{"schema":"postcutoff/event@1","as_of":"2026-10-13T07:38:00+02:00","url":"https://postcutoff.com/e/2026-10-07-bolzano-automated-open-problem-solving-charles-university/","md":"https://postcutoff.com/e/2026-10-07-bolzano-automated-open-problem-solving-charles-university/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-bolzano-automated-open-problem-solving-charles-university","date":"2026-10-07","date_precision":"day","short_title":"Bolzano (Charles University)","deck":"Open-source multi-agent system run unattended on about 3,800 open problems from research papers; about 200 reported solved, four STOC 2026 questions confirmed by the papers' authors","takeaway":"A Prague group pointed an open-source prover–verifier loop built on GPT-5.5 and GPT-5.6 Sol at every open question it could extract from recent papers, with no per-problem human guidance. It reports about 214 solutions out of 3,800 attempts, most of them estimates, and four results that the authors of STOC 2026 papers confirmed.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2610.09769: Bolzano: From Expert-Guided Proof Search to Automated Open-Problem Solving","url":"https://arxiv.org/abs/2610.09769","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"arXiv 2604.16989: Bolzano: Case studies in LLM-assisted mathematical research","url":"https://arxiv.org/abs/2604.16989","type":"paper","group":"primary","domain":"arxiv.org"},{"n":3,"title":"Bolzano project site","url":"https://bolzano.app/","type":"official","group":"primary","domain":"bolzano.app"}],"official":3,"filed":"2026-10-13","updated":"2026-10-13","orgs":["Charles University","Czech Academy of Sciences"],"title":"Bolzano (Charles University): open-source multi-agent system run unattended on about 3,800 open problems from research papers; about 200 reported solved, four STOC 2026 questions confirmed by the papers' authors","summary":"On 7 Oct 2026 Adrián Zámečník, Matěj Kripner, Martin Koutecký, Martin Balko, Jan Grebík, Pavel Hubáček, Robert Šámal and Václav Rozhoň (Charles University, Prague) posted arXiv 2610.09769, accepted at the MATH-AI workshop of NeurIPS 2026. Bolzano is \"an open-source orchestration layer around LLMs\": parallel prover agents, a verifier agent and a shared research record. After eight expert-checked results in an April 2026 report, the team ran it \"without problem-specific human guidance on about 3,800 open problems extracted from four sets of papers, solving about 200 open problems\". For papers accepted to STOC 2026 \"we answered four questions raised in the papers, as confirmed by their authors\". The solved counts outside STOC are the team's estimates, partly based on LLM assessments.","key_facts":["Paper: 'Bolzano: From Expert-Guided Proof Search to Automated Open-Problem Solving', arXiv 2610.09769 [cs.AI], 7 Oct 2026 (v2 9 Oct); 'Accepted at the 6th Workshop on Mathematical Reasoning and AI (MATH-AI), NeurIPS 2026'","Table 1 (problems / model / rounds / promising / solved): arXiv math.CO and cs.DS 1,600 / GPT-5.5 / 4 / 250 / 90; STOC 2026 420 / GPT-5.5 / 4 / 41 / 4; earlier FOCS, SODA and STOC 900 / GPT-5.6 Sol / 2 / 57 / 40; Midsummer Combinatorial Workshop 880 / GPT-5.6 Sol / 2 / 137 / 80; total 3,800 / 485 / 214","Caveat (Table 1 footnote): 'Except for STOC 2026, the solved counts are estimates. These estimates are partially informed by LLM-based assessments of whether the output addresses the source question and whether its argument is substantially correct.' For STOC 2026 only the four results checked by the team and verified by the source papers' authors are counted","Setup: 'the prover, verifier, and summarizer operated without web search, code execution, direct filesystem access, or any tools'; all runs used one prover at the maximum reasoning effort; open problems were extracted from the papers by a separate LLM workflow with literature search","STOC 2026 results, confirmed 'in private correspondence in June and July 2026': a one-pass O(log n)-memory algorithm detecting a semi-random planted biclique with sides at least Cn^(2/3) (question from Garg, Hastings, Pabbaraju and Sharan); an exponential separation between monotone rank programs and monotone span programs (Cavalar et al.); a strongly convex rational quartic in two variables whose zero sublevel set is a single irrational point (Slot, Steurer and Wiedmer); an Ω(log n) circuit-depth lower bound for joining and splitting biased CAT states (Parham)","Footnote: for the convex-quartic and CAT-state results 'the authors reviewed only the main idea, not the full proof'","Paper: 'Bolzano found a few hundred solutions, which we sent to authors of the source papers, some of whom plan to include these results in revised versions of their papers.'","Limits stated by the authors: 'The verifier is another LLM, not a formal proof checker. An accepted but incorrect lemma may propagate through persistent memory'; observed failures 'include missing assumptions, proofs addressing weaker versions of the original questions, invalid arguments and rediscoveries of known results'; there is 'no comparison with direct use of the underlying models'","Earlier report: 'Bolzano: Case studies in LLM-assisted mathematical research', arXiv 2604.16989 (eight problems checked by domain experts; six classed as publishable research and five as essentially autonomous). Project site: bolzano.app"],"key_numbers":[],"tags":["math","theoretical-computer-science","combinatorics","automated-research","multi-agent","open-problems","gpt-5-5","gpt-5-6-sol","open-source","neurips-2026"],"science":{"field":"computer-science","subfield":"theoretical computer science and combinatorics (automated research)","problem":"Can an LLM research loop, run without per-problem human steering, answer the open questions stated in current research papers?","result":"About 3,800 extracted open problems attempted; 485 outputs flagged as promising and about 214 counted as solved (estimate); four answers to questions from STOC 2026 papers confirmed by those papers' authors.","open_since":"","ai_system":["Bolzano","GPT-5.5","GPT-5.6 Sol"],"human_role":"Autonomous solving (no problem-specific guidance); humans built the pipeline, screened flagged outputs and sent results to the source papers' authors","verification":{"key":"expert","label":"Expert-checked","text":"Four STOC 2026 results confirmed by source-paper authors in private correspondence (two only at the level of the main idea); the other counts are estimates partly based on LLM audits; workshop paper"},"status":"pending","shock":"The unit of work is no longer a problem but a conference: every open question in the accepted papers of STOC 2026 was attempted.","short":"Bolzano: unattended open-problem solving at scale"},"body_md":"## What happened\n\nBolzano is an open-source system from Charles University in Prague. In each round several prover agents attack a problem in\nparallel, a verifier agent criticises their output and updates three shared documents (notes, candidate proofs, current outcome),\nand a summarizer records the round. A first report in April 2026 described eight results obtained with expert guidance and checked\nby specialists.\n\nThe new paper asks whether the same loop works with nobody steering. An LLM workflow read four collections of papers (new arXiv\nsubmissions in combinatorics and algorithms, papers accepted to STOC 2026, older FOCS/SODA/STOC papers, and work around the\nMidsummer Combinatorial Workshop), extracted every explicitly stated open question, checked that it was still open, and wrote a\nself-contained task. Each task got a fixed budget of two or four rounds with GPT-5.5 or GPT-5.6 Sol and no tools. The team then\nscreened the outputs, compared promising ones with the original question and sent a few hundred to the authors of the source papers.\n\nThe headline number, \"about 200 open problems\", is an estimate. The firmly checked part is the STOC 2026 experiment: four answers\nthat the papers' own authors confirmed, among them an algorithm matching a known memory lower bound up to logarithmic factors and\nan exponential separation between two models of monotone computation.\n\n## Why it matters\n\nOpenAI's October release of 722 manuscripts came from an internal model and its own problem list. Bolzano is a public,\nopen-source pipeline built on commercially available models by a university group, and it took its problems from other people's\npapers, whose authors could check the answers. The paper is also frank about the bottleneck it exposes: producing candidate\nsolutions is cheap, and judging their correctness, scope and novelty is now the expensive step.","disputed":[{"label":"Verification","text":"Four STOC 2026 results confirmed by source-paper authors in private correspondence (two only at the level of the main idea); the other counts are estimates partly based on LLM audits; workshop paper","sources":[]}],"related":[{"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":80,"official":13,"filed":"2026-10-07","updated":"2026-10-13","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":"2026-10-13","orgs":["OpenAI","Anthropic","Google DeepMind","various mathematicians"]},{"id":"2026-09-21-openai-100-open-problems-claim","url":"https://postcutoff.com/e/2026-09-21-openai-100-open-problems-claim/","date":"2026-09-21","date_precision":"day","short_title":"OpenAI says an internal model resolved 100+ long-standing open problems in 24 days of training (list released 6 Oct: 722 manuscripts)","deck":null,"takeaway":"If substantiated, it would mean open problems are being resolved at industrial scale.","category":"science","category_label":"Science & math","importance":3,"confidence":"low","status":{"key":"pending","labels":["Awaiting review"]},"sources":11,"official":5,"filed":"2026-09-29","updated":"2026-10-10","orgs":["OpenAI"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-13","type":"filed","text":"Created from the arXiv PDF (v2)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-13","run":null,"sources_read":"The arXiv PDF (v2)","updated":"2026-10-13","human_review":null,"version":null},"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":null}