithmic memory can solve in polynomial time with small error.\nL is the deterministic version. Whether randomness helps here (BPL = L?) is one of the main questions in derandomization. The best\nunconditional simulation, Saks and Zhou's (published 1999), needed log^{3/2} n space, and Hoza improved it only slightly in\n2021.\n\nOpenAI's 6 Oct release includes a 100-plus-page manuscript claiming the full equality L = RL = BPL. A Lean formalization was added\nin the 7 Oct update. Our coverage of the release listed the claim but described it as manuscript-only. The community hub\noai-math-revised later ran OpenAI's comparator challenges independently. One volunteer then checked that the definitions the\nproof uses are exactly those of the statement, which closes a gap the comparator run left open.\n\n## Why it matters\n\nIf the formal model is faithful, the result is a machine-checked solution of a flagship complexity problem. As with other formal\nclaims in the release, the question that remains is whether the definitions are right, not whether the proof is right. That\nneeds complexity theorists to read the formal definitions of the machine model. Nobody has published such a reading yet.","disputed":[{"label":"Verification","text":"Lean 4 proof accepted by an independent comparator run (standard axioms only); definitions identical between statement and proof; faithfulness of the formal machine model to the standard classes not independently established; manuscript unrefereed","sources":[]}],"related":[{"id":"2026-10-29-ai-assisted-results-october-29-listing","url":"https://postcutoff.com/e/2026-10-29-ai-assisted-results-october-29-listing/","date":"2026-10-29","date_precision":"day","short_title":"AI-assisted results in the 29 Oct 2026 sweep","deck":"Counterexamples to Wilking's 2007 question found by GPT-6 Astra, an LLM-written Plateau paper by De Lellis and Glaudo, explicit Reed–Solomon codes with optimal list size (Guruswami et al.), rank-four quiver mutation decided by GPT Astra, and a dozen named conjectures","takeaway":"The 29 Oct sweep covered the arXiv announcement of 28 Oct, which the previous run had read only through keyword queries. This time 43 papers with an AI statement surfaced, and a PDF scan of a further 405 new and cross-listed submissions in math, cs.CC, cs.DS and cs.DM found about 180 more with an AI statement, most of them describing more than editing help. Four results have their own entries: OpenAI's abundance claim and Jihao Liu's priority remark, OpenAI's Lean-checked L = RL = BPL, R(3,3,3,3) ≤ 61 and the sharp Banach–Mazur exponent. This page lists the rest. GPT-6 Astra found the counterexamples to Wilking's 2007 question 'with minimal human input' and discovered the main results of a rank-four quiver paper. Claude found examples in opinion dynamics and proposed a minimal-surface construction. An LLM supplied 'the main idea' of a paper by Camillo De Lellis, a member of the IAS advisory group on AI and mathematics, and wrote it.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":23,"official":23,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI","Anthropic","various researchers"]},{"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-log-abundance-nonvanishing-claim","url":"https://postcutoff.com/e/2026-10-06-openai-log-abundance-nonvanishing-claim/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI claims the abundance conjecture; Jihao Liu raises priority","deck":null,"takeaway":"A manuscript in OpenAI's 6 Oct 2026 math release, 'Log abundance in characteristic zero' (dated 24 Sept), claims the log abundance conjecture in every dimension in characteristic zero, and with it canonical nonvanishing for smooth projective varieties. Both are central open problems of the minimal model program, proved only up to dimension three. Companion papers claim Iitaka's subadditivity conjecture and related results. On 7 Oct Jihao Liu (Peking University) submitted a human-led proof that nonvanishing implies abundance. arXiv announced it only on 28 Oct. His comment line says he was 'forced to submit here due to an announcement of the proof of the abundance conjecture by OpenAI'. In a remark he suggests that an earlier draft, sent to 'codex chat' before July 2026, may have fed OpenAI's work. No birational geometer has assessed the claims publicly, there is no Lean proof, and OpenAI has not responded.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":8,"official":5,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI","Peking University"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-29","type":"filed","text":"Created (science agent; paper, CONTENTS, Lean scope note, oai-math-revised NOTES, comparator results and LEAN_CHECKS, HN/web searches)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-29","run":null,"sources_read":null,"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-l-rl-bpl"}