{"schema":"postcutoff/event@1","as_of":"2026-10-09T19:24:00+02:00","url":"https://postcutoff.com/e/2026-10-06-finite-index-over-treeable-jkl-problem/","md":"https://postcutoff.com/e/2026-10-06-finite-index-over-treeable-jkl-problem/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":"2026-10-06-finite-index-over-treeable-jkl-problem","date":"2026-10-06","date_precision":"day","short_title":"Chen, Tserunyan & Tucker-Drob solve the Jackson–Kechris–Louveau 'finite index over treeable' problem (2002); ChatGPT 6 Astra Ultra supplied a 'crucial observation'","deck":null,"takeaway":"A countable Borel equivalence relation that contains a treeable subrelation of finite index is itself treeable. This answers Open Problem 6.4(B) of Jackson–Kechris–Louveau (2002), open even in the index-2 measure-preserving case. The authors had the strategy; GPT supplied the observation that made the stages fit together.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":[{"n":1,"title":"Chen, Tserunyan & Tucker-Drob: Finite index extensions of treeable equivalence relations are treeable (arXiv 2610.08986)","url":"https://arxiv.org/abs/2610.08986","type":"paper","group":"primary","domain":"arxiv.org"}],"official":1,"filed":"2026-10-09","updated":"2026-10-09","orgs":["University of Warsaw","McGill University","University of Florida","OpenAI"],"title":"Chen, Tserunyan & Tucker-Drob solve the Jackson–Kechris–Louveau 'finite index over treeable' problem (2002); ChatGPT 6 Astra Ultra supplied a 'crucial observation'","summary":"On 6 Oct 2026 Ruiyuan Chen (Warsaw), Anush Tserunyan (McGill) and Robin Tucker-Drob (Florida) posted arXiv 2610.08986, dedicated to Alexander Kechris's 80th birthday. They prove that every countable Borel equivalence relation containing a finite-index Borel treeable subequivalence relation is Borel treeable, answering Jackson–Kechris–Louveau's Open Problem 6.4(B) (2002) in its strongest, purely Borel form. As a corollary, strong treeability of locally compact groups passes to finite-index overgroups (new even for countable discrete groups). Their \"Declaration of AI use\" says they consulted ChatGPT 6 Astra Ultra after conceiving the approach and that \"GPT provided a crucial observation which ensures the coherence of the forest obtained at stage n + 1 with that obtained at stage n\".","key_facts":["Theorem 1.2: if F ⊆ E are countable Borel equivalence relations, F has finite index in E and F is Borel treeable, then E is Borel treeable. Previously open even for index 2 in the probability-measure-preserving setting (treeing only on a conull set)","Method: a median-subalgebra filtration on products of trees, extending the median-graph treeing algorithm of Chen–Poulin–Tao–Tserunyan (2025) to well-behaved increasing unions; answers Question 1.1, how to 'canonically combine' finitely many trees on the same vertex set into one tree-like structure","Corollary 1.3: a locally compact second countable group with a strongly treeable finite-index open subgroup is strongly treeable (also for measure and Borel strong treeability); for unimodular groups this implies fixed price","AI disclosure: 'We conceived of the approach to proving Theorem 1.2 described above prior to consulting GPT.' GPT 'supplied the following observation that we missed: for a given Q_{n+1}-class D, the partial equivalence relation Q_n|D is invariantly separable by half-spaces in M_D', which in the final proof 'remains implicit but becomes essentially definitional' (Theorem 4.1(c))","Writing: 'The article was entirely written by the three named authors'; ChatGPT gave editorial suggestions after the first draft","No expert reactions found on X, HN or blogs as of 9 Oct; not in OpenAI's 6 Oct catalogue"],"key_numbers":[],"tags":["math","descriptive-set-theory","borel-equivalence-relations","treeability","ergodic-theory","gpt-6-astra","ai-assisted"],"science":{"field":"mathematics","subfield":"descriptive set theory / measured group theory","problem":"Jackson–Kechris–Louveau Open Problem 6.4(B): is a countable Borel equivalence relation with a finite-index treeable subrelation treeable?","result":"Yes, in the purely Borel setting (hence also measurably); with consequences for strongly treeable groups.","open_since":"2002","ai_system":["ChatGPT 6 Astra Ultra"],"human_role":"Human-led: the authors designed the staged construction; GPT contributed one key coherence observation; the paper was written by the authors","verification":{"key":"preprint","label":"Preprint only","text":"Unrefereed preprint by leading experts in the field"},"status":"pending","shock":"A well-known problem in descriptive set theory was closed with the decisive missing step suggested by a chatbot, as described in unusual detail by the authors.","short":"Finite index over treeable (JKL 2002)"},"body_md":"## What happened\n\nTreeable equivalence relations play the role that free groups play among groups. Jackson, Kechris and Louveau asked in 2002\nwhether treeability survives finite-index extensions. They had shown that free actions of virtually free groups give\ntreeable relations, and the general question became a driver of geometric methods in the field.\n\nChen, Tserunyan and Tucker-Drob build the final tree in stages from a filtration of median algebras on a product of trees.\nTheir AI declaration is unusually precise: they had the staged approach and its index-2 setting, and ChatGPT 6 Astra Ultra\npointed out that each stage's relation is invariantly separable by half-spaces in the next stage's median graph. This was\nthe coherence property they needed, and after reformulation it became a definition.\n\n## Why it matters\n\nIt settles a named problem from a standard reference on countable Borel equivalence relations, with direct consequences in\nmeasured group theory (fixed price, strong treeability). The disclosure is a clear example of the \"AI supplies the missing\nlemma\" pattern in research-level mathematics.","disputed":[{"label":"Verification","text":"Unrefereed preprint by leading experts in the field","sources":[]}],"related":[{"id":"2026-10-06-gch-above-strongly-compact-woodin-question","url":"https://postcutoff.com/e/2026-10-06-gch-above-strongly-compact-woodin-question/","date":"2026-10-06","date_precision":"day","short_title":"Woodin's question answered","deck":"GCH below a strongly compact cardinal implies GCH everywhere (Zhixing You, 4-page proof with GPT-6 Astra)","takeaway":"Woodin's question is a well-known open problem about large cardinals and the continuum function, and a short positive answer would be a notable result in set theory.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-07","updated":"2026-10-07","orgs":["Hebrew University of Jerusalem","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":"2026-10-09","orgs":["OpenAI","Anthropic","Google DeepMind","various mathematicians"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-09","type":"filed","text":"Created from the arXiv PDF"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-09","run":null,"sources_read":"The arXiv PDF","updated":"2026-10-09","human_review":null,"version":null},"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-3-8-flash","name":"Gemini 3.8 Flash","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/chen-tserunyan-tucker-drob-solve"}