{"schema":"postcutoff/event@1","as_of":"2026-10-13T07:38:00+02:00","url":"https://postcutoff.com/e/2026-10-06-multiplier-associativity-hard-for-resolution-gpt-5-6-sol/","md":"https://postcutoff.com/e/2026-10-06-multiplier-associativity-hard-for-resolution-gpt-5-6-sol/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-multiplier-associativity-hard-for-resolution-gpt-5-6-sol","date":"2026-10-06","date_precision":"day","short_title":"Why SAT solvers fail on multiplication","deck":"First exponential resolution lower bound for multiplier associativity; 'main ideas and the core proof … are due to GPT-5.6-Sol' (Liew)","takeaway":"SAT solvers are known to choke on multiplication, but nobody had proved that short proofs do not exist. Vincent Liew's preprint proves an exponential lower bound for resolution proofs that n-bit multiplication is associative, resolving an open problem he had posed with Paul Beame. He credits the main ideas and the core proof to GPT-5.6 Sol; a Lean 4 formalization was generated by Claude Fable 5.1.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2610.08988: Associativity of Multiplication Is Hard for Resolution","url":"https://arxiv.org/abs/2610.08988","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"Lean 4 formalization (dysfunctor/associativity)","url":"https://github.com/dysfunctor/associativity","type":"code","group":"primary","domain":"github.com"}],"official":2,"filed":"2026-10-13","updated":"2026-10-13","orgs":["OpenAI","Anthropic"],"title":"Why SAT solvers fail on multiplication: first exponential resolution lower bound for multiplier associativity; 'main ideas and the core proof … are due to GPT-5.6-Sol' (Liew)","summary":"On 6 Oct 2026 Vincent Liew posted arXiv 2610.08988, \"Associativity of Multiplication Is Hard for Resolution\". It shows that general resolution proofs of the associativity of n-bit multiplication need size 2^Ω((n/log n)^(1/4)), for a broad class of multiplier encodings that includes the array and Wallace-tree multipliers used by SMT solvers. The abstract calls it the first lower bound of this kind and says it \"resolves an open problem of Beame and Liew\". The paper's \"AI Methodology\" section states: \"The main ideas and the core proof of the associativity lower bound are due to GPT-5.6-Sol.\" Claude Fable 5 helped rewrite the proofs and Claude Fable 5.1 generated a Lean 4 formalization of the main theorems. Unrefereed preprint.","key_facts":["Paper: 'Associativity of Multiplication Is Hard for Resolution', arXiv 2610.08988 [cs.CC], 6 Oct 2026, 33 pages; Vincent Liew","Result: resolution proofs of associativity of n-bit multiplication 'require size 2^Ω((n/log n)^(1/4))'; the bound 'holds for a broad class of multiplier encodings based on partial product summation, including the standard array and Wallace-tree multipliers used to bit-blast multiplication in SMT solvers'","Motivation (abstract): 'SAT solvers are empirically known to perform poorly when reasoning about multiplication. Yet for over a decade we have lacked a theoretical explanation for this phenomenon. CDCL SAT solvers implicitly search for resolution proofs, and no lower bound on proof size has ruled out the existence of short proofs that solvers simply fail to find.'","Method: a reduction from a perfect-matching principle on bounded-degree bipartite expander graphs, which Itsykson, Slabodkin and Sokolov proved hard for resolution, to multiplier associativity","Extension: combined with Håstad's recent lower bound for the perfect-matching principle of the odd grid, the same reduction 'yields an exponential lower bound for multiplier associativity in the stronger bounded-depth Frege proof systems'","AI Methodology: 'The main ideas and the core proof of the associativity lower bound are due to GPT-5.6-Sol. This materially affected Sections 5, 7, 8, and 9, and Appendix A. During the verification process, the author reformulated and rewrote the definitions, lemma statements, and proofs with the assistance of Claude Fable 5, and added attributions along the way.'","Also: 'The Lean formalization (Appendix B) was generated by Claude Fable 5.1.' and 'The author verified the correctness and originality of all content, including references, and takes full responsibility for the paper's content.'","Caveat on the Lean proof (paper): the formalization includes its own definitions of the circuits; 'While we have inspected these definitions ourselves, we caution that they are a potential source of error.'","Lean 4 code: github.com/dysfunctor/associativity"],"key_numbers":[],"tags":["computer-science","proof-complexity","sat-solvers","resolution","lower-bounds","multiplication","gpt-5-6-sol","claude-fable-5-1","lean","ai-generated-proof"],"science":{"field":"computer-science","subfield":"proof complexity (SAT solving)","problem":"Do short resolution proofs exist for the associativity of n-bit multiplication (open problem of Beame and Liew), i.e. is the poor performance of SAT solvers on multiplication unavoidable?","result":"Claimed: every resolution proof has size 2^Ω((n/log n)^(1/4)) for partial-product multiplier encodings; an exponential bound also for bounded-depth Frege.","open_since":"","ai_system":["GPT-5.6 Sol","Claude Fable 5","Claude Fable 5.1"],"human_role":"AI-generated core per the author (main ideas and core proof by GPT-5.6 Sol); the author verified, rewrote with Claude Fable 5 and wrote the exposition","verification":{"key":"lean","label":"Lean-verified","text":"Lean 4 formalization of the main theorems generated by Claude Fable 5.1 (circuit definitions inspected by the author, flagged as a possible source of error); unrefereed preprint"},"status":"pending","shock":"The author of the open problem reports that a model supplied the idea and the core proof of its solution.","short":"Multiplier associativity is hard for resolution"},"body_md":"## What happened\n\nModern SAT solvers (CDCL solvers) can be seen as searching for proofs in a simple system called resolution. They are fast on many\nhardware-verification tasks but notoriously slow on anything involving multiplication, such as checking that two multiplier\ncircuits compute the same function or that (xy)z = x(yz) for n-bit numbers. It was not known whether this is a weakness of the\nsolvers' search or a real limit, meaning that short resolution proofs do not exist at all.\n\nLiew's preprint proves the second for associativity: any resolution proof has size exponential in roughly the fourth root of n.\nThe proof embeds a perfect-matching principle on expander graphs, already known to be hard for resolution, into the associativity\nstatement. With a recent result of Håstad the argument extends to bounded-depth Frege systems, which are stronger than resolution.\n\nThe paper says that the main ideas and the core proof come from GPT-5.6 Sol, that the author checked and rewrote them with\nClaude Fable 5, and that Claude Fable 5.1 produced a Lean 4 formalization of the main theorems.\n\n## Why it matters\n\nIt gives the first theoretical explanation for an effect that practitioners in verification have worked around for more than a\ndecade, on a question posed by specialists in proof complexity. It is also a clear instance of the 2026 pattern in theoretical\ncomputer science: the model supplies the reduction, a second model formalizes it, and the human author's work is verification\nand exposition.","disputed":[{"label":"Verification","text":"Lean 4 formalization of the main theorems generated by Claude Fable 5.1 (circuit definitions inspected by the author, flagged as a possible source of error); unrefereed preprint","sources":[]}],"related":[{"id":"2026-10-08-rank-3x3-matrix-multiplication-f2-is-23-lean-tomar","url":"https://postcutoff.com/e/2026-10-08-rank-3x3-matrix-multiplication-f2-is-23-lean-tomar/","date":"2026-10-08","date_precision":"day","short_title":"Laderman's 23 multiplications are optimal for 3×3 matrices over F_2 and over the integers","deck":"AI-found proof checked in Lean (Tomar)","takeaway":"Whether 3×3 matrices can be multiplied with fewer than Laderman's 23 multiplications has been open since 1976. A preprint claims the answer is no for coefficients in F_2 or the integers, ending a 2026 race in which the lower bound rose from 19 to 20, 21, 22 and now 23. The author says generative AI found and proved the result.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":8,"official":8,"filed":"2026-10-12","updated":"2026-10-13","orgs":["independent"]},{"id":"2026-10-06-random-3-sat-threshold-upper-bound-4-268-llm","url":"https://postcutoff.com/e/2026-10-06-random-3-sat-threshold-upper-bound-4-268-llm/","date":"2026-10-06","date_precision":"day","short_title":"Preprint claims a 4.268 upper bound for the random 3-SAT threshold, down from 4.4898, with a Lean 4 proof; 'large language models drove the research'","deck":null,"takeaway":"If it holds, this closes almost all of the gap between what is proved and what physics predicts for one of the best-known problems on random constraint satisfaction.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-13","updated":"2026-10-13","orgs":["Independent"]},{"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"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-13","type":"filed","text":"Created from the arXiv PDF"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-13","run":null,"sources_read":"The arXiv PDF","updated":"2026-10-13","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":null}