{"schema":"postcutoff/event@1","as_of":"2026-10-30T13:42:00+01:00","url":"https://postcutoff.com/e/2026-10-27-uppcc-not-closed-under-intersection-gpt-6-astra/","md":"https://postcutoff.com/e/2026-10-27-uppcc-not-closed-under-intersection-gpt-6-astra/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":"2026-10-27-uppcc-not-closed-under-intersection-gpt-6-astra","date":"2026-10-27","date_precision":"day","short_title":"Unbounded-error communication is not closed under AND, found with GPT-6 Astra","deck":null,"takeaway":"Unbounded-error communication complexity (UPP^cc) is the communication analogue of PP. Beigel, Reingold and Spielman proved in 1995 that PP is closed under intersection. Whether UPP^cc is closed under intersection had been asked since Halstenberg and Reischuk (1988). In a 27 Oct preprint, Mark Bun and Mandar Juvekar (Boston University) give explicit matrices of sign rank Θ(d) whose AND has sign rank 2^Θ(d). The previous gap, by Bun, Mande and Thaler, was only log versus log². The paper began when they asked GPT-6 Astra how to beat that gap. 'To our (in retrospect, naïve) astonishment,' they write, the model 'discovered what appeared to be complete proofs'.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2610.37634: UPP^cc is Not Closed Under Intersection (Bun, Juvekar)","url":"https://arxiv.org/abs/2610.37634","type":"paper","group":"primary","domain":"arxiv.org"}],"official":1,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI","Boston University"],"title":"UPP^cc is not closed under intersection: Mark Bun and Mandar Juvekar answer a question open since 1988 after GPT-6 Astra 'discovered what appeared to be complete proofs'","summary":"\"UPP^cc is Not Closed Under Intersection\" (Mark Bun, Mandar Juvekar; arXiv 2610.37634, 27 Oct 2026) builds, for every large d, an explicit ±1 matrix A with 2^Θ(d²) rows and columns and sign rank Θ(d) such that A ∧ A has sign rank 2^Θ(d), via Forster's method and a continuous analogue on the sphere. Hence the class UPP^cc is not closed under intersection, answering a question raised by Halstenberg and Reischuk (1988) and repeated by Göös–Pitassi–Watson and others. The work started from prompting GPT-6 Astra, which produced what appeared to be complete proofs of the separation; the authors wrote the paper and improved the bounds with further model help.","key_facts":["Abstract: 'For every sufficiently large d, we give an explicit matrix A with 2^Θ(d²) rows and columns and sign rank Θ(d) such that A ∧ A has sign rank 2^Θ(d). … this implies that the communication class UPP^cc is not closed under intersection'","Concrete form (Theorem 1.2): the bounded integer halfspace problem HS_{M,d} has U(HS_{M,d}) = O(log d), while U(HS_{M,d} ∧ HS_{M,d}) = Ω(d) for M ≥ 2^{cd}; on n-bit inputs this is O(log n) versus Ω(√n), and both bounds are tight for this family","Prior state: the only earlier example of growth under intersection was Bun–Mande–Thaler (ACM TOCT 2021), U(F) = O(log n) but U(F ∧ F) = Θ(log² n). The question was first asked explicitly by Halstenberg and Reischuk [HR88] and reiterated in Göös–Pitassi–Watson (2018), Sherstov (2018), BMT21 and BT22, per the paper","AI acknowledgment (verbatim): 'This paper arose out of our prompting OpenAI’s GPT-6 Astra model for ideas on how to improve the separation of [BMT21] beyond the quadratic gap they achieved. To our (in retrospect, naïve) astonishment, it discovered what appeared to be complete proofs that (slightly different parameterizations) of the functions we consider in this paper yield the separation U(f) = O(log n) vs. U(f ∧ f) = Ω(n^{1/4}). Subsequent interactions with GPT-6 Astra helped us understand the argument, simplify it, and improve it quantitatively.'","Writing: 'The text of this manuscript was written by the authors, with additional AI assistance used to locate references and provide feedback on our exposition.' Access came through OpenAI's ChatGPT for Academic Researchers program","Status: unrefereed preprint; no expert comment found as of 29 Oct"],"key_numbers":[],"tags":["computer-science","complexity-theory","communication-complexity","sign-rank","gpt-6-astra","arxiv"],"science":{"field":"computer-science","subfield":"communication complexity (unbounded-error, sign rank)","problem":"Is the unbounded-error communication class UPP^cc closed under intersection? (Halstenberg–Reischuk 1988)","result":"No: explicit matrices of sign rank Θ(d) whose conjunction has sign rank 2^Θ(d), i.e. U(F) = O(log n) but U(F ∧ F) = Ω(√n)","open_since":"1988","ai_system":["GPT-6 Astra"],"human_role":"Authors posed the question to GPT-6 Astra, which produced what appeared to be complete proofs with weaker parameters; the authors then understood, simplified and strengthened the argument with further model help and wrote the paper","verification":{"key":"preprint","label":"Preprint only","text":"Unrefereed arXiv preprint"},"status":"pending","shock":"The authors expected ideas and received, to their 'naïve astonishment', apparently complete proofs of a separation that had resisted the area for decades."},"body_md":"## What happened\n\nIn unbounded-error communication, two players need only beat a coin flip, and the cost of a problem is governed by the sign rank of its\nmatrix. Because error cannot be amplified in this model, closure properties are hard to prove. The class PP of complexity theory was shown\nto be closed under intersection in 1995, but the communication analogue stayed open. The only known example of growth under AND, due to\nBun, Mande and Thaler, raised the cost from log n to log² n.\n\nBun and Juvekar asked GPT-6 Astra how to go beyond that quadratic gap. The model returned arguments that appeared to prove a polynomial\ngap for halfspace problems. Working further with the model, the authors simplified the proof and pushed it to an exponential separation in\nsign rank, which they say is tight for the family.\n\n## Why it matters\n\nIt answers a structural question that has been open for about 38 years and that several recent surveys repeated. The authors' account is\nunusually direct: the core proof came from the model in response to a request for ideas, and the humans' work was to understand, verify,\nsimplify and strengthen it.","disputed":[{"label":"Verification","text":"Unrefereed arXiv preprint","sources":[]}],"related":[{"id":"2026-10-29-ai-assisted-results-october-29-listing-2","url":"https://postcutoff.com/e/2026-10-29-ai-assisted-results-october-29-listing-2/","date":"2026-10-29","date_precision":"day","short_title":"AI-assisted results, 29 Oct 2026 (part 2)","deck":"Claude 'largely carried out' Lipshitz's Khovanov research, ChatGPT found Soundararajan's proof and wrote Etingof's and Mathew's papers, George Elliott's free-product classification 'carried out using ChatGPT', and a first read of Hexagon's backlog","takeaway":"A second pass over the 28 Oct arXiv announcement and a first full read of Hexagon (hexagonmath.org), which the sweep had never covered, found results with large AI roles signed by senior mathematicians. Robert Lipshitz writes that his Khovanov paper's research 'was largely carried out by various Claude models'. Kannan Soundararajan's proof of a problem from Montgomery's CBMS book 'was found by ChatGPT 6'. ChatGPT 6 Pro wrote Pavel Etingof's answer to a 1999 question and produced Akhil Mathew's answer to a question of Drinfeld. George Elliott's classification of reduced free products of nuclear C*-algebras 'was carried out using ChatGPT'. Eight items have their own entries: the unforced 2D Euler non-uniqueness claims, the Erdős–Lovász covering-number bound, UPP^cc under intersection, Kisin's Serre conjecture, Linnik's constant ≤ 3.99, 3D Yang–Mills on a torus, random k-SAT and Chenglong Ma's Hovey–Strickland provenance claim.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":30,"official":29,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI","Anthropic","Google DeepMind","various researchers"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-29","type":"filed","text":"Created (arXiv PDF of 2610.37634)"}],"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":180,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":119,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":210,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":149,"in_training_data":false}],"short_url":null}