Search: “gap-entropy-conjecture”

30 stories

DomainAll Cyber AIWindow24h 3d 7d 30d AllIn the wild Important only Show duplicates

CategoryAll

SeverityAll critical 0 high 0 medium 0 low 0 info 0

Smart search ranks by meaning as well as keywords (one row per story, last 45 days).

A positive resolution of the gap-entropy conjecture

New proof resolves the gap-entropy conjecture for Gaussian bandits, bounding optimal best-arm identification samples by H(log(1/delta)+Ent(I)) up to constants.

A paper proves the gap-entropy conjecture for fixed-confidence best-arm identification with independent unit-variance Gaussian arms, means in [0,1], and a unique optimal arm. It shows the optimal expected sample count, averaged over arm-label permutations, is within absolute constant factors of H(log(1/delta)+Ent(I)), where H sums squared gaps and Ent(I) is the instance's gap-entropy. It also gives an instance-independent algorithm bounded by a constant multiple of this quantity plus a g^-2 loglog(e^e/g) term for the smallest gap g.

arXiv cs.AI / cs.LG / cs.CLSep 9, 17:57 UTC · 6d agoAI research

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.CRSep 9, 07:45 UTC · 7d agoResearch

Thin-shell stability of Gaussian cooling: logconcave sampling with sesteric complexity from a cold start

Thin-shell stability proof along the Gaussian cooling path improves cold-start logconcave sampling complexity to near n^2.5 from n^2.75.

The authors prove that logconcave probability measures along the Gaussian cooling path have thin-shell stability, generalizing the thin-shell theorem. This yields improved complexity for sampling an arbitrary logconcave distribution from a cold start. For (near-)isotropic logconcave distributions the complexity is nearly n^2.5, improving the previous n^2.75 bound and matching the abstract Speedy walk.

arXiv cs.AI / cs.LG / cs.CLSep 14, 17:10 UTC · 1d agoAI research

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 · securitySep 14, 12:23 UTC · 2d agoAI research HN 22↑ · 19 comments

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.CLSep 9, 17:26 UTC · 6d agoAI research

On the Navier–Stokes Millennium Prize Problem

OpenAI says an unreleased model produced a claimed solution to the Navier-Stokes existence and smoothness problem, disputed by an NYU mathematician.

OpenAI used an unreleased model to produce a claimed solution to the Navier-Stokes existence and smoothness problem, one of the seven Millennium Prize Problems carrying a $1,000,000 prize since May 24, 2000. The result is contested: NYU mathematics professor Tristan Buckmaster accused collaborators of skulduggery and rushed out his own competing results with mathematician Levent Alpoge, who works at Anthropic. The dispute is documented in a published PDF describing the competing claims.

Simon WillisonSep 8, 23:39 UTC · 7d agoAI research

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.CLSep 9, 17:54 UTC · 6d agoAI research1

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 · AISep 8, 17:32 UTC · 7d agoAI industry

Operational Roles of QRNG-Derived Quantum Entropy in Bitcoin Proof-of-Work Architectures

arXiv study finds quantum-random entropy adds no Bitcoin PoW success advantage but helps assurance in fault and provenance scenarios.

The paper shows replacing classical entropy with QRNG output does not change honest Bitcoin proof-of-work success probability when candidate headers remain distinct. It introduces a reproducible benchmark measuring an entropy-efficiency factor and a reboot-diversity index, finding QRNG value only in assurance-oriented scenarios involving correlated restart faults, namespace reuse, and entropy provenance. Validation is simulation-based, with hardware-in-the-loop testing identified as future work.

arXiv cs.CRSep 4, 12:47 UTC · 12d agoResearch

A Note on Sphere Packing Bounds for Tuple Lattice Sieving

Proves upper bounds on k-irreducible unit vector set rates, yielding nearly tight asymptotics relevant to tuple lattice sieving in cryptanalysis.

The paper bounds the maximal asymptotic rate of k-irreducible sets of unit vectors via spherical code packing bounds. It shows R_k is sandwiched between (1/2 - o(1)) log2(k)/k and (1 + o(1)) log2(k)/k for large k. These almost-tight bounds inform subexponential complexity analyses of tuple lattice sieving, which underpins security estimates for lattice-based cryptography.

arXiv cs.CRSep 8, 03:20 UTC · 8d agoResearch

OpenAI researcher allegedly pressured mathematician to drop Anthropic co-author from math breakthrough paper

OpenAI researcher Sébastien Bubeck allegedly pressured mathematician Tristan Buckmaster to drop his Anthropic co-author from an AI-assisted Navier-Stokes breakthrough paper.

Mathematician Tristan Buckmaster says OpenAI, after learning of his and Levent Alpöge's AI-assisted progress on the Navier-Stokes equations, pressed him to drop his Anthropic-employed co-author and dictated how any results would be announced. He says Sébastien Bubeck claimed an internal OpenAI model had produced a roughly 100-page proof for Navier-Stokes with forcing and allegedly told him 'Why would you ruin your career?' when he threatened to go public. Buckmaster published a public statement detailing the exchanges; OpenAI has not yet responded. The pair had worked with models including Claude and OpenAI Codex running GPT-5.6 Sol on the Clay Millennium Problem, which carries a $1 million prize.

The DecoderSep 8, 17:23 UTC · 7d agoAI industry1

Privacy Failure in Split-LLM Training, The Returned Gradient Nullifies the Decoys

Researchers show split-LLM training leaks privacy via zero-valued gradients on decoy rows, exposing which activations are real despite passing forward-channel checks.

A systems-security case study of a two-node split-LLM training setup found that the returned output gradient from an Untrusted Cloud Node is exactly zero for decoy rows, revealing which rows are real. Across nine seeds, zero patterns identified real rows in 4,096 of 4,096 frames per run, and an attack on frame contents recovered 0.65 to 1.50 percentage points of extra tokens over a baseline. Both datasets passed forward-channel privacy and quality checks but failed once the returned gradient was included. Row-wise gradient clipping and noise closed the leak for roughly 0.01 nats of held-out cross-entropy, though five unmeasured attack classes remain.

Hugging Face daily papersSep 2, 20:00 UTC · 13d agoAI safety & security

EFI Pairs Without One-Way Puzzles: Oracle Separations from Communication Complexity

Theorists build a classical oracle where one-way puzzles fail yet EFI pairs survive, separating two candidate minimal assumptions of quantum cryptography.

The paper constructs a single classical oracle relative to which one-way puzzles do not exist, even with an unbounded verifier, while an EFI pair survives every classical-query distinguisher holding advice, making one superposition query at the end. Security is proven by reducing adversary knowledge to communication complexity for Vector-in-Subspace, with the superposition query bounded using random matrix theory. Relative to the oracle, quantum polynomial time offers no advantage on tasks with classical inputs and outputs and there is no proof of quantumness, separating the leading minimal assumptions of quantum cryptography.

arXiv cs.CRSep 10, 17:54 UTC · 5d agoResearch

Benign Loss Landscapes Can Coexist with Worst-Case Hardness

Theory paper shows tree tensor networks contain worst-case hard targets yet benign loss landscapes, with difficulty arising from degenerate saddles.

The paper studies tree tensor networks (TTNs), which generalize deep linear networks and Tucker decompositions and embed arbitrary read-once Boolean formulas. It proves that every local minimum that is minimum-norm is global for every realizable target, so bad local minima do not distinguish typical from worst-case problems. Instead, learning difficulty arises from high-order degenerate saddle points caused by rank-deficiency, illustrated via a parity function case study, linking landscape geometry to computational hardness.

arXiv cs.AI / cs.LG / cs.CLSep 11, 16:57 UTC · 4d agoAI research

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 DecoderSep 9, 10:27 UTC · 7d agoAI industry1

Quenched Ensemble Sampling

Quenched Ensemble Sampling generalizes nested sampling's hard energy constraint to repulsive potentials, traversing first-order phase transitions where tempering fails.

Quenched Ensemble Sampling generalizes nested sampling's hard energy constraint into a family of repulsive potentials at the energy boundary, preserving monotone energy descent while making the constrained target amenable to scalable gradient-based kernels. On synthetic phase-transition models it estimates marginal likelihood and draws posterior samples across first-order transitions where popular alternatives such as tempering fail. Applications include marginal likelihood estimation for Bayesian neural network architecture comparison and partition function estimation in a high-dimensional continuous lattice field theory.

arXiv cs.AI / cs.LG / cs.CLSep 14, 17:16 UTC · 1d agoAI research1

Implementing a White-Box Undetectable Backdoor for Random Fourier Features

Researchers implement Goldwasser's CLWE-based undetectable backdoor for Random Fourier Features models in numpy/scipy, confirming practical realizability with no detectable differences from clean models.

The paper provides an end-to-end implementation of the Goldwasser et al. white-box undetectable backdoor for models trained with the Random Fourier Features algorithm, using only numpy and scipy. It derives two samplers for the core GP_d(b_k) distribution: a rejection-sampling proxy and an exact closed-form sampler verified against its analytic form. Statistical indistinguishability tests covering weight-space and functional black-box comparisons found no detectable difference between backdoored and clean models across sparsity ratios. The underlying lattice hardness reduction was not reproduced, and the work demonstrates the threat is realizable with commodity scientific-computing tools rather than specialized cryptographic infrastructure.

arXiv cs.CRSep 14, 22:18 UTC · 1d agoResearch

Schneider Electric Easergy, EcoStruxture, PowerLogic, and Saitel Products (Update A)

CISA updated ICSA-26-169-07: CVE-2026-4827 (CVSS 8.3) insufficient entropy enables session hijacking across dozens of Schneider Electric grid products; fixes available.

CISA republished advisory ICSA-26-169-07 (Update A) for CVE-2026-4827, CWE-331 insufficient entropy in session management, scored CVSS 8.3. Affected lines include Easergy MiCOM relays and C5, EcoStruxure Power Automation (EPAS-GTW, EPAS-UI, iPMFLS), EcoStruxure Power Operation, PowerLogic P5/P7/T300/T500, and Saitel DP/T150 RTUs, with dozens of fixed versions listed. Successful exploitation could enable session hijacking and unauthorized operations on systems in energy, chemical, critical manufacturing, and water sectors. Fixes are available; no exploitation is reported.

CISA AdvisoriesSep 3, 12:00 UTC · 13d agoAdvisory CVE-2026-4827

Rare Not Random Using Token Efficiency for Secrets Scanning

Researcher proposes token efficiency (string length divided by BPE token count) as a better post-regex filter than entropy for secrets scanning, validated on CredData.

The post explores whether Byte-Pair Encoding tokenization can replace Shannon entropy as the primary filter for candidate secrets captured by regex in tools like Gitleaks. It defines 'token efficiency' as string length divided by token count under the cl100k_base tokenizer; secret-like strings such as GitHub tokens tokenize into many small tokens and score low, while natural text scores high. Evaluating labeled secrets from the CredData dataset shows a usable separation, with roughly 2.5 suggested as a minimum cutoff versus Gitleaks' 3.5 entropy threshold. The technique is positioned as a post-regex filtering step rather than a standalone detector.

Lobsters · securitySep 12, 06:42 UTC · 4d agoResearch

Bridging Control, Inference, Transport, and Thermodynamics: From Theory to Applications in Learning

Review connects control theory, optimal transport, probabilistic inference, thermodynamics, and machine learning via free-energy optimization under constraints.

The review unifies five fields: control theory, optimal transport, probabilistic inference, non-equilibrium thermodynamics, and machine learning. The common conceptual thread is optimization of free-energy-like functionals under dynamical or statistical constraints. Selected applications are presented in reinforcement learning, variational inference, and generative modeling. The tutorial-style text assumes no prior familiarity and begins from physics principles.

arXiv cs.AI / cs.LG / cs.CLSep 14, 17:16 UTC · 1d agoAI research

Clay Mathematics Institute says the Navier-Stokes Millennium Prize Problem has "apparently been settled"

Clay Mathematics Institute says the Navier-Stokes Millennium Problem appears settled amid accusations OpenAI misused a mathematician's leaked drafts.

The Clay Mathematics Institute stated the Navier-Stokes Millennium Prize Problem, one of seven problems worth $1 million each, has 'apparently been settled' and the solution is under review. A dispute has erupted around the work: mathematician Tristan Buckmaster accuses OpenAI of redirecting resources to the problem after rumors of his research leaked, using his drafts in training data, and excluding co-author Levent Alpöge, who works at Anthropic. CMI also noted new technologies' increasing ability to accelerate mathematical research.

The DecoderSep 14, 13:01 UTC · 2d agoAI industry1

Nearly Tight Rademacher Bounds for Sparsely Activated Neural Networks

Theory paper derives nearly tight Rademacher complexity bounds for sparsely activated one-hidden-layer ReLU networks.

Building on Awasthi et al. (COLT 2024), the authors bound statistical complexity for networks where each input activates at most k of s hidden units. A support-preserving cover and normalized chaining argument remove the explicit dimension factor, with matching lower bounds up to logarithms. They also derive agnostic minimax excess-risk bounds of order min{1, sqrt(s/(km))} for a normalized bounded loss and show bias bounds comparable to WR restore worst-case rates even on domains where sparsity holds globally.

arXiv cs.AI / cs.LG / cs.CLSep 8, 17:53 UTC · 7d agoAI research

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.CRSep 10, 11:20 UTC · 6d agoResearch1

Drama swirls around OpenAI’s legendary mathematical milestone

OpenAI claims an internal AI model solved the 90-year-old Navier-Stokes problem, sparking a priority dispute with mathematician Tristan Buckmaster.

OpenAI announced a solution to the Navier-Stokes problem, one of the $1 million Millennium Prize Problems, using an internal AI model it says outperforms the newly released GPT-6 Astra alongside 10,000 concurrent agents. NYU professor Tristan Buckmaster, who with Anthropic researcher Levent Alpöge published related findings a day earlier, questioned whether OpenAI accessed drafts from his Codex sessions. OpenAI says no specific user data was accessed, though it cannot rule out that de-identified usage data helped improve its models.

The Verge · AISep 8, 20:53 UTC · 7d agoAI industry1

Searching for New Physics with Reinforcement Learning

Researchers apply reinforcement learning to identify SMEFT operators explaining particle physics anomalies, reproducing and improving known CDF W-mass results.

The paper introduces a reinforcement learning method to search the large Standard Model Effective Field Theory (SMEFT) operator space for explanations of measurement anomalies. It was validated on the CDF W-mass anomaly, reproducing and improving known results, then applied to a harder multi-anomaly scenario. RL efficiently navigates complex loop-level operator correlations that bias human-driven phenomenological analysis.

arXiv cs.AI / cs.LG / cs.CLSep 9, 16:07 UTC · 6d agoAI research

Observational Indistinguishability and Integrity Blind Regions in Hybrid Quantum-Classical Workflows

Framework formalizes integrity blind regions in hybrid quantum-classical workflows, validated across 3,600 label interventions with conformal detection rules.

The paper presents a claim-relative evidence and reference framework for integrity of hybrid quantum-classical workflows, distinguishing structural blind regions caused by observational indistinguishability from finite-batch statistical misses. Experiments over 3,600 label interventions show exact label-path invariance for feature and prediction views. The geometry-aligned construction detects 343 of 2,700 conclusion-changing interventions using the conformal rule and 1,183 of 2,700 with the uncorrected union, with executed conformal clean false-action rates of 0.048-0.059.

arXiv cs.CRSep 15, 13:16 UTC · 1d agoResearch

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 · AISep 10, 11:00 UTC · 6d agoAI industry

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 · securitySep 9, 17:09 UTC · 6d agoAI industry HN 20↑ · 6 comments

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 NewsSep 8, 10:00 UTC · 8d agoAI research

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 papersSep 8, 20:00 UTC · 7d agoModel release