{
  "counts": {
    "adjacent": 15,
    "core": 8,
    "errors": 0,
    "negative": 65,
    "total": 88
  },
  "date": "2026-07-12",
  "errors": [],
  "fresh_content_days": 21,
  "generated_at": "2026-07-12T16:28:44Z",
  "items": [
    {
      "age_days": 5,
      "arxiv_id": "2607.06341",
      "authors": [
        "Shuangxiang Kan",
        "Shuanglong Kan",
        "Sebastian Ertel"
      ],
      "content_date": "2026-07-07",
      "freshness": "fresh",
      "id": "arxiv:2607.06341",
      "kind": "paper",
      "label": "core",
      "matched_signals": [
        "lean_formal_proving_agents",
        "verifier_guided_reasoning"
      ],
      "published": "2026-07-07",
      "score": 9.5,
      "source": "arxiv-ai4math-core",
      "summary": "Formal verification offers the strongest guarantee of software correctness, but it does not scale: the proofs demanded by interactive theorem provers such as Coq require enormous expert effort. Large language models (LLMs) promise to generate these proofs automatically, yet existing approaches wire a fixed, human-designed proof strategy into the system and constrain the model to follow it (retrieving premises and predicting tactics one step at a time, or splitting goals by divide-and-conquer), and still prove only a fraction of their target theorems. We show that imposing such a strategy is unnecessary and limiting. Handing the whole lemma to a general LLM code agent (for example, Claude Code), free to choose its own approach, and wrapping it in a verification harness is both simpler and more effective, achieving full coverage: every targeted lemma proved, with no failures and no Coq expert intervention. The agent writes the proofs under feedback and hard constraints from the harness that keep each one sound (accepted only when the prover's kernel closes it), complete (no obligation left unproved or silently dropped), and terminating (no divergent tactics). We evaluate this harness plus code agent along three dimensions. (1) Core logic: on Iris, the state-of-the-art separation logic for concurrent and memory-manipulating programs, Aria proves all 4,257 lemmas of the four core modules and the 217 lemmas verifying Rust's standard libraries built on it, fully automatically. (2) Comparison with prior LLM provers: on reglang, where prior provers manage barely one in eight, Aria proves all 318. (3) Generality: on iris-lean, the unfinished Lean 4 port of Iris, it proves 72 not-yet-ported lemmas, showing the approach is not specific to Coq. A state-of-the-art model (Claude Opus 4.7) can write proofs for verified software development fully and automatically.",
      "title": "Harnessing Code Agents for Automatic Software Verification",
      "updated": "2026-07-07",
      "url": "https://arxiv.org/abs/2607.06341"
    },
    {
      "age_days": 5,
      "arxiv_id": "2607.06820",
      "authors": [
        "Pavel Snopov",
        "German Magai"
      ],
      "content_date": "2026-07-07",
      "freshness": "fresh",
      "id": "arxiv:2607.06820",
      "kind": "paper",
      "label": "core",
      "matched_signals": [
        "autoformalization",
        "general_ai_math_reasoning",
        "verifier_guided_reasoning"
      ],
      "published": "2026-07-07",
      "score": 9.3,
      "source": "arxiv-ai4math-core",
      "summary": "Recent advances in AI for Mathematics have focused largely on autoformalization and theorem proving, leaving the role of Computer Algebra Systems (CAS) in agentic LLM workflows underexplored. We propose a ReAct-style agentic setup that combines LLM reasoning with verifiable feedback from SageMath, together with Context7 for the up-to-date documentation. We evaluate this agentic setup across frontier models for solving research-level mathematical problems from the RealMath benchmark in a setting that emulates a computational-mathematics research loop. We also propose a refinement to the RealMath benchmark by introducing a multi-step post-processing procedure and a multi-stage validation pipeline, both of which improve the quality and reliability of the extracted problem set. Our experiments reveal substantial performance gains from SageMath access across all evaluated models on +9.7~pp on average, the gains range from 1.5~pp to 27.8~pp and narrow the gap between open-weight and closed models. Qwen~3.7-Max benefits from SageMath the most, while GPT-5.5 achieves the highest solve rate of $75.2\\%$ and the lowest token usage among tool-enabled configurations. Our findings suggest that CAS-augmented agents represent a promising direction for assisting mathematicians in computational exploration, and we believe that this work is a step towards automated conjecture discovery. The project repository is available online.",
      "title": "Evaluating SageMath-Augmented LLM Agents for Computational and Experimental Mathematics",
      "updated": "2026-07-07",
      "url": "https://arxiv.org/abs/2607.06820"
    },
    {
      "age_days": 6,
      "arxiv_id": "2607.04655",
      "authors": [
        "Haocheng Wang",
        "Baiyu Huang",
        "Yingjia Wan",
        "Xiao Zhu",
        "Xiaoyang Liu",
        "Yinya Huang",
        "Zhijiang Guo"
      ],
      "content_date": "2026-07-06",
      "freshness": "fresh",
      "id": "arxiv:2607.04655",
      "kind": "paper",
      "label": "core",
      "matched_signals": [
        "autoformalization",
        "general_ai_math_reasoning",
        "verifier_guided_reasoning"
      ],
      "published": "2026-07-06",
      "score": 7.9,
      "source": "arxiv-ai4math-core",
      "summary": "The veracious semantic alignment in autoformalization is significant for formal mathematical reasoning. However, existing evaluations provide only opaque binary verdicts or scalar scores, offering no interpretable insight into where or why translations fail. This opacity severely limits both human understanding and automated system improvement. To bridge this gap, we introduce FormalRx, a comprehensive diagnostic evaluation framework that transforms autoformalization assessment from black-box judgments into actionable feedback. At its core is SCI Error Taxonomy, a hierarchical classification scheme decomposing autoformalization errors into 28 distinct categories with strict priority ordering. Building on this taxonomy, FormalRx provides four critical diagnostic capabilities: alignment verdicts, error categorization, error localization, and correction. We instantiate the framework with a diagnostic model FormalRx-8B, trained on 56,287 NL-FL pairs with fine-grained diagnostic annotations, and release FormalRx-Test as the first fine-grained diagnostic benchmark. FormalRx-8B achieves F1-scores of 0.88 (verdict) and 0.71 (categorization), along with accuracies of 0.75 (localization) and 0.73 (correction), substantially outperforming both general-purpose LLMs and specialized baselines. By connecting evaluation with actionable insights, FormalRx enables systematic diagnosis and improvement of autoformalization systems.",
      "title": "FormalRx: Rectify and eXamine Semantic Failures in Autoformalization",
      "updated": "2026-07-06",
      "url": "https://arxiv.org/abs/2607.04655"
    },
    {
      "age_days": 4,
      "arxiv_id": "2607.07779",
      "authors": [
        "Eric Jiang",
        "Xiao Liang",
        "Yikai Zhang",
        "Yingjia Wan",
        "Mengting Li",
        "Haikang Deng",
        "Alexander K. Taylor",
        "Justin Baker",
        "Rushil Raghavan",
        "Junyi Zhang",
        "Ying Nian Wu",
        "Andrea L. Bertozzi",
        "Kai-Wei Chang",
        "Raghu Meka",
        "Matthew Sottile",
        "Nanyun Peng",
        "Amit Sahai",
        "Terence Tao",
        "Wei Wang"
      ],
      "content_date": "2026-07-08",
      "freshness": "fresh",
      "id": "arxiv:2607.07779",
      "kind": "paper",
      "label": "core",
      "matched_signals": [
        "general_ai_math_reasoning",
        "lean_formal_proving_agents",
        "seed_author:Nanyun Peng"
      ],
      "published": "2026-07-08",
      "score": 7.5,
      "source": "arxiv-ai4math-core",
      "summary": "Recent developments in AI for Mathematics (AI4Math), especially Large Language Model (LLM)-driven theorem provers, has achieved remarkable success in formal proof generation for well-defined mathematical problems through Interactive Theorem Proving (ITP) languages. However, current systems remain fundamentally limited in tackling frontier research mathematics, such as discovering new theorems or resolving open conjectures, which are often open-ended, under-specified, and involve multiple layers of abstraction. We argue that the next leap in AI4Math systems requires a decisive shift from predefined problem-solvers to research agents that can address frontier mathematical challenges with rigorous formal mathematical reasoning. In this position paper, we provide a systematic review of the field, covering datasets, auto-formalization, and proof synthesis. More importantly, we identify core limitations of existing systems in serving as mathematical research agents, examining issues across datasets, relational structure, mathematical exploration, tool ecosystem, and human-AI collaboration, outlining a strategic road-map for the future of AI4Math.",
      "title": "From Solvers to Research: Large Language Model-Driven Formal Mathematics at the Research Frontier",
      "updated": "2026-07-08",
      "url": "https://arxiv.org/abs/2607.07779"
    },
    {
      "age_days": 3,
      "arxiv_id": "2607.08692",
      "authors": [
        "Arthur F. Ramos",
        "Tulio Soria"
      ],
      "content_date": "2026-07-09",
      "freshness": "fresh",
      "id": "arxiv:2607.08692",
      "kind": "paper",
      "label": "core",
      "matched_signals": [
        "lean_formal_proving_agents",
        "verifier_guided_reasoning"
      ],
      "published": "2026-07-09",
      "score": 6.5,
      "source": "arxiv-ai4math-core",
      "summary": "We present a metagame analysis of the competitive Pokemon Trading Card Game, machine-checked in Lean 4 over real tournament data. The headline game-theoretic results, including Nash equilibrium, replicator dynamics, and the matrix-level type-bridge computation, rely on native_decide, which trusts Lean's compiler rather than its kernel; the trust boundary is made explicit. The artifact spans approximately 31,900 lines, 87 files, and 2,627 theorems, of which roughly 200 directly verify empirical claims, with no sorry, admit, or custom axioms. Analyzing Trainer Hill data from January to February 2026 for events with at least 50 players, over 14 archetypes and their full pairwise matchup matrix, we prove a popularity paradox: the most played deck, Dragapult, with 15.5% metagame share, has only 46.7% expected win rate, while Grimmsnarl, with 5.1% share, achieves 52.7%. A machine-checked Nash equilibrium of the raw game assigns Dragapult 0% weight; exhaustive enumeration over all nonempty support subsets confirms a unique symmetric Nash equilibrium of the constant-sum symmetrization with seven-deck support. Against this equilibrium mix, Dragapult falls 40.4 permil below the game value. Single-step replicator dynamics indicate downward fitness pressure on Dragapult, upward pressure on Grimmsnarl, and strongest extinction pressure on Alakazam. A 10,000-iteration sensitivity analysis confirms qualitative stability, with core support decks appearing in more than 96% of resampled equilibria. The primary contribution is methodological: a reproducible case study showing how formal verification can turn qualitative metagame narratives into machine-checkable, re-runnable strategic science.",
      "title": "From Rules to Nash Equilibria: A Lean 4 Case Study in Game-Theoretic Analysis of a Competitive Trading Card Game",
      "updated": "2026-07-09",
      "url": "https://arxiv.org/abs/2607.08692"
    },
    {
      "age_days": 5,
      "arxiv_id": "2607.06693",
      "authors": [
        "Pedro Abdalla",
        "Jaume de Dios Pont",
        "João P. G. Ramos",
        "Mitchell A. Taylor"
      ],
      "content_date": "2026-07-07",
      "freshness": "fresh",
      "id": "arxiv:2607.06693",
      "kind": "paper",
      "label": "core",
      "matched_signals": [
        "autoformalization",
        "lean_formal_proving_agents"
      ],
      "published": "2026-07-07",
      "score": 6.5,
      "source": "arxiv-ai4math-core",
      "summary": "We prove that, after $L^2$ normalization, stable phase retrieval holds over the $L^2$-spans of independent real-valued centered random variables if and only if all but possibly one coordinate satisfies a uniform two-sided $L^1$ bound. This provides a complete characterization of stable phase retrieval for such subspaces, building upon the pioneering work of Calderbank--Daubechies--Freeman--Freeman and confirming the conjectured characterization communicated to us by those authors. We provide two different proofs of this fact, both based on a decomposition of the $\\ell^2$-coefficients of each random variable. The first is a compactness proof, which makes use of the infinite divisibility of limit laws of tail sums. The second is a quantitative proof, which substitutes the compactness step with an explicit dichotomy based on anticoncentration estimates of Sperner type. This latter proof was partially LLM generated based on the ideas in the first proof and a considerable amount of guidance by the authors. An autoformalization of our main result in Lean 4 is also provided, following the ideas in the quantitative proof.",
      "title": "Stable Phase Retrieval for Spans of Independent Random Variables",
      "updated": "2026-07-07",
      "url": "https://arxiv.org/abs/2607.06693"
    },
    {
      "age_days": 7,
      "arxiv_id": "2607.04321",
      "authors": [
        "Ján Perháč",
        "Vasyl Khashcha",
        "Samuel Novotný"
      ],
      "content_date": "2026-07-05",
      "freshness": "fresh",
      "id": "arxiv:2607.04321",
      "kind": "paper",
      "label": "core",
      "matched_signals": [
        "lean_formal_proving_agents",
        "verifier_guided_reasoning"
      ],
      "published": "2026-07-05",
      "score": 6.5,
      "source": "arxiv-ai4math-core",
      "summary": "We present LogicProof, an interactive web-based theorem prover designed for educational use. The system supports natural deduction and sequent calculus for propositional and first-order logic in both classical and constructive variants. It emphasizes a modern user experience through real-time feedback and interactive visualization of proof trees. We evaluated LogicProof in a study involving 35 students. The results suggest that the system improves understanding of formal proof construction and supports student engagement with logical concepts. Compared to traditional pen-and-paper approaches, students reported faster iteration, easier error correction, and greater confidence in the correctness of their solutions. These findings indicate that LogicProof can serve as a practical supplement to existing teaching methods, particularly in supporting the learning process and reducing barriers associated with formal proof construction.",
      "title": "LogicProof: An Interactive Web-Based Educational Theorem Prover for Natural Deduction and Sequent Calculus across Classical and Constructive Logics",
      "updated": "2026-07-05",
      "url": "https://arxiv.org/abs/2607.04321"
    },
    {
      "age_days": 8,
      "arxiv_id": "2607.03963",
      "authors": [
        "Yanjie Zhao",
        "Hongjie Chen",
        "Li Lu",
        "Zhou Yang",
        "Xiao Cheng",
        "Haoyu Wang"
      ],
      "content_date": "2026-07-04",
      "freshness": "fresh",
      "id": "arxiv:2607.03963",
      "kind": "paper",
      "label": "core",
      "matched_signals": [
        "lean_formal_proving_agents"
      ],
      "published": "2026-07-04",
      "score": 6.5,
      "source": "arxiv-ai4math-core",
      "summary": "Ask a large language model (LLM) whether a pointer dereference is safe, and it can often produce a plausible justification for ``yes''. The difficulty is that a fluent justification is not a proof. This gap is precisely where automated vulnerability detection lives: deciding, for a given operation in source code, whether a memory safety defect such as a null dereference, use-after-free, or double free can actually occur. We trace the unreliability of LLM-based vulnerability detection to a mechanism, the premature discharge of safety obligations, and argue that the remedy is not better prompting but a separation of roles: the component that interprets the code must not also be the one that decides a safety obligation is met. In this paper, we present LeanGuard, a neuro-symbolic framework that assigns each act to the side equipped for it. On the neural side, an LLM serves strictly as a semantic filter over candidate facts extracted from the abstract syntax tree (AST): it prunes spurious facts and keeps the real ones, but never discharges an obligation or decides the verdict on its own. On the symbolic side, the surviving facts are compiled into a verification model in Lean 4 (a formal proof assistant whose kernel accepts a conclusion only when it is formally proved), where every dangerous operation must be matched by a guard that provably covers it in scope; absent such a guard, the obligation stays open rather than being argued away. Because a function rarely arrives with full context, this symbolic model is necessarily partial: an unproved obligation is not yet a defect. An evidence-aware adjudicator therefore weighs the symbolic and neural verdicts by the quality of each. We instantiate the framework on five CWE classes to ask how far this division of labor can be pushed.",
      "title": "Neuro-Symbolic Reasoning for Vulnerability Detection",
      "updated": "2026-07-04",
      "url": "https://arxiv.org/abs/2607.03963"
    },
    {
      "age_days": 5,
      "arxiv_id": "2607.05810",
      "authors": [
        "Yueke Zhang",
        "Yifan Zhang",
        "Zihan Fang",
        "Kevin Leach",
        "Wei Zhang",
        "Yu Huang"
      ],
      "content_date": "2026-07-07",
      "freshness": "fresh",
      "id": "arxiv:2607.05810",
      "kind": "paper",
      "label": "adjacent",
      "matched_signals": [
        "reasoning_rl_distillation",
        "verifier_guided_reasoning"
      ],
      "published": "2026-07-07",
      "score": 4.9,
      "source": "arxiv-ai4math-core",
      "summary": "Code generation with large language models (LLMs) remains unreliable because generated programs can appear correct while still violating key semantic requirements in the natural language specification. Existing feedback-based methods improve over coder-only generation, but they often rely on unstructured critique or execution signals that do not explicitly identify what the code is semantically missing. We present SCOPE, a prover-initialized subgoal critic for code generation. SCOPE adapts a Lean-oriented prover model to produce three parseable feedback fields for downstream code generation: subgoals, gap analysis, and a robustness checklist. Our approach combines supervised fine-tuning, process-aligned reinforcement learning (RL), and feedback-guided inference, with two complementary rewards during RL: a dense reward for structured critique quality and a sparse reward based on whether the critique improves the coder's execution score. Experiments show that SCOPE improves over the compared feedback baselines. On LiveCodeBench V6, SCOPE achieves 39.4% pass@1, compared with 36.6% for Reflexion and 20.6% for the coder-only baseline. On BigCodeBench (Hard), it reaches 42.6%, surpassing Reflexion at 36.5% and coder-only generation at 34.5%. Further analysis shows that SCOPE's gains are concentrated in tasks with concrete semantic constraints and that its code corrections are more localized than Reflexion's.",
      "title": "SCOPE: Leveraging Subgoal Critiques for Code Generation",
      "updated": "2026-07-07",
      "url": "https://arxiv.org/abs/2607.05810"
    },
    {
      "age_days": 3,
      "arxiv_id": "2607.08656",
      "authors": [
        "Jaume de Dios Pont",
        "Lukas Liehr",
        "Mitchell A. Taylor"
      ],
      "content_date": "2026-07-09",
      "freshness": "fresh",
      "id": "arxiv:2607.08656",
      "kind": "paper",
      "label": "adjacent",
      "matched_signals": [
        "lean_formal_proving_agents"
      ],
      "published": "2026-07-09",
      "score": 3.5,
      "source": "arxiv-ai4math-core",
      "summary": "We prove that the Cantor measure with base $b$ does not admit a Fourier frame whenever $b > 1$ is an odd integer. In particular, this answers a question of Strichartz on the existence of a Fourier frame for the middle third Cantor measure. A formalization of our main result in Lean 4 is also provided.",
      "title": "Cantor measures with odd base do not admit Fourier frames",
      "updated": "2026-07-09",
      "url": "https://arxiv.org/abs/2607.08656"
    },
    {
      "age_days": 4,
      "arxiv_id": "2607.07857",
      "authors": [
        "Sirui Lu",
        "Erickson Tjoa",
        "J. Ignacio Cirac"
      ],
      "content_date": "2026-07-08",
      "freshness": "fresh",
      "id": "arxiv:2607.07857",
      "kind": "paper",
      "label": "adjacent",
      "matched_signals": [
        "autoformalization"
      ],
      "published": "2026-07-08",
      "score": 3.5,
      "source": "arxiv-ai4math-core",
      "summary": "We build a team of specialized large language-model agents and present an agent-driven workflow for research-level formalization in theoretical physics, with the autoformalization of the fundamental theorem of matrix-product states as a demonstration. The agents, coordinated through a structured mathematical blueprint and periodic human review, orchestrated and executed the full formalization autonomously. For some statements, the agents were able to explore new proof routes that are not part of the standard literature. Along the way the agents produced extensive tensor-network and quantum-information libraries not previously available in Mathlib, Lean's mathematical library. As a physical application, the formalization also extends towards symmetry-protected topological phases in one dimension. We find that the main bottleneck in large-scale autoformalization is enforcing mathematical intent and we provide a detailed study of the full process and various subtleties involved. We release the codebase as the library \\href{https://github.com/LionSR/TNLean}{TNLean}, together with a \\nChapters{}-chapter \\href{https://lionsr.github.io/TNLean/blueprint/}{blueprint} of the formalization effort.",
      "title": "Multi-agent Autoformalization of Tensor Network Theory",
      "updated": "2026-07-08",
      "url": "https://arxiv.org/abs/2607.07857"
    },
    {
      "age_days": 5,
      "arxiv_id": "2607.05987",
      "authors": [
        "Yuwei Lyu",
        "Kai Li"
      ],
      "content_date": "2026-07-07",
      "freshness": "fresh",
      "id": "arxiv:2607.05987",
      "kind": "paper",
      "label": "adjacent",
      "matched_signals": [
        "lean_formal_proving_agents"
      ],
      "published": "2026-07-07",
      "score": 3.5,
      "source": "arxiv-ai4math-core",
      "summary": "We formalize in Lean 4 a complete combinatorial route from Scarf's theorem to Brouwer's fixed point theorem and to the existence of mixed Nash equilibria in finite games. The development follows Ivanov's indexed-order formulation of Scarf's theorem, formalizes the room--door incidence structure and parity argument, instantiates the theorem on finite grids of the standard simplex, and carries out the compactness and continuity argument needed to obtain a fixed point. We then extend the result to finite products of simplices by an explicit embedding--projection construction and use this product theorem to prove mixed Nash equilibrium existence via the Nash map. As a secondary by-product, we derive BrouwerBench, a preliminary 80-item Lean-grounded benchmark for probing proof-structure understanding within this single formal development.",
      "title": "Formalizing Scarf, Brouwer, and Nash in Lean",
      "updated": "2026-07-07",
      "url": "https://arxiv.org/abs/2607.05987"
    },
    {
      "age_days": 5,
      "arxiv_id": "2607.06089",
      "authors": [
        "Serhii Zabolotnii"
      ],
      "content_date": "2026-07-07",
      "freshness": "fresh",
      "id": "arxiv:2607.06089",
      "kind": "paper",
      "label": "adjacent",
      "matched_signals": [
        "lean_formal_proving_agents"
      ],
      "published": "2026-07-07",
      "score": 3.5,
      "source": "arxiv-ai4math-core",
      "summary": "We study binary classification under shared-generator elliptical class-conditional distributions. The log-likelihood ratio is an additive function of the two squared Mahalanobis radii, with radial link $\\varphi=\\log g$; QDA is recovered only when this link is affine. We derive the Bayes radial-link family from the within-class radius law and estimate it by a finite fractional-power stochastic-polynomial projection instead of tuning a generic spline. The link is identifiable from the radius law, the plug-in estimator is $\\sqrt{n}$-consistent and asymptotically normal under finite-moment regularity conditions, and the induced classifier is asymptotically Bayes-optimal in an iterated sieve limit. The structural bridge, GAM membership, and identity-link/affine-generator dichotomy are verified in Lean 4 without unproven placeholders. Against the global Mahalanobis-GAM of Ghosh et al. (2025), reimplemented with mgcv REML splines at equal input budget, the derived link is never significantly worse on three UCI benchmarks and is decisively better on breast_cancer ($[+0.009,+0.021]$ global, $[+0.109,+0.136]$ global+local). Across six real financial series under temporal-dependence-robust validation, it is never significantly worse than the fitted GAM and is significantly better on three of five heavy-tailed series plus the light-tailed control. Relative to QDA, it improves the heaviest-tailed series (oil $[+0.024,+0.070]$, S&P 500 $[+0.038,+0.126]$, JPY/USD $[+0.009,+0.047]$) and ties elsewhere. A closed-form rate simulation corroborates the $\\sqrt{n}$ rate and the predicted excess-risk dichotomy between QDA's approximation-limited floor and the derived link's vanishing excess risk. The contribution is no significant loss relative to a tuned global GAM without spline smoothing-parameter selection, plus improved accuracy over QDA where generator curvature matters.",
      "title": "Closed-form fractional radial links for elliptical Mahalanobis discriminant analysis",
      "updated": "2026-07-07",
      "url": "https://arxiv.org/abs/2607.06089"
    },
    {
      "age_days": 5,
      "arxiv_id": "2607.06379",
      "authors": [
        "Michael R. Douglas"
      ],
      "content_date": "2026-07-07",
      "freshness": "fresh",
      "id": "arxiv:2607.06379",
      "kind": "paper",
      "label": "adjacent",
      "matched_signals": [
        "lean_formal_proving_agents"
      ],
      "published": "2026-07-07",
      "score": 3.5,
      "source": "arxiv-ai4math-core",
      "summary": "Mathematicians have embraced interactive theorem provers with growing enthusiasm -- building large shared libraries and machine-checking a string of landmark results. Theoretical physics is different: most of its results are not theorems but justified by arguments the community trusts without a rigorous proof. For many -- the one we treat here among them -- no rigorous proof is within reach. For 4d Yang--Mills theory, deriving exact rigorous results from first principles would first require constructing the interacting theory nonperturbatively, which is a sizable piece of one of the Clay Millennium prize problems. We argue here that an interactive theorem prover can be used to verify some non-rigorous physics arguments. The method is to postulate a short list of explicit, named physical postulates, which imply the physical results by virtue of a machine-checkable proof. The trust that remains then rests on that short, inspectable list, and the prover can report, for any downstream result, exactly which assumptions it used. We carry this out for the Seiberg--Witten solution of ${N}=2$ $SU(2)$ super-Yang--Mills -- the genus-one case -- formalized in Lean 4; the higher-genus $SU(N)$ generalization is developed in the same repository as an axiomatized skeleton and left to future work. We describe what is proved, what is assumed, how the assumptions are checked -- external review and an independent numerical oracle -- and why this discipline is a sound standard for validating AI-generated results in theoretical physics. What we offer is a discipline, reviewable on its own terms: a reader may take the Seiberg--Witten mathematics on trust and still assess the formalization method.",
      "title": "Axioms for physical reasoning: codifying the Seiberg--Witten solution in Lean",
      "updated": "2026-07-07",
      "url": "https://arxiv.org/abs/2607.06379"
    },
    {
      "age_days": 6,
      "arxiv_id": "2607.07727",
      "authors": [
        "Wen G. Gong"
      ],
      "content_date": "2026-07-06",
      "freshness": "fresh",
      "id": "arxiv:2607.07727",
      "kind": "paper",
      "label": "adjacent",
      "matched_signals": [
        "verifier_guided_reasoning"
      ],
      "published": "2026-07-06",
      "score": 3.5,
      "source": "arxiv-ai4math-core",
      "summary": "We present SPL (Structured Prompt Language), a declarative language that composes deterministic and probabilistic computation modes in a single specification. While existing frameworks separate these -- orchestration systems (AutoGen, CrewAI, LangGraph) for LLM calls, symbolic tools (SymPy, SageMath, Lean) for computation -- SPL unifies them. It provides GENERATE/EVALUATE for probabilistic computation and SOLVE/ASSERT for deterministic computation, sharing syntax, variable bindings, and runtime routing. A .spl specification runs unchanged across local nodes (Ollama), cloud APIs (OpenRouter, Anthropic), and distributed grids (Momagrid), with model and verifier selection deferred to invocation time. We validate SPL through an extensive 78-recipe cookbook and a controlled 1,200-run experiment (10 models x 20 problems x 2 arms x 3 repetitions; the 20 problems span 6 difficulty tiers). The solver arm achieves 82-93% machine-verified correctness (sonnet-4-6: 85%, gemma4:e2b: 93%) while the LLM-only arm measures output production without mathematical verification, making the comparison one of verified correctness against unverified fluency. A backend difficulty gradient emerges (SymPy 78%, Sage 54%), and the dominant failure mode is solver_error (kernel-rejected expressions), not format non-compliance.",
      "title": "SPL: Orchestrating Workflows with Declarative Deterministic-Probabilistic Composition",
      "updated": "2026-07-06",
      "url": "https://arxiv.org/abs/2607.07727"
    },
    {
      "age_days": 6,
      "arxiv_id": "2607.05492",
      "authors": [
        "Kazumi Kasaura",
        "Kei Tsukamoto",
        "Kento Mori",
        "Risa Mizuno",
        "Takahiro Namatame",
        "Yuta Oriike",
        "Masaya Taniguchi",
        "Sho Sonoda",
        "Hayata Yamasaki"
      ],
      "content_date": "2026-07-06",
      "freshness": "fresh",
      "id": "arxiv:2607.05492",
      "kind": "paper",
      "label": "adjacent",
      "matched_signals": [
        "lean_formal_proving_agents"
      ],
      "published": "2026-07-06",
      "score": 3.5,
      "source": "arxiv-ai4math-core",
      "summary": "Quantum information theory is built on entropic quantities; among them, the sandwiched Rényi relative entropy is a fundamental divergence with various applications, and its data processing inequality (DPI) under quantum channels is a cornerstone result. In this work, we present a Lean 4 library for quantum information, designed as a reusable formal infrastructure for theoretical analysis. As a central demonstration of the library, we formalize the DPI for the sandwiched Rényi relative entropy for positive semidefinite operators on finite-dimensional quantum systems. The library provides a basis-independent operator-theoretic framework for finite-dimensional quantum mechanics compatible with the standard mathematical library Mathlib, including reusable interfaces for finite-dimensional systems, states, channels, tensor products, partial traces, Choi operators, Kraus representations, and Stinespring representations. It also builds infrastructure for noncommutative trace inequalities, including operator monotonicity and convexity via the real continuous functional calculus, block-operator positivity, Hilbert-Schmidt operator spaces, Jensen's operator inequality, generalized perspectives, operator power means, and Lieb-Ando trace inequalities. On top of this framework, we formalize entropy-specific ingredients for the DPI: variational formulas for the sandwiched quasi-entropy via Young and reverse-Young inequalities, tensor-product compatibility of real powers, and Haar measures on unitary groups. Together, these components yield a Lean formalization of the DPI, give strong subadditivity as a corollary, and provide the last missing component needed to complete the Lean formalization of the generalized quantum Stein's lemma. More broadly, the development provides machine-checkable foundations for future formalized and AI-assisted research in quantum information theory.",
      "title": "Lean-Quantum: Toward AI-Assisted Formalization of Quantum Information",
      "updated": "2026-07-06",
      "url": "https://arxiv.org/abs/2607.05492"
    },
    {
      "age_days": 7,
      "arxiv_id": "2607.04534",
      "authors": [
        "Manuel Israel Cázares",
        "Wenlin Zhang",
        "Haobo Ma"
      ],
      "content_date": "2026-07-05",
      "freshness": "fresh",
      "id": "arxiv:2607.04534",
      "kind": "paper",
      "label": "adjacent",
      "matched_signals": [
        "lean_formal_proving_agents"
      ],
      "published": "2026-07-05",
      "score": 3.5,
      "source": "arxiv-ai4math-core",
      "summary": "We present an empirical study of structural routing failure in large language models (LLMs) over a formally verified algebraic corpus. The task requires selecting the correct proof-mechanism label from a fixed closed template set for compact mathematical objects drawn from the FiberRing formalization in Lean 4, where each item is anchored to a Lean-verified artifact and assigned a label from the corresponding certificate family. Our central finding is a mechanism-level routing ceiling: under blind conditions, gpt-oss-120b achieves 80.3% template accuracy on 22 FiberRing items (n=66; temperature=0, seed=0), while Llama 3.3 70B reaches 68.2%. Exposing a mechanism-bearing Lean verdict/witness cue (Condition A2) raises accuracy to 90.9% and 81.8% -- gaps of +10.6 and +13.6 pp termed cue-induced routing uplift. The dominant failure is a CRT-to-ring-equivalence misroute: gpt-oss-120b misroutes 7 of 12 CRT items (58.3%) blind, zero under A2. A cross-model dissociation in Llama is notable: verdict accuracy is identical in both conditions (95.5%), while template accuracy improves 13.6 pp -- confirming that truth inference and proof-mechanism classification are separable capacities. A cross-corpus extension (Set B; 6 POM/CollisionKernel items, 72 evaluations) provides a small cross-module check: CRT-granularity compression reappears with different labels, and an inverse cross-model dissociation emerges. These findings extend the router hypothesis (Cazares 2026) to formal algebraic structures. The full pipeline, manifest, and results are at https://github.com/bytepro-ai/fiber-routing-eval.",
      "title": "Mechanism-level routing failure in LLMs over Lean-verified algebraic structures",
      "updated": "2026-07-05",
      "url": "https://arxiv.org/abs/2607.04534"
    },
    {
      "age_days": 9,
      "arxiv_id": "2607.03406",
      "authors": [
        "Simon Jeanteur",
        "Lorenzo Veronese",
        "Magdalena Solitro",
        "Matteo Maffei"
      ],
      "content_date": "2026-07-03",
      "freshness": "fresh",
      "id": "arxiv:2607.03406",
      "kind": "paper",
      "label": "adjacent",
      "matched_signals": [
        "verifier_guided_reasoning"
      ],
      "published": "2026-07-03",
      "score": 3.5,
      "source": "arxiv-ai4math-core",
      "summary": "Computer-aided formal verification is a widely used approach for the symbolic analysis of cryptographic protocols. However, many modern protocols rely on features that remain challenging for existing techniques. In particular, reasoning about state, time-dependent behavior, inductively defined data structures, unbounded executions, and conditional secrecy requires a level of expressiveness that is difficult to reconcile with effective automation. As a result, protocol verification has largely followed two disjoint paths: fully automated methods with limited expressiveness, or interactive proofs in general-purpose theorem provers that offer flexibility but only limited, non-specialized automation. We present an orthogonal approach that bridges this gap by combining compositional type-based reasoning with trace-based reasoning, enabling modular verification of stateful and unbounded protocols. Guided by the language-and-automation co-design (LAC) principle, our approach delivers protocol-specific automation while retaining high expressiveness. We implement this framework as the LeanDY library for the Lean proof assistant, building on and extending the design of DY*, and combining protocol-specific automation with interactive proofs. Our framework supports, in a unified setting, a broad class of functional and security requirements, including secrecy and authentication for stateful protocols, as well as recursive conditional secrecy for protocols using XOR. We formalize SegWit-style blockchain primitives in LeanDY and demonstrate its expressiveness by carrying out an in-depth formalization of payment channels on top of this blockchain model, verifying punishment mechanisms and properties that depend on chain liveness.",
      "title": "LeanDY: Type-Based and Trace-Based Symbolic Protocol Verification in Lean",
      "updated": "2026-07-07",
      "url": "https://arxiv.org/abs/2607.03406"
    },
    {
      "age_days": 10,
      "arxiv_id": "2607.01734",
      "authors": [
        "Simon Guilloud",
        "Sankalp Gambhir",
        "Samuel Chassot"
      ],
      "content_date": "2026-07-02",
      "freshness": "fresh",
      "id": "arxiv:2607.01734",
      "kind": "paper",
      "label": "adjacent",
      "matched_signals": [
        "autoformalization"
      ],
      "published": "2026-07-02",
      "score": 3.5,
      "source": "arxiv-ai4math-core",
      "summary": "We present a case study in reformalization, a variant of autoformalization in which the input proof is not natural language but a formal development in a different proof assistant. Concretely, we report three reformalizations of the Jordan Curve Theorem: from Mizar to Lean, from HOL Light to Lean, and from HOL Light to Agda. We analyse the results and identify pipeline design choices that matter for practical reformalization tasks.",
      "title": "Reformalization of the Jordan Curve Theorem",
      "updated": "2026-07-02",
      "url": "https://arxiv.org/abs/2607.01734"
    },
    {
      "age_days": 3,
      "arxiv_id": "2607.08366",
      "authors": [
        "José A. R. Fonollosa"
      ],
      "content_date": "2026-07-09",
      "freshness": "fresh",
      "id": "arxiv:2607.08366",
      "kind": "paper",
      "label": "adjacent",
      "matched_signals": [
        "general_ai_math_reasoning"
      ],
      "published": "2026-07-09",
      "score": 1.9,
      "source": "arxiv-ai4math-core",
      "summary": "Fix $n \\ge 2$. A set $A = \\{a_0 < a_1 < \\dots < a_{n-1}\\}$ of $n$ residues in $\\Z_N$ is \\emph{valid mod $N$} if the all-ones multiset is the \\emph{only} size-$n$ multiset drawn from $A$ whose sum is $p := \\sum_i a_i \\pmod N$. For the super-increasing set $A = \\{2^k - 1 : 0 \\le k \\le n-1\\}$ we determine the least valid modulus exactly: $\\Nmin(n) = 2^{\\,n} - 2^{\\lfloor \\log_2 n \\rfloor}$ for all $n \\ge 2$. Both directions of the proof are elementary, resting on a sharp minimal-digit-sum estimate for representations by binary coins, and the full theorem has been machine-checked in Lean~4/Mathlib for all $n$ (https://github.com/jarfo/min-modulus). We conjecture that no size-$n$ residue set admits a smaller valid modulus. This validity condition is exactly what makes the permanent of an $n \\times n$ matrix equal to a single coefficient of a row-product polynomial modulo $x^N - 1$, extractable by a size-$N$ discrete Fourier (or number-theoretic) transform; the theorem thus identifies the smallest transform, $N \\approx 2^n$, for which this evaluation is exact. That application -- and the resulting common framework for the classical formulas of Ryser and Glynn and this transform -- is developed in a companion paper [2].",
      "title": "Minimum modulus for the unique multiset-sum problem",
      "updated": "2026-07-09",
      "url": "https://arxiv.org/abs/2607.08366"
    },
    {
      "age_days": 4,
      "arxiv_id": "2607.06944",
      "authors": [
        "Shanwen Wang",
        "Yijun Yuan"
      ],
      "content_date": "2026-07-08",
      "freshness": "fresh",
      "id": "arxiv:2607.06944",
      "kind": "paper",
      "label": "adjacent",
      "matched_signals": [
        "general_ai_math_reasoning"
      ],
      "published": "2026-07-08",
      "score": 1.9,
      "source": "arxiv-ai4math-core",
      "summary": "Let $p$ be a prime number. We introduce a sparseness condition on the supports of $p$-adic Hahn series, and prove that this condition implies transcendence over $\\breve{\\mathbf Q}_p$, the completed maximal unramified extension of $\\mathbf{Q}_p$. As an application, we prove the order-type conjecture of $\\mathbf{Q}_p$-algebraic $p$-adic Hahn series with bounded support under the condition that the support has only finitely many accumulation points. All results in this paper have been fully formalized in the Lean theorem prover (v 4.31.0), building over Mathlib.",
      "title": "$p$-adic Hahn series with sparse support",
      "updated": "2026-07-08",
      "url": "https://arxiv.org/abs/2607.06944"
    },
    {
      "age_days": 5,
      "arxiv_id": "2607.05739",
      "authors": [
        "Ken Ono"
      ],
      "content_date": "2026-07-07",
      "freshness": "fresh",
      "id": "arxiv:2607.05739",
      "kind": "paper",
      "label": "adjacent",
      "matched_signals": [
        "general_ai_math_reasoning"
      ],
      "published": "2026-07-07",
      "score": 1.9,
      "source": "arxiv-ai4math-core",
      "summary": "For $n\\ge1$, we let $$x_n:=\\tan\\bigl(\\sum_{k=1}^{n}\\arctan k\\bigr).$$ In 2008, Amdeberhan, Medina, and Moll conjectured that $x_n\\not \\in \\mathbb{Z}$ for every $n\\ge5$. This was known for a set of positive integers of density $\\tfrac{120}{817}\\approx0.1469$. We prove that an integer value $x_n=m$ satisfies $|m|\\ge e^{(1/2+o(1))\\,n\\log n}$, which we use to deduce that $$\\#\\{\\,1\\leq n\\le N:x_n\\in\\mathbb{Z}\\,\\}=O(\\log N). $$ In particular, the conjecture holds for a density-one set of $n$. The results in this note were formalized in Lean/Mathlib and produced autonomously by AxiomProver from natural-language statements.",
      "title": "Integer values of $\\tan(\\arctan 1+\\arctan 2+\\cdots+\\arctan n)$ are rare",
      "updated": "2026-07-07",
      "url": "https://arxiv.org/abs/2607.05739"
    },
    {
      "age_days": 11,
      "arxiv_id": "2607.01544",
      "authors": [
        "Kenny Lau",
        "Seewoo Lee",
        "Ken Ono"
      ],
      "content_date": "2026-07-01",
      "freshness": "fresh",
      "id": "arxiv:2607.01544",
      "kind": "paper",
      "label": "adjacent",
      "matched_signals": [
        "general_ai_math_reasoning"
      ],
      "published": "2026-07-01",
      "score": 1.9,
      "source": "arxiv-ai4math-core",
      "summary": "The theory of $q$-series and basic hypergeometric series plays a crucial role at the intersection of combinatorics, number theory, and representation theory. From the classical partition identities of Euler and Jacobi to modern developments in class field theory, vertex operator algebras, and the Monstrous Moonshine conjecture, $q$-series provide the analytic framework for a wide range of profound applications. In this paper, we discuss the formalization of this theory in the Lean proof assistant, a process that requires careful design of scalable and versatile structures to reconcile formal algebraic identities with analytic convergence properties. We address these foundational challenges by focusing on the construction of $q$-Pochhammer symbols, $q$-binomial coefficients, Bailey's Lemma and similar primitives. To demonstrate the utility of this work, we provide fully verified proofs of the Jacobi Triple Product formula and the celebrated Rogers-Ramanujan identities, which serve as both historical and technical benchmarks for the field. This work establishes a rigorous computational foundation for the future formalization of mock theta functions, modular forms, and the diverse algebraic structures that underpin their applications across mathematics and physics.",
      "title": "Formalized $q$-series: The Rogers-Ramanujan Identities and Beyond",
      "updated": "2026-07-01",
      "url": "https://arxiv.org/abs/2607.01544"
    },
    {
      "age_days": 0,
      "authors": [
        "Nailin Guan"
      ],
      "content_date": "2026-07-12",
      "freshness": "fresh",
      "id": "github:leanprover-community/mathlib4:7c59a0a6cbff",
      "kind": "github_update",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-07-12",
      "repo": "leanprover-community/mathlib4",
      "score": 0.8,
      "source": "mathlib4-github",
      "summary": "Recent commit on leanprover-community/mathlib4.",
      "title": "leanprover-community/mathlib4: feat(RingTheory/Ideal): prime avoidance for maximal ideal (#40814)",
      "updated": "2026-07-12",
      "url": "https://github.com/leanprover-community/mathlib4/commit/7c59a0a6cbff267024f1c701907f704c690782fb"
    },
    {
      "age_days": 0,
      "authors": [
        "Noah Walker"
      ],
      "content_date": "2026-07-12",
      "freshness": "fresh",
      "id": "github:leanprover-community/mathlib4:6ee55367d7ab",
      "kind": "github_update",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-07-12",
      "repo": "leanprover-community/mathlib4",
      "score": 0.8,
      "source": "mathlib4-github",
      "summary": "Recent commit on leanprover-community/mathlib4.",
      "title": "leanprover-community/mathlib4: feat(Algebra/GroupWithZero): prove that `Is{Left,Right}CancelMulZero` is Dedekind-finite (#41453)",
      "updated": "2026-07-12",
      "url": "https://github.com/leanprover-community/mathlib4/commit/6ee55367d7ab04d9a9d24751d6b493b7e023e27d"
    },
    {
      "age_days": 0,
      "authors": [
        "mathlib-update-dependencies[bot]"
      ],
      "content_date": "2026-07-12",
      "freshness": "fresh",
      "id": "github:leanprover-community/mathlib4:bd2f30dcd35b",
      "kind": "github_update",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-07-12",
      "repo": "leanprover-community/mathlib4",
      "score": 0.8,
      "source": "mathlib4-github",
      "summary": "Recent commit on leanprover-community/mathlib4.",
      "title": "leanprover-community/mathlib4: chore: update Mathlib dependencies 2026-07-12 (#41650)",
      "updated": "2026-07-12",
      "url": "https://github.com/leanprover-community/mathlib4/commit/bd2f30dcd35b91fe8cf67d333de797947f6173e2"
    },
    {
      "age_days": 0,
      "authors": [
        "Sebastien Gouezel"
      ],
      "content_date": "2026-07-12",
      "freshness": "fresh",
      "id": "github:leanprover-community/mathlib4:9de183cf4048",
      "kind": "github_update",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-07-12",
      "repo": "leanprover-community/mathlib4",
      "score": 0.8,
      "source": "mathlib4-github",
      "summary": "Recent commit on leanprover-community/mathlib4.",
      "title": "leanprover-community/mathlib4: chore: fix nsmul and zsmul non-reducible diamond in SplittingField (#40708)",
      "updated": "2026-07-12",
      "url": "https://github.com/leanprover-community/mathlib4/commit/9de183cf4048aa3d2198571106af8d53ac8ff4fd"
    },
    {
      "age_days": 0,
      "authors": [
        "mathlib-nolints[bot]"
      ],
      "content_date": "2026-07-12",
      "freshness": "fresh",
      "id": "github:leanprover-community/mathlib4:79aee35d9696",
      "kind": "github_update",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-07-12",
      "repo": "leanprover-community/mathlib4",
      "score": 0.8,
      "source": "mathlib4-github",
      "summary": "Recent commit on leanprover-community/mathlib4.",
      "title": "leanprover-community/mathlib4: chore(scripts): update nolints.json (#41627)",
      "updated": "2026-07-12",
      "url": "https://github.com/leanprover-community/mathlib4/commit/79aee35d9696d759b73eed71d7dde666750bc35e"
    },
    {
      "age_days": 0,
      "authors": [
        "Yongle Hu"
      ],
      "content_date": "2026-07-12",
      "freshness": "fresh",
      "id": "github:leanprover-community/mathlib4:2800ac2bd3c2",
      "kind": "github_update",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-07-12",
      "repo": "leanprover-community/mathlib4",
      "score": 0.8,
      "source": "mathlib4-github",
      "summary": "Recent commit on leanprover-community/mathlib4.",
      "title": "leanprover-community/mathlib4: chore(RingTheory/Polynomial/Basic): remove unused `haveI` and defeq abuse (#41608)",
      "updated": "2026-07-12",
      "url": "https://github.com/leanprover-community/mathlib4/commit/2800ac2bd3c2653f48647df6cb15b22c6e6e5039"
    },
    {
      "age_days": 0,
      "authors": [
        "Scott Carnahan"
      ],
      "content_date": "2026-07-12",
      "freshness": "fresh",
      "id": "github:leanprover-community/mathlib4:c0cbc05a6f61",
      "kind": "github_update",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-07-12",
      "repo": "leanprover-community/mathlib4",
      "score": 0.8,
      "source": "mathlib4-github",
      "summary": "Recent commit on leanprover-community/mathlib4.",
      "title": "leanprover-community/mathlib4: chore(LinearAlgebra/TensorProduct/Map): add symm_lTensor and symm_rTensor (#41550)",
      "updated": "2026-07-12",
      "url": "https://github.com/leanprover-community/mathlib4/commit/c0cbc05a6f61999c6e6e8afdcbbe0b3daff95a27"
    },
    {
      "age_days": 0,
      "authors": [
        "mathlib-splicebot[bot]"
      ],
      "content_date": "2026-07-12",
      "freshness": "fresh",
      "id": "github:leanprover-community/mathlib4:b95982f413bf",
      "kind": "github_update",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-07-12",
      "repo": "leanprover-community/mathlib4",
      "score": 0.8,
      "source": "mathlib4-github",
      "summary": "Recent commit on leanprover-community/mathlib4.",
      "title": "leanprover-community/mathlib4: chore(Geometry/Manifold/ContMDiff/Atlas): generalise `mem_maximalAtlas_of_contMDiffOn` and friends (#40720)",
      "updated": "2026-07-12",
      "url": "https://github.com/leanprover-community/mathlib4/commit/b95982f413bf08801f23ed434feeb52dad009330"
    },
    {
      "age_days": 1,
      "authors": [
        "Leonardo de Moura"
      ],
      "content_date": "2026-07-11",
      "freshness": "fresh",
      "id": "github:leanprover/lean4:2aaf675cfd53",
      "kind": "github_update",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-07-11",
      "repo": "leanprover/lean4",
      "score": 0.8,
      "source": "lean4-github",
      "summary": "Recent commit on leanprover/lean4.",
      "title": "leanprover/lean4: perf: try to skip `check` at `applyAbstractResult?` (#14361)",
      "updated": "2026-07-11",
      "url": "https://github.com/leanprover/lean4/commit/2aaf675cfd53143c757178c1a5954ded3e930ff5"
    },
    {
      "age_days": 1,
      "authors": [
        "Henrik Böving"
      ],
      "content_date": "2026-07-11",
      "freshness": "fresh",
      "id": "github:leanprover/lean4:630f130affce",
      "kind": "github_update",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-07-11",
      "repo": "leanprover/lean4",
      "score": 0.8,
      "source": "lean4-github",
      "summary": "Recent commit on leanprover/lean4.",
      "title": "leanprover/lean4: perf: inline unsafe term helpers (#14365)",
      "updated": "2026-07-11",
      "url": "https://github.com/leanprover/lean4/commit/630f130affce3a7a0065014dd4ea4603c42c06b2"
    },
    {
      "age_days": 1,
      "authors": [
        "Mac Malone"
      ],
      "content_date": "2026-07-11",
      "freshness": "fresh",
      "id": "github:leanprover/lean4:12c859a4d735",
      "kind": "github_update",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-07-11",
      "repo": "leanprover/lean4",
      "score": 0.8,
      "source": "lean4-github",
      "summary": "Recent commit on leanprover/lean4.",
      "title": "leanprover/lean4: fix: lake: no lib files for exe template (#14366)",
      "updated": "2026-07-11",
      "url": "https://github.com/leanprover/lean4/commit/12c859a4d735fbcdb3ebbddb42dcc8619bcc7a7d"
    },
    {
      "age_days": 1,
      "authors": [
        "Mac Malone"
      ],
      "content_date": "2026-07-11",
      "freshness": "fresh",
      "id": "github:leanprover/lean4:49a89d6148a4",
      "kind": "github_update",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-07-11",
      "repo": "leanprover/lean4",
      "score": 0.8,
      "source": "lean4-github",
      "summary": "Recent commit on leanprover/lean4.",
      "title": "leanprover/lean4: feat: lake: `presetup` module facet (#14300)",
      "updated": "2026-07-11",
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      "freshness": "fresh",
      "id": "github:leanprover-community/mathlib4:a08ff0d0f4a0",
      "kind": "github_update",
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      "matched_signals": [],
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      "repo": "leanprover-community/mathlib4",
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      "source": "mathlib4-github",
      "summary": "Recent commit on leanprover-community/mathlib4.",
      "title": "leanprover-community/mathlib4: chore: refactor bench suite (#41587)",
      "updated": "2026-07-10",
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    {
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      "authors": [
        "Garmelon"
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      "content_date": "2026-07-10",
      "freshness": "fresh",
      "id": "github:leanprover-community/mathlib4:286a924d324f",
      "kind": "github_update",
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      "matched_signals": [],
      "published": "2026-07-10",
      "repo": "leanprover-community/mathlib4",
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      "source": "mathlib4-github",
      "summary": "Recent commit on leanprover-community/mathlib4.",
      "title": "leanprover-community/mathlib4: chore: measure longest (re-)build path instructions (#41331)",
      "updated": "2026-07-10",
      "url": "https://github.com/leanprover-community/mathlib4/commit/286a924d324fcb0280beb30182341d0f0d52ef01"
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    {
      "age_days": 2,
      "authors": [
        "Kim Morrison"
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      "content_date": "2026-07-10",
      "freshness": "fresh",
      "id": "github:leanprover-community/mathlib4:c368140668f5",
      "kind": "github_update",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-07-10",
      "repo": "leanprover-community/mathlib4",
      "score": 0.8,
      "source": "mathlib4-github",
      "summary": "Recent commit on leanprover-community/mathlib4.",
      "title": "leanprover-community/mathlib4: chore(Topology/DiscreteSubset): fix typo eventualy -> eventually in lemma name (#41569)",
      "updated": "2026-07-10",
      "url": "https://github.com/leanprover-community/mathlib4/commit/c368140668f5fa16a1bd977448c1f665d48c3df4"
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    {
      "age_days": 2,
      "authors": [
        "Thomas Browning"
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      "content_date": "2026-07-10",
      "freshness": "fresh",
      "id": "github:leanprover-community/mathlib4:19e77d298b5e",
      "kind": "github_update",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-07-10",
      "repo": "leanprover-community/mathlib4",
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      "summary": "Recent commit on leanprover-community/mathlib4.",
      "title": "leanprover-community/mathlib4: chore(CategoryTheory/Sites/Descent/Precoverage): deprecate duplicate theorem (#41553)",
      "updated": "2026-07-10",
      "url": "https://github.com/leanprover-community/mathlib4/commit/19e77d298b5e3a9df932543a36d10b9874b4145b"
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    {
      "age_days": 3,
      "authors": [
        "Leonardo de Moura"
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      "freshness": "fresh",
      "id": "github:leanprover/lean4:4c1cf2f9806a",
      "kind": "github_update",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-07-09",
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      "source": "lean4-github",
      "summary": "Recent commit on leanprover/lean4.",
      "title": "leanprover/lean4: perf: construct `IntModule.OfNatModule.Q type` manually (#14346)",
      "updated": "2026-07-09",
      "url": "https://github.com/leanprover/lean4/commit/4c1cf2f9806ac782dc2bcedba9aba7229ce99ac0"
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    {
      "age_days": 3,
      "authors": [
        "Leonardo de Moura"
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      "content_date": "2026-07-09",
      "freshness": "fresh",
      "id": "github:leanprover/lean4:da19ea085d7c",
      "kind": "github_update",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-07-09",
      "repo": "leanprover/lean4",
      "score": 0.8,
      "source": "lean4-github",
      "summary": "Recent commit on leanprover/lean4.",
      "title": "leanprover/lean4: perf: assign synthesized instances directly when the goal type is ground (#14330)",
      "updated": "2026-07-09",
      "url": "https://github.com/leanprover/lean4/commit/da19ea085d7cdba0131ec59b4f678b0910304878"
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    {
      "age_days": 3,
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        "Sebastian Ullrich"
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      "content_date": "2026-07-09",
      "freshness": "fresh",
      "id": "github:leanprover/lean4:804bb1281b43",
      "kind": "github_update",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-07-09",
      "repo": "leanprover/lean4",
      "score": 0.8,
      "source": "lean4-github",
      "summary": "Recent commit on leanprover/lean4.",
      "title": "leanprover/lean4: fix: run shell main on a thread with the configured stack size (#14343)",
      "updated": "2026-07-09",
      "url": "https://github.com/leanprover/lean4/commit/804bb1281b4365af2195fb716b7982a7f37eee4d"
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    {
      "age_days": 3,
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        "Joachim Breitner"
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      "content_date": "2026-07-09",
      "freshness": "fresh",
      "id": "github:leanprover/lean4:e599b3e12cc3",
      "kind": "github_update",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-07-09",
      "repo": "leanprover/lean4",
      "score": 0.8,
      "source": "lean4-github",
      "summary": "Recent commit on leanprover/lean4.",
      "title": "leanprover/lean4: fix: `partial_fixpoint` error on nested recursive calls (#14335)",
      "updated": "2026-07-09",
      "url": "https://github.com/leanprover/lean4/commit/e599b3e12cc3da4f9240cd649529d6edaf03abaf"
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    {
      "age_days": 3,
      "authors": [
        "Julia Markus Himmel"
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      "content_date": "2026-07-09",
      "freshness": "fresh",
      "id": "github:leanprover/lean4:41ace69675a8",
      "kind": "github_update",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-07-09",
      "repo": "leanprover/lean4",
      "score": 0.8,
      "source": "lean4-github",
      "summary": "Recent commit on leanprover/lean4.",
      "title": "leanprover/lean4: feat: lint on `open` statements within `namespace` which do not open all candidate namespaces (#14325)",
      "updated": "2026-07-09",
      "url": "https://github.com/leanprover/lean4/commit/41ace69675a807d954d1a6938ced70e26bcded17"
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    {
      "age_days": 3,
      "authors": [
        "Julia Markus Himmel"
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      "content_date": "2026-07-09",
      "freshness": "fresh",
      "id": "github:leanprover/lean4:5ae4723270cd",
      "kind": "github_update",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-07-09",
      "repo": "leanprover/lean4",
      "score": 0.8,
      "source": "lean4-github",
      "summary": "Recent commit on leanprover/lean4.",
      "title": "leanprover/lean4: feat: error on self-deprecation (#14333)",
      "updated": "2026-07-09",
      "url": "https://github.com/leanprover/lean4/commit/5ae4723270cd968db64e8f307ac41a3eaa592466"
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    {
      "age_days": 3,
      "authors": [
        "Garmelon"
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      "content_date": "2026-07-09",
      "freshness": "fresh",
      "id": "github:leanprover/lean4:69d6d5229239",
      "kind": "github_update",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-07-09",
      "repo": "leanprover/lean4",
      "score": 0.8,
      "source": "lean4-github",
      "summary": "Recent commit on leanprover/lean4.",
      "title": "leanprover/lean4: chore: repeat some benchmarks to avoid outliers (#14342)",
      "updated": "2026-07-09",
      "url": "https://github.com/leanprover/lean4/commit/69d6d5229239dadbff0853c8d8629a5097da8a2b"
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    {
      "age_days": 3,
      "authors": [
        "Julia Markus Himmel"
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      "content_date": "2026-07-09",
      "freshness": "fresh",
      "id": "github:leanprover/lean4:387908cb61ec",
      "kind": "github_update",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-07-09",
      "repo": "leanprover/lean4",
      "score": 0.8,
      "source": "lean4-github",
      "summary": "Recent commit on leanprover/lean4.",
      "title": "leanprover/lean4: chore: reduce namespace pollution (#14308)",
      "updated": "2026-07-09",
      "url": "https://github.com/leanprover/lean4/commit/387908cb61eca4b499242240eda2dde1ebc589f9"
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    {
      "age_days": 3,
      "authors": [
        "Sebastian Ullrich"
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      "content_date": "2026-07-09",
      "freshness": "fresh",
      "id": "github:leanprover/lean4:9d4bba94d5ae",
      "kind": "github_update",
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      "matched_signals": [],
      "published": "2026-07-09",
      "repo": "leanprover/lean4",
      "score": 0.8,
      "source": "lean4-github",
      "summary": "Recent commit on leanprover/lean4.",
      "title": "leanprover/lean4: chore: disable `interpreter.prefer_native` for further build stages as well (#13757)",
      "updated": "2026-07-09",
      "url": "https://github.com/leanprover/lean4/commit/9d4bba94d5aeb0de2c3d2236899b2206d3ddba39"
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      "age_days": 3,
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      "content_date": "2026-07-09",
      "freshness": "fresh",
      "id": "github:leanprover/lean4:03af2ed90848",
      "kind": "github_update",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-07-09",
      "repo": "leanprover/lean4",
      "score": 0.8,
      "source": "lean4-github",
      "summary": "Recent commit on leanprover/lean4.",
      "title": "leanprover/lean4: chore: `LAKE_CACHE_EXTRA_ARGS` (#14334)",
      "updated": "2026-07-09",
      "url": "https://github.com/leanprover/lean4/commit/03af2ed90848f8153aac5fa52467f1ea759f954b"
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    {
      "age_days": 3,
      "authors": [
        "Sebastian Ullrich"
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      "content_date": "2026-07-09",
      "freshness": "fresh",
      "id": "github:leanprover/lean4:1bb99fe3a25e",
      "kind": "github_update",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-07-09",
      "repo": "leanprover/lean4",
      "score": 0.8,
      "source": "lean4-github",
      "summary": "Recent commit on leanprover/lean4.",
      "title": "leanprover/lean4: chore: CI: show failed tests' output in test summary (#14337)",
      "updated": "2026-07-09",
      "url": "https://github.com/leanprover/lean4/commit/1bb99fe3a25e0b93a9a7a518bb1d27475e2f2745"
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    {
      "age_days": 4,
      "authors": [
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      "content_date": "2026-07-08",
      "freshness": "fresh",
      "id": "github:leanprover/lean4:6b303d979a1b",
      "kind": "github_update",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-07-08",
      "repo": "leanprover/lean4",
      "score": 0.8,
      "source": "lean4-github",
      "summary": "Recent commit on leanprover/lean4.",
      "title": "leanprover/lean4: feat: add hint to `unusedVariables` linter (#14259)",
      "updated": "2026-07-08",
      "url": "https://github.com/leanprover/lean4/commit/6b303d979a1bb740ef0c11d91dff0e3c8f86f36e"
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      "age_days": 4,
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      "content_date": "2026-07-08",
      "freshness": "fresh",
      "id": "github:leanprover/lean4:c1d6de5ef78e",
      "kind": "github_update",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-07-08",
      "repo": "leanprover/lean4",
      "score": 0.8,
      "source": "lean4-github",
      "summary": "Recent commit on leanprover/lean4.",
      "title": "leanprover/lean4: feat: add Insert instances for NameMap/NameSet (#14153)",
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      "url": "https://github.com/leanprover/lean4/commit/c1d6de5ef78e1ef320d89599031ab42bd3d8012d"
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    {
      "age_days": 4,
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      "content_date": "2026-07-08",
      "freshness": "fresh",
      "id": "github:leanprover/lean4:d14ff0ba3412",
      "kind": "github_update",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-07-08",
      "repo": "leanprover/lean4",
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      "source": "lean4-github",
      "summary": "Recent commit on leanprover/lean4.",
      "title": "leanprover/lean4: chore: fix namespaceing in `DeprecatedModule` (#14310)",
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      "url": "https://github.com/leanprover/lean4/commit/d14ff0ba34122b569f8e4a19265d5d2458fc00ac"
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    {
      "age_days": 4,
      "authors": [
        "Henrik Böving"
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      "content_date": "2026-07-08",
      "freshness": "fresh",
      "id": "github:leanprover/lean4:306bbbc83e53",
      "kind": "github_update",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-07-08",
      "repo": "leanprover/lean4",
      "score": 0.8,
      "source": "lean4-github",
      "summary": "Recent commit on leanprover/lean4.",
      "title": "leanprover/lean4: chore: disable binary stripping in release configuration (#14139)",
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    {
      "age_days": 3,
      "arxiv_id": "2607.08006",
      "authors": [
        "Naveen Ravi",
        "Nathan Wichmann",
        "Md. Wasi-ur- Rahman",
        "Aurelien Bouteiller",
        "Yıltan Hassan Temuçin",
        "Avinash Kethineedi",
        "Johnathan Alsop",
        "Brandon Potter",
        "Shubhendra Pal Singhal",
        "Jun Shirako",
        "Akihiro Hayashi",
        "Vivek Sarkar",
        "Lawrence C. Stewart",
        "Michael Beebe",
        "Benjamin Michalowicz",
        "Jeongnim Kim",
        "Thiago Teixeria",
        "Mark F. Brown",
        "Aaron Welch",
        "Oscar Hernandez",
        "Wendy Poole",
        "Steve Poole"
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      "content_date": "2026-07-09",
      "freshness": "fresh",
      "id": "arxiv:2607.08006",
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      "matched_signals": [],
      "published": "2026-07-09",
      "score": 0.5,
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      "summary": "Leadership-class HPC systems are now accelerator-centric, with GPUs providing most floating-point throughput and memory bandwidth. As next-generation systems increasingly integrate accelerators through high-speed memory fabrics and system interconnects, exposing larger tightly coupled device domains, \\ac{PGAS} models such as OpenSHMEM provide a natural abstraction for expressing fine-grained remote memory operations across these devices. While OpenSHMEM 1.x offers a lean PGAS model for irregular communication, atomics, fine-grained synchronization, and collectives, its memory model lacks portable semantics for accelerator architectures. As a result, existing GPU-enabled OpenSHMEM implementations differ in memory management, capability discovery, and operation semantics, limiting portability and ecosystem cohesion. This risks fracturing the community that OpenSHMEM was originally created to unify. This paper proposes an OpenSHMEM Auxiliary Specification for GPU-Aware Communication, designed as a lightweight, backward-compatible extension to OpenSHMEM 1.x. The auxiliary specification introduces a minimal memory model extension via a GPU-scoped memory space abstraction, along with capability queries and well-defined semantics for using \\acs{GPU}-attached buffers in RMA, atomic, synchronization, and collective operations. This is initially conceived through the lens of a host-initiated interface, although it provides a general set of semantics that also allow for optional device-initiated support. A central goal of this effort is to demonstrate that GPU-aware OpenSHMEM semantics can be specified and implemented across GPUs from multiple vendors, providing a practical and rapidly implementable step toward unification under a vendor-neutral specification while informing the design of future OpenSHMEM specifications.",
      "title": "Toward a Unified GPU-Aware OpenSHMEM Specification",
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      "url": "https://arxiv.org/abs/2607.08006"
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      "authors": [
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        "Eulalie Boucher",
        "Simon Lang",
        "Tomas Kral",
        "Paul Poli",
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        "Matthew Chantry",
        "Anthony McNally"
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      "content_date": "2026-07-08",
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      "summary": "Earth system reanalysis datasets are foundational for weather and climate research and provide the gridded training data used by most machine learning weather prediction systems. Here we show results from a prototype system that suggest that machine learning models trained only on Earth system observations can potentially be used to generate multi-decade global reanalyses without using physics-based numerical models. The resulting gridded fields capture large-scale atmospheric structure and variability across multiple timescales, while exhibiting signs of physical coherence in several key dynamical diagnostics. Evaluations of the prototype against held-out independent atmospheric observations indicate that the root mean square vector error of upper-level winds is close to that of ERA5 when compared at a consistent resolution, and that the standard deviation of the error at the surface is between that of 4th- and 5th-generation ECMWF reanalyses (ERA-Interim and ERA5). Furthermore, while traditional reanalysis production is computationally expensive, typically taking several years to produce, the reanalysis presented here was generated during the course of a single working day. These results suggest that observation-trained machine learning models offer a promising new approach for reanalysis production from observations alone.",
      "title": "Global reanalysis from observations alone with machine learning",
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      "authors": [
        "Jordan Painter",
        "Dipankar Srirag",
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        "Diptesh Kanojia",
        "Aditya Joshi",
        "Lu Yin"
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      "content_date": "2026-07-08",
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      "id": "arxiv:2607.07669",
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      "score": 0.5,
      "source": "arxiv-ai4math-core",
      "summary": "Large language models increasingly \\emph{understand} dialectal English, yet still \\emph{produce} only standard, US-leaning English, leaving dialectal generation, the harder half of the problem, largely unaddressed. We introduce \\textbf{DiaLLM}, which continually pretrains three open-weight language model families on the International Corpus of English and applies implicit and explicit post-training paradigms, each combined with three model alignment strategies, giving the first controlled comparison of these components across Australian, Indian, and Northern British English. Our results reveal that dialectal robustness and generation are \\emph{dissociated}: benchmarks are shaped by continual pretraining and SFT, while alignment visibly reshapes generation in ways benchmarks do not capture. Explicit variety-targeted adaptation produces output reliably recognised as dialectal and preferred over broad alignment, yet the method that most aggressively optimises the dialectal reward is not preferred by human evaluators. Independent linguistic analysis corroborates this reward-quality gap, most clearly on two of the three families. No single alignment method dominates, and closing the gap will require richer reward designs and continued investment in dialectal resources. We release all code, checkpoints, and preference datasets.",
      "title": "DiaLLM: An Investigation into the Robustness-Generation Gap in English Dialect Adaptation",
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      "url": "https://arxiv.org/abs/2607.07669"
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      "authors": [
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        "Jamal Agbanwa"
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      "freshness": "fresh",
      "id": "arxiv:2607.06627",
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      "summary": "Many Diophantine equations can be reduced to the question of whether, for a given non-degenerate quadratic form $F$ and a univariate polynomial $P$ with integer coefficients, $P(x)$ can be represented by $F$ for infinitely many values of $x$. We develop a method for answering this question for certain cubic and quartic polynomials $P$, as well as for certain polynomials of the form $P(x)=R(Q(x))$, where $R(t)$ and $Q(x)$ are polynomials of degree $3$ and $2$, respectively. Applying this method with $F(y,z)=y^2+z^2$, $R(t)=t^3-4$ and $Q(x)=x^2$, we conclude that $x^6-4$ is a sum of two squares infinitely often. In turn, this implies that the equation $y^2+x^3y+z^2+1=0$ has infinitely many integer solutions. Prior to this work, it was the shortest equation for which it was open whether its integer solution set is finite or infinite. We conclude with a list of the new shortest equations whose finiteness problem remains open. All main results of this paper has been formalized in Lean using Aristotle.",
      "title": "On the polynomial values represented by quadratic forms",
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      "url": "https://arxiv.org/abs/2607.06627"
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      "authors": [
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        "Zichen Wang",
        "Zaiwen Wen"
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      "freshness": "fresh",
      "id": "arxiv:2607.05874",
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      "score": 0.5,
      "source": "arxiv-ai4math-core",
      "summary": "Existence proofs for many matrix decompositions share a recursive routine: a local transformation prepares the matrix, a slice is selected, a recursive solution is obtained, and the result is lifted and transported back. Formalizing this routine uniformly in dependent type theory is difficult because recursive subproblems may change index types, and reconstruction must preserve structural predicates across block embeddings and reindexings. We develop a Lean~4 framework that separates decomposition schemas, transformations, reduction strategies, measures, lifting, transport, and subtype induction. The framework uses general index types, packages square and rectangular matrices in universe types, and provides a decomposition driver that assembles strategy data into subtype-induction instances. It has been instantiated across PLU, LU, LDL/Cholesky, QR variants, Gauss rank normal form, Hessenberg reductions, Schur variants, normal spectral decomposition, SVD, bidiagonalization, tridiagonalization, UTV, Smith normal form, rational canonical form, and Jordan-type forms at varying levels of statement strength. Across these instances, repeated decomposition proofs are best treated not as separate tasks but as instances of a more general inductive statement whose interface records a certified proof path compatible with the chosen decomposition statement.",
      "title": "A Unified Framework for Formalizing Matrix Decomposition Proofs",
      "updated": "2026-07-07",
      "url": "https://arxiv.org/abs/2607.05874"
    },
    {
      "age_days": 6,
      "arxiv_id": "2607.05626",
      "authors": [
        "Muhammad Rizwan",
        "David Nabergoj",
        "Jure Demšar"
      ],
      "content_date": "2026-07-06",
      "freshness": "fresh",
      "id": "arxiv:2607.05626",
      "kind": "paper",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-07-06",
      "score": 0.5,
      "source": "arxiv-ai4math-core",
      "summary": "Background: Depression frequently co-occurs with ADHD and autism spectrum disorder (ASD), but population-level differences in symptom expression between these groups remain underexplored. Objective: We examined whether social media users with ADHD and ASD differ in how they express DSM-5 depressive symptoms in their tweets, and whether differences persist across varying levels of depressive-content filtering. Methods: We analysed 1,282,437 tweets from 792 users (622 ADHD; 170 ASD) with self-reported diagnoses on Twitter. Tweets were pre-filtered for depressive relevance using zero-shot NLI, then classified into nine DSM-5 symptoms using MentalRoBERTa fine-tuned on ReDSM5. Profiles were mean-centered per user. We applied L1-penalised logistic regression with cross-validation to distinguish ADHD from ASD users, complemented by Pearson correlations for symptom co-occurrence, and tested robustness across five filtering thresholds using bootstrapping. Results: MentalRoBERTa achieved macro-F1 of 0.901 on a held-out set, outperforming the original ReDSM5 benchmark. ADHD vs ASD classification yielded stable but modest performance (cross-validated ROC-AUC 0.645-0.653). Cognitive issues, sleep issues, appetite change, and fatigue leaned toward ADHD, while suicidal ideation and anhedonia leaned toward ASD. A largely shared symptom co-occurrence structure emerged between groups; no pair met our criterion for a robust disorder-specific difference. Conclusions: Population-level differences in depression-related language between ADHD and ASD social media users were consistently observed across thresholds, reflecting reproducibility rather than clinical validity. Findings are exploratory and do not establish differing phenomenology at the individual level.",
      "title": "Population-Level Profiling of DSM-5 Depressive Symptoms Among Self-Reported ADHD and ASD Users on Twitter: An Exploratory Study Using Advanced NLP and Statistical Analysis",
      "updated": "2026-07-06",
      "url": "https://arxiv.org/abs/2607.05626"
    },
    {
      "age_days": 7,
      "arxiv_id": "2607.04077",
      "authors": [
        "David Victor Feldman"
      ],
      "content_date": "2026-07-05",
      "freshness": "fresh",
      "id": "arxiv:2607.04077",
      "kind": "paper",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-07-05",
      "score": 0.5,
      "source": "arxiv-ai4math-core",
      "summary": "Let $S_N = \\sum_{j=1}^N j^{-3}$ and $R_N = ζ(3) - S_N$. The simple continued fraction of $\\sqrt{R_N}$ has partial quotients of generic size $O(N)$. We prove that at the sequence of indices $N_k = (Q_{2k+1}-1)/2$, where $Q_{2k+1}$ are companion Pell numbers, the continued fraction begins \\[ \\sqrt{R_{N_k}} = \\bigl[0;\\; M_k-1,\\; 1,\\; 6M_k^3+12M_k-2,\\; 1,\\; \\ldots\\,\\bigr], \\] with $M_k = P_{2k+1}$ (Pell numbers), and the third partial quotient grows cubically while generic ones are linear. We determine all partial quotients through the fifth: \\begin{align*} \\PQ_0 &= M_k - 1, & \\PQ_2 &= 6M_k^3 + 12M_k - 2, & \\PQ_4 &= \\Bigl\\lfloor\\frac{10M_k - 261}{261}\\Bigr\\rfloor, \\PQ_1 &= 1, & \\PQ_3 &= 1, & \\PQ_5 &= \\Bigl\\lfloor\\frac{261}{r_k}\\Bigr\\rfloor + ε_k, \\end{align*} where $r_k = (10M_k) \\bmod 261$ satisfies the recurrence $r_{k+1} \\equiv 6r_k - r_{k-1} \\pmod{261}$, and $ε_k = -1$ at the $k$ with $r_k \\mid 261$ (the two residue classes $k \\equiv 57, 62 \\pmod{60}$), and $ε_k = 0$ otherwise. All six formulas follow from the Euler--Maclaurin expansion of $1/\\sqrt{R_{N_k}}$, carried to sufficient precision, combined with the Pell identity $Q_{2k+1}^2 - 2M_k^2 = -1$. The delicate first step, $\\PQ_0 = M_k - 1$, is proved by rationalizing the irrational factor $\\sqrt{2}$ in the Euler--Maclaurin expansion; we complement this proof with a heuristic derivation via Gosper's bihomographic continued-fraction algorithm that exposes the underlying mechanism. All claimed results have been formalized in LEAN with the aid of Aristotle.",
      "title": "Anomalous Partial Quotients in the Continued Fraction of $\\sqrt{ζ(3)-S_N}$",
      "updated": "2026-07-05",
      "url": "https://arxiv.org/abs/2607.04077"
    }
  ],
  "lookback_days": 21,
  "schema": "ai4math-radar-run-v1",
  "timezone": "America/Los_Angeles"
}