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How well do agents use test/verification techniques?

Dan Luu's eval finds coding-agent testing instructions (TDD, formal methods, PBT, skills) mostly fail to beat defaults on Zstd implementation correctness.

The author ran 26 prompt conditions plus 4 skills on a Zstd-in-Rust implementation eval using codex with GPT-5.6, testing TDD, fuzzing, property-based testing, formal methods (Lean 4, TLA+, Verus, Kani, SMT solvers) and community skills. Nothing dramatically outperformed the default no-instruction condition, which did above average; at xhigh effort, fuzzing and PBT conditions did slightly better than formal methods. Pre-registered predictions included TDD underperforming and popular test skills (ECC, Hegel, Trail of Bits) not outperforming. Results are averages of 80 runs per condition plotted against cost.

Hacker News · securitySep 8, 02:58 UTC · 9d agoAI research HN 131↑ · 50 comments

StochBench: A Domain-Specific Benchmark for Stochastic Processes in Lean

StochBench introduces 450 graduate-level stochastic-processes problems in Lean 4; an Opus 4.8-based agent proved 34.9% under a 15-minute limit.

StochBench is a Lean 4 benchmark of 450 graduate stochastic-processes problems, each paired with its natural-language source, covering Markov chains, renewal processes, martingales, Brownian motion, stochastic calculus, weak convergence, and Poisson and continuous-time Markov processes. The benchmark addresses field-specific applied mathematics underrepresented in Mathlib, unlike competition-math-dominated suites such as IMO and Putnam collections. An Opus 4.8-based agent achieved a 34.9% proof rate (157/450) under a 15-minute per-problem limit, showing the benchmark remains challenging for advanced provers.

Hugging Face daily papersSep 7, 20:00 UTC · 9d 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 · 7d agoAI research1

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 · 9d agoAI research

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 · securitySep 8, 16:57 UTC · 8d agoAI research HN 21↑ · 5 comments1

🔬“We have foundation models for language, not for physics” — Anima Anandkumar, Bren Professor of Computing

Caltech professor Anima Anandkumar discusses Neural Operators and FourCastNet for physics modeling, arguing inductive biases beat pure token scaling.

Anima Anandkumar, Bren Professor at Caltech and co-founder of Accelerated Understanding, describes Fourier Neural Operators that learn in frequency and spherical-harmonic domains to model weather, fusion, and fluid or heat flow. Her team built FourCastNet 3, a global weather model competitive with physics-based simulations that runs on consumer-grade GPUs. She also introduced TorchLean, a framework for writing PyTorch-style networks inside the Lean proof assistant for formal verification, and was appointed to the United Nations Scientific Advisory Board. She argues physical domains resist scaling due to tiny datasets and context lengths in the hundreds of billions, so progress comes from built-in structure and physical priors.

Latent SpaceAug 26, 15:15 UTC · 21d agoAI research1