{"schema":"postcutoff/event@1","as_of":"2026-10-30T13:42:00+01:00","url":"https://postcutoff.com/e/2026-10-23-tokareva-bent-functions-conjecture-disproved-minimax-agents/","md":"https://postcutoff.com/e/2026-10-23-tokareva-bent-functions-conjecture-disproved-minimax-agents/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":"2026-10-23-tokareva-bent-functions-conjecture-disproved-minimax-agents","date":"2026-10-23","date_precision":"day","short_title":"Tokareva's bent-function conjecture disproved by LLM agents","deck":null,"takeaway":"Every Boolean function of degree ≤ n/2 is a sum of two bent functions: that conjecture of Natalia Tokareva, listed among Carlet's open problems, fails in every even dimension n ≥ 8. The counterexample x₁x₂(x₃x₄+x₅x₆+x₇x₈) was found by 19 rounds of up to 100 MiniMax M3.1-Flash-Preview agents (1,889 reports) with separate audit rounds. Two key equations are checked in Lean.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2610.32239: A counterexample to Tokareva's conjecture on sums of two bent functions","url":"https://arxiv.org/abs/2610.32239","type":"paper","group":"primary","domain":"arxiv.org"}],"official":1,"filed":"2026-10-26","updated":"2026-10-26","orgs":["MiniMax","NSUCRYPTO"],"title":"Tokareva's conjecture on sums of two bent functions disproved by swarms of MiniMax M3.1 Flash agents during the NSUCRYPTO 2026 olympiad","summary":"In arXiv 2610.32239 (23 Oct 2026) Yansong Feng, Hao Jiang, Guancheng Li, Tianyi Li and Lili Tang disprove Tokareva's conjecture. They show that x₁x₂(x₃x₄+x₅x₆+x₇x₈) on F₂⁸ is not a sum of two bent functions, and extend the construction to every even n ≥ 8 and, via Shaporenko's reduction, to Tokareva's derivative conjecture for n ≥ 10. The paper says the counterexamples and proofs \"were obtained through large-scale automated reasoning with LLM agents while solving Problem 10 in NSUCRYPTO 2026\", using MiniMax M3.1-Flash-Preview. The authors checked every statement and formalized Equations 4.2 and 4.7 in Lean 4.","key_facts":["Conjecture (Tokareva; Carlet's 'Open Problem 2'): for even n, every Boolean function on F₂ⁿ of algebraic degree ≤ n/2 is a sum of two bent functions. Previously verified for even n ≤ 6 and some special classes. A positive answer would have closed the gap in asymptotic order between bounds on the number of bent functions","Result: f₈ = x₁x₂(x₃x₄+x₅x₆+x₇x₈) ∉ B₈+B₈; counterexamples in every even dimension n ≥ 8; explicit counterexamples to the derivative conjecture for even n ≥ 10","AI statement (verbatim): 'The counterexamples and proofs in this paper were obtained through large-scale automated reasoning with LLM agents while solving Problem 10 in NSUCRYPTO 2026 (MiniMax M3.1-Flash-Preview, 19 rounds of up to 100 independent agents between 12 and 18 October 2026, 1889 reports in all), paired with 8 rounds of independent audits (227 reports).'","Audit caught an error (verbatim): 'an early attempt to prove that f8 ∉ B8+B8 was discarded during the audit phase because it mistakenly treated the Walsh transform as additive. Another agent then established Equation (3.1) and confirmed that f8 is indeed a counterexample'","In the first 10 rounds the agents decomposed every cubic on F₂⁸ and the first quartic candidate into two bent functions, which narrowed the search","Verification: 'Every statement in this paper was subsequently checked by the authors. We also provide a formal verification for Equation (4.2) and Equation (4.7) in Lean 4.'","Context: NSUCRYPTO is the international cryptography olympiad of Novosibirsk State University; its 'unsolved problems' track poses open research problems. A small, fast model from MiniMax (a Chinese lab) did the search, not a frontier model"],"key_numbers":[],"tags":["math","cryptography","boolean-functions","bent-functions","counterexample","agents","minimax","open-weights","lean","arxiv"],"science":{"field":"mathematics","subfield":"Boolean functions / cryptography","problem":"Tokareva's conjecture on sums of two bent functions (and her derivative conjecture)","result":"Disproved for every even n ≥ 8 (derivative conjecture for even n ≥ 10) with an explicit quartic counterexample","open_since":"","ai_system":["MiniMax M3.1-Flash-Preview"],"human_role":"Agents found the counterexamples and proofs during an olympiad; humans checked every statement and formalized two key equations in Lean","verification":{"key":"preprint","label":"Preprint only","text":"Unrefereed arXiv preprint; partial Lean 4 formalization (Eq. 4.2, 4.7)"},"status":"pending","shock":""},"body_md":"## What happened\n\nBent functions are Boolean functions that are as far from linear as possible, which makes them useful in cryptography. Rothaus\nshowed that they have degree at most n/2, so a sum of two of them does too. Natalia Tokareva asked whether every function of\ndegree at most n/2 arises this way. Claude Carlet recorded the question as an open problem.\n\nA team of five posted a no answer on 23 Oct. They worked on it as Problem 10 of the 2026 NSUCRYPTO olympiad with a swarm of\nMiniMax M3.1-Flash-Preview agents. Over a week, the agents ran 19 rounds with up to 100 independent instances each, and eight\nfurther rounds audited their output. One early \"proof\" was thrown out because it treated the Walsh transform as additive. A\nlater agent proved the needed identity, and the authors generalised the example to all even dimensions.\n\n## Why it matters\n\nThis is a disproof of a named conjecture in cryptographic Boolean-function theory, made by many runs of a cheap model with a\nseparate audit stage rather than by one frontier model. The record notes that the audit stage caught a false proof.","disputed":[{"label":"Verification","text":"Unrefereed arXiv preprint; partial Lean 4 formalization (Eq. 4.2, 4.7)","sources":[]}],"related":[{"id":"2026-10-26-ai-assisted-results-october-26-listing","url":"https://postcutoff.com/e/2026-10-26-ai-assisted-results-october-26-listing/","date":"2026-10-26","date_precision":"day","short_title":"AI-assisted results in the 26 Oct 2026 sweep","deck":"Claude-written Lean proof of a frame-indifferent rank-one convex function that is not quasiconvex, a sculptor and Claude Code prove Takemura's conjecture, Claude's contribution to deterministic root-finding, and Zenodo records","takeaway":"Monday's arXiv announcement (papers submitted 22–23 Oct) brought four results with their own entries: OpenAI's propeller conjecture reproved by Heilman, Tokareva's conjecture disproved by MiniMax agents, the triangle lower tail with GPT-6 Astra and Claude, and graph-connectivity communication complexity with ChatGPT. This page lists the next tier, from Lean proofs written entirely by Claude to papers where the AI only checked the text.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":22,"official":22,"filed":"2026-10-26","updated":"2026-10-26","orgs":["Anthropic","OpenAI","various researchers"]},{"id":"2026-10-19-minimax-m3-1-flash","url":"https://postcutoff.com/e/2026-10-19-minimax-m3-1-flash/","date":"2026-10-19","date_precision":"day","short_title":"MiniMax releases M3.1 Flash, revealed as OpenRouter's stealth 'Space Bunny'","deck":"$0.15/$0.60 per 1M tokens, DeepSWE 73.7%, weights due Oct 23","takeaway":"It continues the Chinese price war on agent models: on the same weekend DeepSeek shipped V4.1-Flash-1017 under MIT, and M3.1 Flash undercuts MiniMax's own M3 by two-thirds.","category":"model-release","category_label":"Model releases","importance":3,"confidence":"high","status":{"key":"confirmed","labels":["Confirmed"]},"sources":5,"official":3,"filed":"2026-10-19","updated":"2026-10-23","orgs":["MiniMax"]},{"id":"2026-10-13-ai-assisted-results-october-2026","url":"https://postcutoff.com/e/2026-10-13-ai-assisted-results-october-2026/","date":"2026-10-13","date_precision":"day","short_title":"October 2026: the arXiv stream of conjectures settled with disclosed AI help continues","deck":null,"takeaway":"In the first nine days of October 2026 the stream did not slow down: each arXiv listing brought several named conjectures with a disclosed AI role, and two new patterns appeared.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":17,"official":17,"filed":"2026-10-13","updated":"2026-10-29","orgs":["OpenAI","Anthropic","Google DeepMind","various mathematicians"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-26","type":"filed","text":"Created (arXiv PDF read)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-26","run":null,"sources_read":"ArXiv PDF read","updated":"2026-10-26","human_review":null,"version":{"date":"2026-10-27"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":176,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":115,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":206,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":145,"in_training_data":false}],"short_url":null}