Search: “mathematical-proof”

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Unsolved Problem by Fields Medalist Breached by Two High School Students

Two high school students used Claude Opus 5 and GPT-5.6 Sol to help solve an open Lorentzian polynomials problem, posting a 75-page arXiv proof.

Aayush Bathija and Prince Rohatgi of Oak Park High School, mentored by UCLA postdoc Daniel Soskin, published the 75-page paper 'Bounded Ratios for Lorentzian Polynomials' (arXiv 2609.05341), solving an open problem in Fields Medalist June Huh's Lorentzian polynomial theory. The main structural theorem extends bounded coefficient-ratio characterization from quadratic to arbitrary-degree polynomials via discrete convexity conditions. The students used Claude Opus 5 and GPT-5.6 Sol for exploration and proof ideas but independently verified all arguments; the result follows an open letter from 25 Fields Medalists voicing concerns about AI's impact on mathematical rigor.

Hacker News · security·Sep 14, 12:23 UTC · 2d agoAI researchHN 22↑ · 19 comments

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MathKernel: An evidence-aware multi-engine mathematics kernel and MCP server

MathKernel is an open-source, evidence-aware multi-engine mathematics kernel that exposes verification workflows to AI agents via an MCP server.

MathKernel, published on GitHub, is a mathematics kernel that combines multiple computation engines with evidence-aware outputs. It ships as an MCP server, enabling AI agents and coding assistants to perform and verify calculations. The project drew moderate attention on Hacker News.

Hacker News · security·Sep 7, 00:21 UTC · 9d agoAI tools & infraHN 36↑ · 5 comments

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Characterizing Language Generation in the Limit: Finite Witnesses and a Separation-Width Hierarch

New work characterizes language generation in the limit via finite witnesses, proves a full separation-width hierarchy, and formalizes all results in Lean.

The paper fully characterizes when language generation in the limit is possible for arbitrary families over a countable universe: each target must admit a finite positive witness such that targets activated by any finite sample share an infinite common intersection. It defines positive separation width and proves every level of the resulting hierarchy occurs, with countable families admitting singleton witnesses and unions of families with infinite common cores requiring unbounded finite witnesses. The characterization, a universal normalization, and a diagonal capture lemma are machine-checked in the Lean proof assistant, with the development maintained on GitHub.

arXiv cs.AI / cs.LG / cs.CL·Sep 9, 17:54 UTC · 6d agoAI research1

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Controversy over OpenAI's Maths Breakthrough

OpenAI claims its internal model proved the Navier-Stokes equations 'blow up' — a Millennium Prize Problem — amid allegations it borrowed mathematicians' methods.

OpenAI announced that an internal model produced a proof, certified in the Lean proof assistant, showing the Navier-Stokes equations can 'blow up,' implying infinite fluid speeds — a claimed solution to one of the seven $1-million Millennium Prize Problems. Mathematician Tristan Buckmaster alleged OpenAI, after learning of progress by him and Anthropic employee Levent Alpöge on 'blowing up' the related Euler equations, adopted a similar 'forcing' method; OpenAI's Sébastien Bubeck denied this, saying the model independently solved Euler by different means and produced the full Navier-Stokes proof over one weekend. Mathematicians including Diego Córdoba, co-developer of the forcing approach, remain cautious, and the community is still evaluating the competing proofs.

Hacker News · security·Sep 8, 16:57 UTC · 7d agoAI researchHN 21↑ · 5 comments1

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Understanding the Usability of Cryptographic Verification Tools

Survey of Tamarin and ProVerif users reveals usability barriers: debugging non-termination, model validation, and opaque proof failures hinder cryptographic protocol verification.

The paper presents an exploratory human-centered survey of researchers, graduate students, and practitioners with hands-on experience using Tamarin, ProVerif, and related cryptographic protocol verification tools. Findings reveal usability barriers across the verification workflow, including difficulties debugging non-termination and performance issues plus the lack of systematic methods for validating formal models against real protocols. When proofs fail without concrete attacks, users commonly simplify models, add helper lemmas, and revisit modeling abstractions. Participants called for actionable diagnostics, clearer explanations of results, visualization, and automation for recurring proof tasks.

arXiv cs.CR·Sep 14, 20:40 UTC · 1d agoResearch

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AI Doesn't Mean the End of Mathematics—at Least Not Yet

Schneier and Rafi argue frontier AI models produce notable mathematical results but cannot yet build genuinely new conceptual frameworks.

Bruce Schneier and Kasra Rafi, writing in The Guardian, argue current AI models are not yet as capable as experienced academic mathematicians despite striking results. They cite OpenAI's disproof of the unit distance conjecture, Anthropic's published cryptanalysis results, and Claude's attempt at the Riemann hypothesis as achievements in counterexample search and recombining known techniques. They contend AI has not yet developed substantial new conceptual frameworks, though they expect that capability sooner rather than later.

Schneier on Security·Aug 28, 11:03 UTC · 19d agoAI research1

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Beyond the Turing threshold: Productive grammars generate essentially undecidable languages

A theoretical paper designs formal grammars that emulate Post's productive sets, generating languages that are provably beyond Turing decidability.

The paper elaborates on Emil Post's productive sets, which are not even semi-computable, and builds formal grammars that emulate their construction over natural numbers. The resulting languages are shown to be essentially undecidable, placing them beyond Turing decidability. This is pure computability and formal language theory with limited direct security relevance.

arXiv cs.CR·Sep 10, 11:20 UTC · 6d agoResearch1

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AI models' written reasoning steps correspond to distinct internal patterns, a new study finds

KAIST and Naver AI Lab researchers show LLM reasoning steps like extraction and computation map to distinct activation patterns, strongest in middle layers.

Researchers at KAIST and Naver AI Lab defined eight recurring reasoning operations, including extraction, decomposition, formula recall, deduction, and computation, and showed they correspond to separable activation patterns in Qwen2.5-7B, Qwen3-8B, and Gemma4-31B on math tasks, with GPT-5 labeling solution segments. The separation peaks in middle layers, holds even when a computation step produces a wrong answer, and goes beyond surface-level token choice. Findings replicated on Llama-3-8B, and classifiers trained on Qwen3-8B transferred to GPQA-Diamond and MATH-500. The authors note that using internal states for error detection or mid-generation steering remains future work.

The Decoder·Sep 12, 13:39 UTC · 3d agoAI research2

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Mathematicians want proof OpenAI didn’t use their work

Mathematician Andreas Thom publicly accused OpenAI of opacity over whether ChatGPT conversations contributed to its non-sofic groups mathematics result.

A second mathematician, Andreas Thom, accused OpenAI of 'dishonest' behavior and insufficient transparency about training data after OpenAI announced a result in non-sofic groups, Thom's area of expertise. He emailed OpenAI researchers Sébastien Bubeck and Mark Sellke asking whether his ChatGPT interactions fed training or reasoning, but found the answers did not rule out indirect use. The dispute follows Tristan Buckmaster's questions about the Millennium Prize Navier-Stokes solution, where OpenAI denied using specific user data but could not rule out de-identified usage data. Researchers told The Verge they worry such competition with AI labs will make mathematics more secretive.

The Verge · AI·Sep 10, 11:00 UTC · 6d agoAI industry

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OpenAI fought dirty on career-making math problem, says NYU mathematician

NYU mathematician Tristan Buckmaster alleges OpenAI learned of his team's Navier-Stokes approach and raced ahead using massive compute to claim a full proof first.

NYU mathematics professor Tristan Buckmaster and Anthropic mathematician Levent Alpöge announced preliminary proofs toward the Navier-Stokes existence and smoothness problem, a $1 million Clay Millennium Prize problem, developed using OpenAI's Codex and Claude. They allege OpenAI learned of their progress and that an OpenAI team then used an 'insane amount of compute' to announce a full proof first. OpenAI research lead Sebastian Bubeck denies the claims as 'false and inflammatory'. Buckmaster also raised concerns that OpenAI could have learned from his Codex interactions, which the company may use for model training.

TechCrunch · AI·Sep 8, 17:32 UTC · 7d agoAI industry

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What OpenAI’s latest controversy tells us about the future of math

OpenAI says its agents solved the Navier–Stokes Millennium Problem using an internal model, amid uncredited-work accusations from mathematicians Buckmaster and Alpöge.

OpenAI announced that its AI agents produced a proof that the full Navier–Stokes existence and smoothness problem can break down, using an internal model that outperforms the recently released Astra. NYU's Tristan Buckmaster and Anthropic's Levent Alpöge had posted a proof for a simplified version the previous day after nearly a year of work with public OpenAI and Anthropic models. OpenAI denies using their transcripts or training on them; chief research officer Mark Chen reiterated the denial, and the company says it will not claim the $1 million Clay Mathematics Institute prize. The episode fuels debate over attribution norms as frontier labs concentrate mathematical breakthroughs.

MIT Technology Review · AI·Sep 9, 03:10 UTC · 7d agoAI industry

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An Open Recipe for IMO Gold: Training Nemotron for Olympiad Mathematics

Open post-training pipeline turns Nemotron 3 Ultra checkpoints into an IMO 2026 gold-medal system, scoring 30/42 without formal provers or external tools.

Starting from Nemotron 3 Ultra, researchers trained two specialist checkpoints using supervised fine-tuning and reinforcement learning for natural-language olympiad proof generation. Three checkpoints power an iterative generate-verify-refine search plus a separate high-compute selection stage, operating entirely in natural language with no formal prover, external tools, or internet access. The system scored 30 of 42 points at IMO 2026, reaching the gold-medal threshold. The release includes the post-trained checkpoints, training data, training and inference code, submitted solutions, and Nemotron-IMO-Bench with 200 novel olympiad-level problems.

Hugging Face daily papers·Sep 8, 20:00 UTC · 7d agoModel release

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OpenAI’s feud with mathematicians is only escalating

25 Fields Medalists sign an open letter warning AI labs threaten math attribution; NYU's Tristan Buckmaster accuses OpenAI of pressuring him over collaborator credit.

Twenty-five Fields Medal-winning mathematicians signed an open letter arguing rushed AI proofs raise severe attribution and plagiarism questions and could destroy the culture of open research. NYU professor Tristan Buckmaster accused OpenAI of pressuring him not to credit an Anthropic-employed collaborator, and OpenAI withdrew sponsorship of a Caltech math event after researcher criticism. OpenAI's marathon-weekend proof remains unverified, and mathematicians fear their Codex usage may be fed into OpenAI's new models. The letter follows the June Leiden Declaration on LLM proofs.

TechCrunch · AI·Sep 11, 20:57 UTC · 4d agoAI industry

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Nonmaximal sums of maximally monotone operators under Rockafellar's constraint qualification

Mathematical paper constructs counterexamples on c0 and l1 disproving Rockafellar's conjecture that sums of maximally monotone operators remain maximally monotone.

The authors build counterexamples where two maximally monotone operators satisfy the interior-domain condition yet their sum is not maximally monotone, refuting Rockafellar's sum conjecture. One counterexample is constructed on c0 and another on l1 with its usual norm. A general construction theorem computes the monotone polar of a class of graphs, gives necessary and sufficient conditions for maximal monotonicity, and shows how a positive rank-one perturbation yields a nonmaximal sum.

arXiv cs.AI / cs.LG / cs.CL·Sep 9, 17:26 UTC · 6d agoAI research

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Towards a Deterministic Math Solver for Clinical Language Models

Paper shows handing arithmetic to a deterministic Python solver beats direct model calculation at 32B but not reliably at 7B on MedCalc-Bench.

Researchers test a Program-Solve interface where clinical LLMs write case-specific Python executed by a restricted local solver instead of doing arithmetic directly. On MedCalc-Bench Verified (1,100 cases, 55 calculators), Qwen2.5-32B-AWQ scored 90.53% with solver handoff versus 83.47% with direct arithmetic (+7.05 points), while Qwen2.5-7B gained an unreliable +3.29 points with a confidence interval spanning zero. The authors audited the benchmark against clinical guidelines and flagged 16 of 55 calculators for version, use, or coefficient concerns.

Hugging Face daily papers·Sep 8, 20:00 UTC · 7d agoAI research

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Thought without systematicity? Evaluating reasoning models on rule induction tasks

Study finds reasoning models often fail on structurally equivalent variants of tasks they solve, suggesting their reasoning lacks systematicity.

The paper extends rule induction tasks from cognitive science using task isomorphisms such as recombination and substitution to test systematicity in reasoning models. Despite solving tasks correctly, models frequently fail on structurally equivalent variants of the same task. The authors conclude many model behaviors lack systematicity, making it difficult to establish cognitive abilities beyond the specific evaluation contexts.

Hugging Face daily papers·Sep 11, 20:00 UTC · 4d agoAI research

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Recurrent GraphNeural NetworkswithSet-BasedAggregation

Paper proves two-directional equivalence between recurrent GNNs with set-based aggregation and Boolean closure of reachability/safety properties in modal mu-calculus, checkable from weights.

The authors study recurrent graph neural networks with set-based aggregation and identify sufficient conditions, checkable directly from network weights, for compiling networks into logical formulas and formulas into networks. They establish an effective two-directional equivalence with the Boolean closure of reachability and safety properties, the fragment BΣ°1 of the modal μ-calculus, shown to be the exact expressive level of stabilization over finite vocabulary. The correspondence needs no counting logic, external halting signal, or non-effective acceptance condition, yielding a verifiable path from weights to symbolic explanations.

arXiv cs.AI / cs.LG / cs.CL·Sep 14, 17:39 UTC · 1d agoAI research

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Claude Fable Solves a Historical Cipher

Bruce Schneier's blog highlights that the Claude Fable AI model solved a historical cipher, demonstrating LLM capabilities in cryptanalysis.

Bruce Schneier's blog post discusses the Claude Fable AI model successfully deciphering a historical cipher. The post frames the result as a notable example of LLMs applied to classical cryptanalysis. The published text provides limited technical detail beyond the headline.

Schneier on Security·Sep 9, 11:08 UTC · 7d agoAI research

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You've Got a BUD in Me: Authenticated Reads from Per-Block Write Logs

Researchers propose BUD, per-block write-log digests enabling blockchain validators to serve historical membership and exclusion proofs far cheaper than state-wide tries.

The paper introduces Block Update Digests (BUD), which authenticate each block's write log with predecessor pointers, plus a SuperBUD and exponential hierarchy to turn long unchanged intervals into short proofs. Soundness against adversarial provers and up to f Byzantine validators is proven under archive, attestation, and committee evidence assumptions. Benchmarks show a 50x state-size increase raises the base-BUD path only 1.24x versus 3.1x for in-memory and 69.5x for disk-backed Merkle Patricia tries, with read payloads below 800 bytes and p99 warm verification at 146 microseconds.

arXiv cs.CR·Sep 10, 08:46 UTC · 6d agoResearch

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Atlas: Efficient Verifiable Semantic Search

Atlas delivers zero-knowledge proofs for HNSW semantic search, verifying RAG retrieval in under a second on SIFT1M and 2.0 seconds at 100M vectors.

Atlas lets a search provider prove that a query was answered correctly against a committed HNSW index without revealing the index, addressing provider deviations like truncation or bias. It combines offline preprocessing, a fixed-size-state restructuring of HNSW with a correctness proof, and timestep-tagged batching of per-step arguments. The system proves queries in under a second on SIFT1M and 2.0 seconds at 100 million vectors while preserving plaintext HNSW recall, and proven retrieval maintains end-to-end RAG answer quality at lower cost than prior verifiable retrieval systems.

arXiv cs.CR·Sep 10, 17:30 UTC · 5d agoResearch1

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One Symptom, Three Levers: A Critical Review of On-Policy Self-Distillation

A review paper frames on-policy self-distillation collapse as governed by three levers: token weighting, privileged information, and guidance decay.

The paper critically reviews On-Policy Self-Distillation (OPSD), where a language model trains on its own generations scored token-by-token by a teacher conditioned on privileged information such as reference solutions or environment feedback. It identifies collapse, the progressive narrowing of producible reasoning paths, as the dominant failure mode and analyzes it through three levers: signal weighting, the nature of privileged information, and teacher dynamics. The review is restricted to mathematical reasoning, reports no new experiments, and offers a shared vocabulary separating settled findings from disputed ones.

Hugging Face daily papers·Aug 25, 20:00 UTC · 21d agoAI research

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OpenAI's millennium proof dispute raises the question of whether researchers can trust AI labs

Mathematician Tristan Buckmaster accused OpenAI of pressuring him and possibly training on his drafts amid OpenAI's race to claim a Navier-Stokes millennium proof.

OpenAI published a blog post and Sam Altman defended the team behind its AI-generated proof of the Navier-Stokes Millennium Problem after mathematician Tristan Buckmaster accused the company of academic misconduct. Buckmaster and co-author Levent Alpöge, who works at Anthropic, allege OpenAI pressured Buckmaster, sidelined Alpöge, and may have trained on drafts they entered into OpenAI's systems. OpenAI acknowledges it cannot rule out that de-identified data from their product usage helped improve its models. Mathematician Terence Tao warned the episode could discourage researchers from sharing work, reversing centuries of open science.

The Decoder·Sep 9, 10:27 UTC · 7d agoAI industry1

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Is OpenAI Taking Everyone for Fools?

OpenAI faces accusations it scooped NYU mathematicians' Navier-Stokes proof, possibly using their data, amid skepticism about GPT-6 Astra claims.

NYU mathematicians Tristan Buckmaster and Levent Alpöge published solutions to decades-old blowup problems for incompressible Euler, Boussinesq, and porous media equations on the same day OpenAI claimed its internal model solved the Navier-Stokes existence and smoothness problem. OpenAI admitted its effort began September 1st after hearing a related rumor and said it cannot rule out that de-identified data from the researchers' use of its products, such as private Codex sessions, helped improve its models. The column questions OpenAI's transparency, noting the company had just released GPT-6 Astra with claims including that AGI has been achieved, following recent controversies over its agent hacking Hugging Face and a German wiki site.

Hacker News · security·Sep 9, 17:09 UTC · 6d agoAI industryHN 20↑ · 6 comments

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On the Navier–Stokes Millennium Prize Problem

OpenAI announced an AI-generated solution to the Navier-Stokes Millennium Prize Problem, including a writeup and a formal Lean proof.

OpenAI shared what it describes as an AI-generated solution to the Navier-Stokes Millennium Prize Problem, one of the Clay Mathematics Institute's seven Millennium Prize Problems concerning fluid dynamics. The announcement includes a writeup and a machine-checkable formal proof in the Lean theorem prover. Details on the model, methodology and independent verification were not provided in the announcement text.

OpenAI News·Sep 8, 10:00 UTC · 8d agoAI research

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A Misalignment of AI in Mathematics

25 Fields Medallists including Terence Tao issue a declaration warning that AI companies' benchmark-driven mathematics goals are misaligned with science and society.

Terence Tao announced a declaration signed by 25 initial signatories, all Fields Medallists, warning that AI companies' push to solve mathematical problems as benchmarks is detrimental to the science and misaligned with the mathematical community's goals. The signatories argue that rushed, headline-driven releases of LLM solutions to major problems raise attribution and plagiarism questions and could erode the human process that develops and transmits mathematical ideas. They frame the issue as a broader misalignment between AI outputs and the purpose of intellectual work, affecting other sciences and society at large. The declaration is posted on a public page, invites further signatures in the manner of the Leiden declaration, and has been covered by The Economist.

Hacker News · AI·updated Sep 11, 17:45 UTC · 4d ago·first Sep 11, 17:30 UTC · 4d agoAI safety & security 2 sourcesHN 98↑ · 50 comments1

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ZGCM-1: A Fully Open and Extremely Efficient Foundation Model for Math and Agentic Search

ZGCM-1 is a fully open 7B foundation model with 256K context that stays competitive with frontier models on math reasoning and agentic search.

ZGCM-1 is a fully open 7B dense foundation model trained from scratch using an efficiency-focused recipe: interleaved gated sliding-window and full attention, a stable FP8 Muon optimizer, and MDP-based mid-training with context scaling across 16K, 64K, and 256K. On mathematical reasoning and agentic search suites it remains competitive with much larger frontier models such as Qwen3-235B-A22B and GLM-5.1. The recipe yields a ~4.2x improvement in 16K pre-training time-to-loss, and all weights, checkpoints, training code, data recipes, and W&B logs are open-sourced.

Hugging Face daily papers·Sep 10, 20:00 UTC · 5d agoModel release

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Beyond Solver Verdicts: Generative Reward Models for Autoformalization

Researchers introduce Generative Verification (GenV), a generative reward model achieving 0.961 AUROC in detecting unfaithful autoformalization that preserves solver verdicts.

The paper formalizes Verdict-Preserving-Unfaithfulness (VPU), a failure mode in neurosymbolic autoformalization where an incorrect encoding executes successfully and matches the expected solver verdict, and proves verdict-only verification is bounded to chance-level detection. The proposed Generative Verification (GenV) distills an offline Z3-equivalence oracle into a reference-free, continuous reference-equivalence score within the language model's vocabulary space. The oracle-mined verifier (GenV+HN) achieves 0.961 AUROC, generalizes zero-shot across unseen translators and formal styles, and yields an 11.3-point downstream accuracy gain in agentic test-time compute allocation. Mechanistic analysis with decision-projected logit lenses and sparse autoencoders shows the generative readout extracts precise spatial error coordinates without explicit localization training.

Hugging Face daily papers·Sep 9, 20:00 UTC · 6d agoAI research1

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On APN Functions with Boomerang Uniformity One over $\mathbb F_{3^n}$: Differential and Boomerang Spectra and CCZ-Inequivalence

Cryptographic construction yields infinite APN function families with boomerang uniformity one over odd-characteristic fields, proving CCZ-inequivalence to power functions.

For q=3^n with odd n>1, every sign-switch of a perfect nonlinear Dembowski-Ostrom polynomial is proven APN with boomerang uniformity one or two, attaining uniformity one for (q-3)/2 parameters. This gives the first general construction of infinite APN families achieving boomerang uniformity one over odd-characteristic finite fields. The authors determine common differential and complete boomerang spectra, ruling out CCZ equivalence with power and Ness-Helleseth binomials, and exhibit three pairwise CCZ-inequivalent PN functions for infinitely many n, with the smallest degree n=45.

arXiv cs.CR·Sep 8, 16:19 UTC · 7d agoResearch

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Proximity Gaps for Gabidulin Codes and Applications

Researchers prove proximity-gap bounds for rank-metric and Gabidulin codes, enabling the first polynomial commitment scheme framework based on rank-metric error-correcting codes.

The paper proves every linear rank-metric code admits a proximity gap for deltas up to (d-1)/(3n) with error at most q^(e+1)/q^m, and improves the gap to (d-1)/(2n) for Gabidulin codes with error at most 10q^(n-1)/q^m, matching bounds for Reed-Solomon codes. A constructed infinite family of constant-rate Gabidulin codes shows the (d-1)/(2n) bound is tight, and a counterexample establishes a lower bound on the error at the d/(3n) gap. Applications include an IOPP for interleaved Gabidulin codes adapted from the Ligero IOPP and a q-linearized polynomial commitment scheme adapted from Ligero-based PCS, reportedly the first PCS framework based on rank-metric codes.

arXiv cs.CR·Sep 9, 07:45 UTC · 7d agoResearch

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Guppy: Efficient Light Clients via Recursive Zero-Knowledge Proofs

Guppy lets blockchain light clients verify full state via recursive zero-knowledge proofs without validators maintaining state commitments, processing thousands of updates per second.

Guppy is a light-client protocol in which validators commit only to state updates while an off-chain, untrusted service secured by recursive zero-knowledge proofs maintains a verifiable Merkle tree over the full state. A hash-chain commitment moves validator signature verification out of the proving circuit, and a parallel recursive proving pipeline keeps latency growth logarithmic with throughput. A Plonky2-based implementation maintains a tree of size 2^30 while processing thousands of updates per second, adding only 2-4 seconds of latency without increasing block-construction complexity.

arXiv cs.CR·Sep 7, 20:41 UTC · 8d agoResearch

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