{"schema":"postcutoff/event@1","as_of":"2026-10-30T13:42:00+01:00","url":"https://postcutoff.com/e/2026-10-29-deterministic-factoring-n-o1-lean-claude-rudich-rousseau/","md":"https://postcutoff.com/e/2026-10-29-deterministic-factoring-n-o1-lean-claude-rudich-rousseau/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":"2026-10-29-deterministic-factoring-n-o1-lean-claude-rudich-rousseau","date":"2026-10-29","date_precision":"day","short_title":"Deterministic factoring in N^o(1) steps, Lean-checked, ideas by Claude","deck":null,"takeaway":"The best proven deterministic factoring algorithms ran in time N^(1/5+o(1)) (Harvey; Harvey–Hittmeir). On 29 Oct Isaac Rudich and Louis-Martin Rousseau (Polytechnique Montréal) posted on Hexagon a Lean 4 proof that one deterministic Turing machine finds the smallest prime factor of every N within C_k·2^(n/k) steps for every k, i.e. N^o(1). The 13-line statement uses only Mathlib and is registered in Palomar; the proof is about 36,000 lines. The authors say every idea and all Lean code came from Claude agents, which they asked to apply the ETH refutation to factoring; they audited the statement, not the proof. The algorithm is 'galactic' and slower than the randomised and heuristic methods used in practice; what is new is a proven deterministic N^o(1) bound.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"Hexagon 2610.00600: Deterministic Factoring in Time N^o(1) (Rudich, Rousseau)","url":"https://hexagonmath.org/abs/2610.00600v1","type":"paper","group":"primary","domain":"hexagonmath.org"},{"n":2,"title":"Palomar registry entry PALOMAR-2026-10-28-000015","url":"https://palomar-registry.org/entry?id=PALOMAR-2026-10-28-000015&version=1","type":"code","group":"primary","domain":"palomar-registry.org"}],"official":2,"filed":"2026-10-29","updated":"2026-10-29","orgs":["Anthropic","Polytechnique Montréal"],"title":"Deterministic integer factoring in time N^o(1), down from N^(1/5): a Lean certificate written by Claude agents from Anthropic's ETH-refutation counting engine, posted by Isaac Rudich and Louis-Martin Rousseau","summary":"\"Deterministic Factoring in Time N^o(1)\" (Hexagon 2610.00600, 29 Oct 2026; Isaac Rudich, Louis-Martin Rousseau) gives a Lean 4 certificate (Palomar PALOMAR-2026-10-28-000015) that a deterministic multitape Turing machine computes minFac(N) in C_k·2^(n/k) steps on n-bit inputs for every k ≥ 1, so full factorization takes N^o(1) time. The reduction counts divisors with prescribed leading bits using the counting engine of Anthropic's 2^o(n) #3-SAT paper, re-proved from scratch. The authors state that all ideas and code were produced by Claude agents and that they checked only the statement.","key_facts":["Claim (Theorem 1, Lean name subexponentialRate): one deterministic Turing machine with finite alphabets outputs minFac(N) and, for every k ≥ 1, halts within C_k·2^(n/k) steps on every n-bit input; running time 2^o(n) = N^o(1). Full factorization in N^o(1) follows (Theorem 2, not in Lean)","Before (paper): trial division N^(1/2), Lehman N^(1/3), Pollard–Strassen N^(1/4), Hittmeir N^(2/9), Harvey O(N^(1/5) log^(16/5) N), improved by Harvey–Hittmeir by a factor (log log N)^(3/5), 'the record for every N'. 'We know of no earlier deterministic bound of the form N^o(1), conditional or not.' Randomised (Lenstra–Pomerance L_N[1/2]) and heuristic (number field sieve L_N[1/3]) methods remain far faster","AI role (paper, verbatim): 'We asked AI to apply the result of the ETH paper to factoring, and then to conjecture how much the exponent could be easily pushed down. The first request produced Theorem 1. ... All the ideas were AI-generated: the reduction, the choice of method, the proof plans and all of the Lean code were produced by AI agents (Claude, Anthropic). Our role was to pose the questions and design the statement.' And: 'We have audited this result, not the proof.' Hexagon tools: Lean 4.35.0, Claude Opus 5.5","Lean: the audit surface is Challenge.lean (13 lines of statement, imports only Mathlib's Turing-machine, prime and log files); Solution.lean proves it; #print axioms gives propext, Classical.choice, Quot.sound; the machine model is Mathlib's TM2 with a finite-alphabet condition added for every stack. Proof about 36,000 lines (35,000 in the Hexagon abstract)","Uses Sections 3–5 of Anthropic's ETH refutation ('A 2^o(n) Time Algorithm for 3-SAT'), re-proved from scratch: the grouping lemma, the eight-square crossing identity and the exponent recursion; 'Nothing of [OpenAI's paper] enters the proof', and nothing is imported from either Lean development (Anthropic's runs to about 152,000 lines)","Conjecture 1 (stated in Lean as SqrtLogLogConjecture): factoring in time exp(O(log N/√(log log N)))","Context: written 'less than 24 hours after the ETH refutations were posted' as 'a stepping stone for those working to understand the engine'; a follow-up paper is to explain the proof. No expert reaction or HN thread found on 29 Oct"],"key_numbers":[],"tags":["computer-science","complexity","factoring","number-theory","eth","lean","palomar","hexagon","claude","claude-opus-5-5"],"science":{"field":"computer-science","subfield":"computational number theory / complexity","problem":"Best proven running time of deterministic integer factoring (record N^(1/5+o(1)), Harvey 2021; Harvey–Hittmeir)","result":"Lean-proved: deterministic factoring in time N^o(1) (C_k·2^(n/k) for every k)","open_since":"","ai_system":["Claude Opus 5.5"],"human_role":"Authors posed the question and designed and audited the 13-line Lean statement; Claude agents produced the reduction, method, proof plan and all Lean code; the authors did not read the proof","verification":{"key":"lean","label":"Lean-verified","text":"Lean 4 proof registered in Palomar (PALOMAR-2026-10-28-000015); unrefereed Hexagon preprint; no human-readable proof yet"},"status":"pending","shock":"Three days after the ETH refutations were posted, a Lean certificate claims the proven exponent for deterministic factoring drops from 1/5 to o(1), as a side application written by Claude."},"body_md":"## What happened\n\nFactoring is easy to verify and hard to do. With randomness or unproved heuristics, the number field sieve and its relatives are fast.\nWithout randomness and with full proofs, the record stood at about N^(1/5), from Harvey's 2021 algorithm and a small improvement by\nHarvey and Hittmeir.\n\nRudich and Rousseau asked Claude agents to apply the counting engine of the ETH refutations to factoring. The agents reduced factoring\nto counting the divisors of N with prescribed leading bits and counted them with the engine from Anthropic's #3-SAT paper, re-proved in\na separate Lean development. The result is a single deterministic machine that beats 2^(n/k) for every k. The authors checked the\n13-line statement and registered the proof in Palomar. They say they did not read the 36,000-line proof and that no human-readable\nexplanation of the engine exists yet.\n\n## Why it matters\n\nIf the certificate holds, it is the first concrete consequence of the claimed October ETH refutations outside SAT, and it moves a textbook bound in computational number\ntheory. It has no practical effect: by the authors' own comparison it is far slower than the randomised and heuristic methods already used\nto factor real numbers, so on its own it gives attackers of RSA nothing new. Its value is the short, checkable statement, which gives\nreaders a way into the engine.","disputed":[{"label":"Verification","text":"Lean 4 proof registered in Palomar (PALOMAR-2026-10-28-000015); unrefereed Hexagon preprint; no human-readable proof yet","sources":[]}],"related":[{"id":"2026-10-29-ai-assisted-results-october-29-listing-3","url":"https://postcutoff.com/e/2026-10-29-ai-assisted-results-october-29-listing-3/","date":"2026-10-29","date_precision":"day","short_title":"AI-assisted results, 29 Oct (part 3): Hexagon's evening batch","deck":null,"takeaway":"Hexagon published 58 new works on 28–29 Oct that we had not cited. Four have their own entries: the claimed sharp Berry–Esseen constant (Kołodziejek–Rodziewicz), the existence of the q-colourability threshold (Aronow–Lopatto, machine-written), deterministic factoring in N^o(1) (Rudich–Rousseau, Lean, ideas by Claude) and the refutation of Moschovakis's uniform-process lower bounds for gcd (Davies, Lean). Others claim the quantum Jellium = uniform-electron-gas energy (Csirik), C₂-cofiniteness of all finite-group orbifolds (Riesen), analytic floating bodies of equilibrium in every dimension ≥ 3 (Price–de Groot, GPT-6 Pro) and Cowling's conjecture for almost connected groups (Arhancet). From the backlog: S(13) = 45 comparators for sorting 13 inputs and an area law for 3D U(1) Wilson lattice gauge theory (Tomar, Lean), Enomoto's settlement of the third Brauer–Thrall conjecture over algebraically closed fields (GPT-6 Astra) and Soberón's counterexample to Matschke's table conjecture.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":22,"official":22,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI","Anthropic","Google","various researchers"]},{"id":"2026-10-27-anthropic-claude-eth-refutation-3sat-subexponential","url":"https://postcutoff.com/e/2026-10-27-anthropic-claude-eth-refutation-3sat-subexponential/","date":"2026-10-27","date_precision":"day","short_title":"Anthropic's Claude also refutes ETH","deck":null,"takeaway":"Hours after OpenAI's ETH announcement, Anthropic posted 'A 2^o(n) Time Algorithm for 3-SAT', credited to Claude. It is a deterministic #k-SAT algorithm built on tensor-network contraction, with a different approach from OpenAI's paper, a near-identical bound and a Lean 4 proof that uses only the standard axioms. The paper says the core algorithm was found on 16 Oct with 14M output tokens and no human input. OpenAI reports its model found its disproof that same day. Neither proof has had expert review.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":7,"official":6,"filed":"2026-10-27","updated":"2026-10-29","orgs":["Anthropic"]},{"id":"2026-10-26-openai-eth-false-subexponential-sat-claim","url":"https://postcutoff.com/e/2026-10-26-openai-eth-false-subexponential-sat-claim/","date":"2026-10-26","date_precision":"day","short_title":"OpenAI claims ETH is false","deck":null,"takeaway":"Anthropic independently posted a different Lean-checked refutation the next day (separate entry). OpenAI's model output claims a deterministic algorithm that counts the satisfying assignments of any k-CNF in time 2^(n/(log n)^(1/2+o(1))). If correct, it refutes ETH, SETH, Gap-ETH and NSETH, the assumptions behind hundreds of fine-grained hardness results. OpenAI says the main theorem is formalized in Lean. A working group led by ETH's proposers, Russell Impagliazzo and Ramamohan Paturi, with Ryan Williams, will examine the proof. No expert has assessed it in public yet.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":32,"official":10,"filed":"2026-10-26","updated":"2026-10-30","orgs":["OpenAI"]},{"id":"2026-08-18-palomar-lean-registry","url":"https://postcutoff.com/e/2026-08-18-palomar-lean-registry/","date":"2026-08-18","date_precision":"day","short_title":"Palomar launches: a registry of Lean-verified mathematics to curb misrepresented AI proof claims","deck":null,"takeaway":"Formal verification became the main way to trust AI mathematics in 2026.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":6,"official":6,"filed":"2026-09-29","updated":"2026-10-29","orgs":["Lean FRO","ICARM"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-29","type":"edit","text":"Coordinator spot-check against the Hexagon API record (title, abstract, 13-line statement, ~35,000-line proof match); wording made neutral about the still unverified ETH refutations"},{"date":"2026-10-29","type":"filed","text":"Created (Hexagon API record and PDF of 2610.00600; HN search found no thread)"}],"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":182,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":121,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":212,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":151,"in_training_data":false}],"short_url":"https://postcutoff.com/s/deterministic-factoring-n-o"}