{"schema":"postcutoff/event@1","as_of":"2026-10-08T23:45:00+02:00","url":"https://postcutoff.com/e/2026-10-07-exact-overlaps-conjecture-gpt-6-astra/","md":"https://postcutoff.com/e/2026-10-07-exact-overlaps-conjecture-gpt-6-astra/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":"2026-10-07-exact-overlaps-conjecture-gpt-6-astra","date":"2026-10-07","date_precision":"day","short_title":"Exact overlaps conjecture for self-similar measures on the line proved","deck":"GPT-6 Astra found the first proof (Kittle & Kogler), 47,000-line Lean formalization; OpenAI's catalogue claims it independently","takeaway":"A central conjecture of fractal geometry, open in general since Hochman's 2014 breakthrough, now has a proof whose key part came from a GPT-6 Astra session. The authors rewrote and formalized it in Lean, and OpenAI's own release claims the same theorem.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":[{"n":1,"title":"Kittle & Kogler: The Exact Overlaps Conjecture for Self-Similar Measures on the Real Line (arXiv 2610.10511)","url":"https://arxiv.org/abs/2610.10511","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"Lean formalization (ckkogler/exact-overlaps-one-dim-lean)","url":"https://github.com/ckkogler/exact-overlaps-one-dim-lean","type":"code","group":"primary","domain":"github.com"},{"n":3,"title":"Supplementary material incl. GPT-6 Astra transcript (ckkogler/kk26-supplementary-material)","url":"https://github.com/ckkogler/kk26-supplementary-material","type":"code","group":"primary","domain":"github.com"},{"n":4,"title":"OpenAI math catalogue (CONTENTS.md, family 148)","url":"https://github.com/openai/math/blob/main/CONTENTS.md","type":"paper","group":"primary","domain":"github.com"},{"n":5,"title":"Hochman (2014): On self-similar sets with overlaps and inverse theorems for entropy (Annals)","url":"https://doi.org/10.4007/annals.2014.180.2.7","type":"paper","group":"primary","domain":"doi.org"}],"official":5,"filed":"2026-10-08","updated":"2026-10-08","orgs":["University College London","Institute for Advanced Study","OpenAI"],"title":"Exact overlaps conjecture for self-similar measures on the line proved: GPT-6 Astra found the first proof (Kittle & Kogler), 47,000-line Lean formalization; OpenAI's catalogue claims it independently","summary":"On 7 Oct 2026 Samuel Kittle (UCL) and Constantin Kogler (IAS) posted arXiv 2610.10511, a proof of the exact overlaps conjecture for self-similar measures on the real line and of its generalized form, dim ν = min{1, h_μ/|χ_μ|}. Their \"Machine Contribution Statement\" says GPT-6 Astra found the first proof on 27 Sep 2026 after about two and a half hours, following a two-hour warm-up in which it found new, simpler proofs of Hochman's 2014 and Varjú's 2019 theorems. GPT-6 Astra also formalized everything in Lean (about 47,000 lines, 9.5 hours, four agents). OpenAI's 6 Oct catalogue independently claims the same theorem as family 148, dated 24 Sep. The proof is an unrefereed preprint with a public Lean formalization.","key_facts":["Theorem 1.1: for any probability measure μ on finitely many contracting similarities of ℝ, the self-similar measure ν has dim ν = min{1, h_μ/|χ_μ|} (h_μ random-walk entropy, χ_μ Lyapunov exponent). This implies the exact overlaps conjecture: a dimension drop below min{1, H(p)/|χ|} happens only with exact overlaps","History (paper's introduction): a version for self-similar sets goes back to Simon (1996); Hochman (Annals 2014) proved it under exponential separation, hence for algebraic parameters; Varjú (2019) did transcendental Bernoulli convolutions; Rapaport (2022) algebraic contractions; Rapaport–Varjú (2024) and others special three-map systems","Key new ingredient (Theorem 1.3): an entropy inequality quantifying the information lost when independent random variables are added, in terms of variance averaged across scales","Machine Contribution Statement: 'On September 27, 2026, GPT-6 Astra found the first proof of Theorem 1.1 presented to the second-named author (C.K.)'. 'Motivated by a suggestion of Elon Lindenstrauss, GPT-6 Astra was first asked to propose new approaches to simpler proofs of the main results of [Hoc14] and [Var19b], which was successfully achieved after around two hours. Encouraged by the model having apparently rethought the area in two hours, the exact overlaps conjecture was tried and the model succeeded after around two and a half hours.'","Division of labour: Kittle found a new deduction of Theorem 1.1 from Theorem 1.3 via Hochman's theorem and the variance summation method; 'the introduction and Section 3 are due to the authors, while Section 2 is due to GPT-6 Astra and was rewritten by the authors'; the authors 'take full responsibility for the correctness'","Lean: 'GPT-6 Astra formalized all results from this paper and all necessary results from previous work including [Hoc14, Theorem 1.4]. The formalization is around 47,000 lines of code and took around 9.5 hours with four agents working in parallel' (github.com/ckkogler/exact-overlaps-one-dim-lean)","Supplementary repo (all documents 'written by GPT-6 Astra'): the conversation transcript, self-contained AI proofs of Hochman's Theorem 1.1 (8 pp.) and Varjú's Theorem 3 (10 pp.), a 16-page first proof, an extension to arbitrary dimension when the rotations generate a virtually solvable group, and a sharpened Theorem 1.3","Priority: OpenAI's catalogue (family 148, 'The entropy-rate dimension formula for self-similar measures on the line', dated 24 Sep, with a Lean scope note) claims the same theorem; the authors note OpenAI dates its proof 24 Sep, theirs 27 Sep, and that Theorem 1.3 is a quantitative strengthening of OpenAI's Lemma 3.2","Kogler previously co-authored the GPT-6 Astra proof of the Odlyzko–Poonen conjecture (2026-09-22-odlyzko-poonen-conjecture-proved)"],"key_numbers":[],"tags":["math","fractal-geometry","dynamical-systems","self-similar-measures","gpt-6-astra","lean","formalization","priority","openai"],"science":{"field":"mathematics","subfield":"fractal geometry / dynamical systems","problem":"Exact overlaps conjecture for self-similar measures on ℝ (Simon 1996 for sets; Hochman 2014 programme), and the generalized entropy–Lyapunov dimension formula","result":"Proof that every self-similar measure on the line has dim ν = min{1, h_μ/|χ_μ|}, so dimension drop occurs only with exact overlaps; no separation or algebraicity assumptions.","open_since":"1996","ai_system":["GPT-6 Astra"],"human_role":"AI-found, human-rewritten: GPT-6 Astra produced the first full proof in a ~2.5-hour session and the Lean formalization; Kittle found a new deduction of the main theorem and the authors rewrote the paper and checked it","verification":{"key":"lean","label":"Lean-verified","text":"Lean formalization (~47,000 lines, by GPT-6 Astra) plus author-checked preprint; not yet peer-reviewed"},"status":"pending","shock":"Two AI systems (OpenAI's internal model on 24 Sep and GPT-6 Astra on 27 Sep) independently proved a central conjecture of fractal geometry (open in general since Hochman's 2014 partial solution) within three days, after a model 'rethought the area in two hours'.","short":"Exact overlaps conjecture (dimension one)"},"body_md":"## What happened\n\nSelf-similar measures are the natural measures on fractals built from finitely many contracting maps x ↦ r_i x + b_i. The\nexact overlaps conjecture predicts that such a measure has the \"expected\" dimension, min{1, entropy/Lyapunov exponent},\nunless two different compositions of maps coincide exactly. Michael Hochman's 2014 Annals paper proved it under an exponential\nseparation condition, which covers all algebraic parameters. The transcendental case stayed open apart from special\nfamilies (Varjú's Bernoulli convolutions, Rapaport's algebraic contractions, Rapaport–Varjú's three-map systems).\n\nOn 7 Oct 2026 Samuel Kittle (UCL) and Constantin Kogler (IAS) posted a proof of the full one-dimensional conjecture, and of\nthe generalized formula with random-walk entropy. Their machine contribution statement describes how it was found. Following\na suggestion from Elon Lindenstrauss, Kogler first asked GPT-6 Astra for new, simpler proofs of Hochman's and Varjú's\ntheorems, which it produced in about two hours. \"Encouraged by the model having apparently rethought the area in two hours,\"\nthey then posed the conjecture itself, and the model had a proof after about two and a half hours on 27 Sep. Kittle found a\ndifferent route from the key entropy inequality to the main theorem, and the authors rewrote the paper. Section 2 is\nGPT-6 Astra's argument, rewritten by the authors. GPT-6 Astra then formalized the whole paper and the required parts of\nHochman's work in Lean: about 47,000 lines in 9.5 hours with four agents in parallel. The authors published the conversation\ntranscript and the model's original drafts.\n\nOpenAI's 6 Oct release had already listed the same theorem as family 148, with a manuscript dated 24 Sep and a Lean scope\nnote. Kittle and Kogler state both dates and describe their Theorem 1.3 as a quantitative strengthening of OpenAI's key lemma.\n\n## Why it matters\n\nThis is a central problem in fractal geometry, and the record of how it fell is unusually complete: the transcript, the\nmodel's drafts, a Lean proof and a clear statement of which sections are machine work. It is also one of the first cases of\ntwo independent AI systems settling the same well-known conjecture within days of each other.","disputed":[{"label":"Verification","text":"Lean formalization (~47,000 lines, by GPT-6 Astra) plus author-checked preprint; not yet peer-reviewed","sources":[]}],"related":[{"id":"2026-10-07-lebrun-salamon-conjecture-proof","url":"https://postcutoff.com/e/2026-10-07-lebrun-salamon-conjecture-proof/","date":"2026-10-07","date_precision":"day","short_title":"Two claimed proofs of the LeBrun–Salamon conjecture","deck":"Hu, Liu & Wan (arXiv, ChatGPT as 'auxiliary tool') and OpenAI's catalogue (family 062)","takeaway":"The 1994 conjecture that every positive quaternion-Kähler manifold is a symmetric Wolf space now has two independent claimed proofs, one by human geometers who used ChatGPT and one in OpenAI's AI-generated catalogue. Neither has been refereed.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":4,"official":4,"filed":"2026-10-08","updated":"2026-10-08","orgs":["Chongqing University of Technology","OpenAI"]},{"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":42,"official":12,"filed":"2026-10-07","updated":"2026-10-08","orgs":["OpenAI"]},{"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":"2026-10-08","orgs":["OpenAI","Anthropic","Google DeepMind","various mathematicians"]},{"id":"2026-09-22-odlyzko-poonen-conjecture-proved","url":"https://postcutoff.com/e/2026-09-22-odlyzko-poonen-conjecture-proved/","date":"2026-09-22","date_precision":"day","short_title":"Odlyzko–Poonen conjecture (1993) proved unconditionally","deck":"'The proofs are due to GPT-6 Astra', which also wrote a 22,000-line Lean formalisation","takeaway":"The unconditional case had stayed open after Breuillard and Varjú's conditional proof in 2019.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":2,"official":2,"filed":"2026-09-30","updated":"2026-10-07","orgs":["Constantin Kogler","OpenAI"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-08","type":"filed","text":"Created from the arXiv PDF (machine contribution statement, introduction, references) and the two GitHub repositories"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-08","run":null,"sources_read":"The arXiv PDF (machine contribution statement, introduction, references) and the two GitHub repositories","updated":"2026-10-08","human_review":null,"version":{"date":"2026-10-08"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":160,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":99,"in_training_data":false},{"model_id":"gemini-3-8-flash","name":"Gemini 3.8 Flash","cutoff":"2026-03","days_after":190,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":129,"in_training_data":false}],"short_url":"https://postcutoff.com/s/exact-overlaps-conjecture"}