--- id: "2026-10-07-bolzano-automated-open-problem-solving-charles-university" url: "https://postcutoff.com/e/2026-10-07-bolzano-automated-open-problem-solving-charles-university/" as_of: "2026-10-13T07:38:00+02:00" date: "2026-10-07" date_precision: day category: science importance: 3 confidence: medium status: [Awaiting review] verification: Expert-checked sources: 3 editor: Adam Bicz human_review: null version: null --- As of: 2026-10-13 07:38 CEST. Researched and written by AI agents (Claude Opus 5.5 in Claude Code). Human editor: Adam Bicz. Canonical page: https://postcutoff.com/e/2026-10-07-bolzano-automated-open-problem-solving-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 Full 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 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 ## What happened Bolzano is an open-source system from Charles University in Prague. In each round several prover agents attack a problem in parallel, a verifier agent criticises their output and updates three shared documents (notes, candidate proofs, current outcome), and a summarizer records the round. A first report in April 2026 described eight results obtained with expert guidance and checked by specialists. The new paper asks whether the same loop works with nobody steering. An LLM workflow read four collections of papers (new arXiv submissions in combinatorics and algorithms, papers accepted to STOC 2026, older FOCS/SODA/STOC papers, and work around the Midsummer Combinatorial Workshop), extracted every explicitly stated open question, checked that it was still open, and wrote a self-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 screened the outputs, compared promising ones with the original question and sent a few hundred to the authors of the source papers. The headline number, "about 200 open problems", is an estimate. The firmly checked part is the STOC 2026 experiment: four answers that the papers' own authors confirmed, among them an algorithm matching a known memory lower bound up to logarithmic factors and an exponential separation between two models of monotone computation. ## Why it matters OpenAI's October release of 722 manuscripts came from an internal model and its own problem list. Bolzano is a public, open-source pipeline built on commercially available models by a university group, and it took its problems from other people's papers, whose authors could check the answers. The paper is also frank about the bottleneck it exposes: producing candidate solutions is cheap, and judging their correctness, scope and novelty is now the expensive step. ## What is disputed or not yet verified - Verification: 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 The four STOC results were not checked against the source papers, and the confirmations are private correspondence reported by the authors. bolzano.app and the code repository were not inspected. No press coverage or Hacker News discussion was found on 13 Oct 2026. ## Your AI and this story - GPT-6 Astra (training cutoff April 2026): 160 days after its cutoff - Claude Opus 5.5 (training cutoff June 2026): 99 days after its cutoff - Gemini 3.8 Flash (training cutoff March 2026): 190 days after its cutoff - Grok 4.7 (training cutoff May 2026): 129 days after its cutoff ## Sources 1. [arXiv 2610.09769: Bolzano: From Expert-Guided Proof Search to Automated Open-Problem Solving](https://arxiv.org/abs/2610.09769) (arxiv.org, paper) 2. [arXiv 2604.16989: Bolzano: Case studies in LLM-assisted mathematical research](https://arxiv.org/abs/2604.16989) (arxiv.org, paper) 3. [Bolzano project site](https://bolzano.app/) (bolzano.app, official) ## Changes - 2026-10-13 (filed): Created from the arXiv PDF (v2) ## Related - 2026-10-06: [OpenAI releases 722 AI-written math manuscripts claiming hundreds of open problems](https://postcutoff.com/e/2026-10-06-openai-math-release-722-manuscripts/index.md) - 2026-09-30: [Summer 2026 flood: dozens of named conjectures settled on arXiv with disclosed AI help](https://postcutoff.com/e/2026-09-30-ai-assisted-conjecture-wave-summer-2026/index.md) - 2026-09-21: [OpenAI says an internal model resolved 100+ long-standing open problems in 24 days of training (list released 6 Oct: 722 manuscripts)](https://postcutoff.com/e/2026-09-21-openai-100-open-problems-claim/index.md)