{
  "counts": {
    "adjacent": 10,
    "core": 4,
    "errors": 0,
    "negative": 76,
    "total": 90
  },
  "date": "2026-10-06",
  "errors": [],
  "fresh_content_days": 21,
  "generated_at": "2026-10-06T20:29:52Z",
  "items": [
    {
      "age_days": 5,
      "arxiv_id": "2610.01799",
      "authors": [
        "Kuo Zhou",
        "ZiXion Yang",
        "Lu Zhang"
      ],
      "content_date": "2026-10-01",
      "freshness": "fresh",
      "id": "arxiv:2610.01799",
      "kind": "paper",
      "label": "core",
      "matched_signals": [
        "general_ai_math_reasoning",
        "lean_formal_proving_agents",
        "verifier_guided_reasoning"
      ],
      "published": "2026-10-01",
      "score": 12.3,
      "source": "arxiv-ai4math-core",
      "summary": "Skill evolution offers a promising way to improve large language model agents without updating their parameters, but its use in formal theorem proving remains underexplored. Existing methods mainly target natural-language reasoning, improving skills by analyzing successful and failed trajectories and incrementally revising solving strategies. Although the Lean verifier provides reliable execution feedback, when all sampled trajectories fail, existing skill evolution methods lack successful trajectories from which to infer effective update directions. Furthermore, these methods also focus mainly on the root instruction file, thus underexploring the evolution of reference knowledge including mathematical concepts and proving techniques. To address these limitations, we propose a mutation-enhanced skill self-evolution framework for building skill-augmented Lean provers. The framework jointly evolves a high-level solving policy and its reference knowledge through progressive and mutation-based updates. Progressive evolution derives local improvements from successful and failed trajectories, while mutation is triggered when no complete proof can be generated, sampling mathematical concepts to produce and select new skill candidates under verifier feedback. We evaluate our method on MiniF2F, PutnamBench, the 2025 International Mathematical Olympiad (IMO 2025), and the 2026 USA Mathematical Olympiad (USAMO 2026). Under the same backbone model, trajectorysampling budget, and test-time compute, our method achieves proof success rates of 100.0%, 90.6%, 4/6, and 4/6, respectively, with GPT-5.5, outperforming the baseline methods. Further analysis shows that concept-guided mutation outperforms random-text-guided mutation by 6.9 and 8.2 percentage points on MiniF2F and PutnamBench, respectively, while solving one additional problem on both IMO 2025 and USAMO 2026.",
      "title": "SkillEvoLean: Mutation-enhanced skill evolution for Lean provers",
      "updated": "2026-10-01",
      "url": "https://arxiv.org/abs/2610.01799"
    },
    {
      "age_days": 1,
      "arxiv_id": "2610.05701",
      "authors": [
        "Yuxuan Jiang",
        "Aditya Vempaty",
        "Ashish Jagmohan"
      ],
      "content_date": "2026-10-05",
      "freshness": "fresh",
      "id": "arxiv:2610.05701",
      "kind": "paper",
      "label": "core",
      "matched_signals": [
        "lean_formal_proving_agents",
        "verifier_guided_reasoning"
      ],
      "published": "2026-10-05",
      "score": 9.5,
      "source": "arxiv-ai4math-core",
      "summary": "Recursive self-improvement (RSI) enables agents to iteratively optimize their workflows via execution feedback. However, standard RSI typically operates as a first-order optimizer: it repeatedly patches surface-level parameters in response to immediate failure symptoms, often leading to trial-and-error thrashing without resolving underlying mechanisms. To address this limitation, we introduce SO-RSI, a framework that elevates workflow optimization to a second-order diagnostic inquiry, investigating why failures occur before committing to structural interventions. SO-RSI passively monitors execution traces for three structural anomalies (recurrence, opposing edits, and expectation mismatch) to trigger targeted mechanism investigations. By executing lightweight diagnostic probes and maintaining persistent inquiry memory across RSI rounds, SO-RSI accumulates causal evidence to guide systematic workflow edits rather than parameter patches. Across Lean 4 proof generation and Verus-based verifiable code generation, SO-RSI improves final held-out pass rates over Naive RSI by 21.8 and 25.8 percentage points under matched 24-hour search budgets. Behavioral analyses further confirm that SO-RSI substantially suppresses failure recurrence and eliminates unproductive zero-progress optimization loops.",
      "title": "Second-Order Problem Solving for Recursive Self-Improvement in Formal Verification",
      "updated": "2026-10-05",
      "url": "https://arxiv.org/abs/2610.05701"
    },
    {
      "age_days": 4,
      "arxiv_id": "2610.03994",
      "authors": [
        "Traian Serbanuta",
        "Jun Xu",
        "Andrei Stefanescu",
        "Cosmin Radoi"
      ],
      "content_date": "2026-10-02",
      "freshness": "fresh",
      "id": "arxiv:2610.03994",
      "kind": "paper",
      "label": "core",
      "matched_signals": [
        "lean_formal_proving_agents",
        "verifier_guided_reasoning"
      ],
      "published": "2026-10-02",
      "score": 6.5,
      "source": "arxiv-ai4math-core",
      "summary": "VeriContest is a benchmark of 1007 competitive-programming problems in Rust, each with a Verus specification, a judge-accepted solution, and a Verus proof. Its authors report that proof generation is the bottleneck for frontier models: given the specification and the code, the best model produces an accepted Verus proof for 13.95% of the problems on the first attempt. We report on solving the same proof-generation task with Rust-Prover, a verifier for Rust backed by Lean 4. The Verus specification and the Rust code are restated and translated into Lean, each specification becomes a theorem, and agents prove the theorems with Lean's kernel as the final check. All 1325 theorems of all 1007 problems were proved. 1259 of them were proved in one run of under 32 hours on Claude Opus 5.5, at a median of 3.2 minutes and $1.17 per proof, and 70% of them on the first iteration. The restated specifications were checked against the benchmark's test suites, and reviewed where no suite applies. None was wrong or weakened. The translated Lean programs were run on 21,413 of the benchmark's test cases and produced the same output as the Rust programs on every one. Across four Claude and four GPT models at five reasoning-effort settings, every current frontier model proves nearly all of a ten-theorem sample at every setting, and more effort raises the cost without raising the number of proofs. The cheapest Claude setting, Sonnet 5.5 at low effort, proves all of the 50 hardest theorems. We also rerun the benchmark's own Verus protocol with Claude Opus 5.5 on the 50 problems with the longest reference proofs. Opus 5.5 alone fails to prove one of them.",
      "title": "Solving VeriContest with a Lean-Backed Rust Verifier",
      "updated": "2026-10-02",
      "url": "https://arxiv.org/abs/2610.03994"
    },
    {
      "age_days": 2,
      "arxiv_id": "2610.05367",
      "authors": [
        "Prithwish Jana",
        "Viet Bach Hoang",
        "Logan Luna",
        "Viresh Pati",
        "Akash Singirikonda",
        "Cy Xie",
        "Lisa Carbone",
        "Wuyang Chen",
        "Walter Moreira",
        "Joe Stubbs",
        "Sriram Vishwanath",
        "Vijay Ganesh"
      ],
      "content_date": "2026-10-04",
      "freshness": "fresh",
      "id": "arxiv:2610.05367",
      "kind": "paper",
      "label": "core",
      "matched_signals": [
        "reasoning_rl_distillation",
        "tool_use_agents",
        "verifier_guided_reasoning"
      ],
      "published": "2026-10-04",
      "score": 6.3,
      "source": "arxiv-ai4math-core",
      "summary": "Proof auto-formalization translates natural-language (NL) theorems and proofs into a formal language (FL) such as Lean, enabling mechanical verification. Despite rapid progress, research-level proofs often depend on concepts missing from leading proof assistant libraries (e.g., Lean's Mathlib), and successful compilation does not guarantee that a translation preserves the theorem's meaning or the proof's reasoning. Furthermore, aligned NL-FL training data are scarce, and leading agents often rely on costly frontier models and manually engineered harnesses. To address the above issues, we present AIProver, an agentic framework for autonomous proof auto-formalization and proof synthesis (AFPS) that jointly post-trains a 119B open-weight language model and evolves its agentic, tool-calling harness with HarnessEvolve. Verifiers assess type correctness, proof completeness, and semantic correctness, returning rewards and diagnostic certificates that drive model fine-tuning and alternating reinforcement learning via symbolic feedback and HarnessEvolve, a certificate-driven evolutionary search over the whole harness control flow that re-tailors the harness to the updated model. For research-level training and evaluation, we introduce LoCoBench, 58.9k instances from Mathlib, CSLib, Mizar Math Library, and a bounded-arithmetic textbook, with a 771-instance validation split whose theorem-proof pairs have no public Lean formalization. Against 39 frameworks spanning AFPS agents, frontier LLMs, and coding agents, AIProver lifts pass@4 semantic correctness over its Leanstral-1.5 base from 15.7% to 36.7% and outperforms every other open-weight system and Aristotle. As a Claude Code and Codex skill, it lifts their semantic correctness from 41.9% and 34.1% to 79.8% and 62.4%, respectively. Further, it is also 24% cheaper than Numina-Lean-Agent, pushing the accuracy-cost frontier of research-level AFPS.",
      "title": "AIProver: Agentic Auto-Formalization of Mathematical Research via Certificate-Driven Evolving Harness",
      "updated": "2026-10-04",
      "url": "https://arxiv.org/abs/2610.05367"
    },
    {
      "age_days": 1,
      "arxiv_id": "2610.06531",
      "authors": [
        "Lilin Yan",
        "Hongwei Zhao"
      ],
      "content_date": "2026-10-05",
      "freshness": "fresh",
      "id": "arxiv:2610.06531",
      "kind": "paper",
      "label": "adjacent",
      "matched_signals": [
        "general_ai_math_reasoning",
        "lean_formal_proving_agents"
      ],
      "published": "2026-10-05",
      "score": 4.9,
      "source": "arxiv-ai4math-core",
      "summary": "Let all pairwise distances of an arbitrary configuration of points in $\\mathbb{R}^{\\ell}$ be known exactly. We prove that every second-order stationary point of the complete-graph squared-stress objective over configurations in $\\mathbb{R}^{k}$ has zero residual whenever $k > \\ell$. The result holds for every number of points, without a genericity assumption, and allows repeated points and degenerate ground-truth configurations. It resolves the one-extra-dimension conjecture for complete, uniformly weighted data. The proof combines positivity of the residual stress, polar normalization of the candidate configuration, a row-wise rank-nullity argument, and propagation of zero curvature through the full Hessian. The theorem, including the passage from derivatives of the original quartic objective to its matrix formulation, has been formalized in Lean 4. The accompanying verification records include a kernel check with zero trust level and a transitive axiom audit.",
      "title": "One Extra Dimension Suffices for the Complete-Graph Squared-Stress",
      "updated": "2026-10-05",
      "url": "https://arxiv.org/abs/2610.06531"
    },
    {
      "age_days": 5,
      "arxiv_id": "2610.01939",
      "authors": [
        "Ruiyang Si",
        "Jianxin Bi",
        "Shunyu Yang",
        "Rui Ni",
        "Wenbo Huang",
        "Qiang Wang",
        "Shulong Jiang",
        "Duomin Wang",
        "Xiuyu Li",
        "Haiwen Feng",
        "Zhen Dong",
        "Daquan Zhou"
      ],
      "content_date": "2026-10-01",
      "freshness": "fresh",
      "id": "arxiv:2610.01939",
      "kind": "paper",
      "label": "adjacent",
      "matched_signals": [
        "tool_use_agents",
        "verifier_guided_reasoning"
      ],
      "published": "2026-10-01",
      "score": 4.9,
      "source": "arxiv-ai4math-core",
      "summary": "Vision language model (VLM) agents can control robots through visual feedback and action primitives, but repeated model invocations and redundant observations incur substantial token overhead. We introduce PyRUA-Lean, an interactive code-execution framework that couples feedback-driven primitive composition with selective observation: the agent composes classical robot primitives and learned vision-language-action (VLA) policies into Python cells that perform conditional checks and local retries, returning only explicitly requested images and state feedback for replanning. Across 700 simulated task instances from LIBERO-PRO, RoboTwin 2.0, and RoboCasa365, we compare PyRUA-Lean with a tool-calling baseline using the same GPT-6 Astra planner and underlying robot primitives. Under equal LLM-call budgets, PyRUA-Lean increases overall success from 63.1% to 71.7%. On instances solved by both agents, it uses 49% fewer LLM calls and 65% fewer input tokens.",
      "title": "Fewer Tokens, Better Action: GPT-6 Astra Robot Agents with 14% Higher Success Rate but 65% Fewer Tokens",
      "updated": "2026-10-01",
      "url": "https://arxiv.org/abs/2610.01939"
    },
    {
      "age_days": 5,
      "arxiv_id": "2610.02035",
      "authors": [
        "Teng Zhang"
      ],
      "content_date": "2026-10-01",
      "freshness": "fresh",
      "id": "arxiv:2610.02035",
      "kind": "paper",
      "label": "adjacent",
      "matched_signals": [
        "general_ai_math_reasoning",
        "lean_formal_proving_agents"
      ],
      "published": "2026-10-01",
      "score": 4.9,
      "source": "arxiv-ai4math-core",
      "summary": "In this paper, we prove Borcea's 2-variance conjecture. The Lean 4 formalization of the main results are also provided.",
      "title": "Borcea's 2-variance conjecture",
      "updated": "2026-10-01",
      "url": "https://arxiv.org/abs/2610.02035"
    },
    {
      "age_days": 5,
      "arxiv_id": "2610.02009",
      "authors": [
        "Teng Zhang"
      ],
      "content_date": "2026-10-01",
      "freshness": "fresh",
      "id": "arxiv:2610.02009",
      "kind": "paper",
      "label": "adjacent",
      "matched_signals": [
        "general_ai_math_reasoning",
        "lean_formal_proving_agents"
      ],
      "published": "2026-10-01",
      "score": 4.9,
      "source": "arxiv-ai4math-core",
      "summary": "Let $m$ denote the normalized Haar measure on the unit circle $\\mathbb{T}$. For $0<r<\\infty$, define $\\lVert f\\rVert_{r}:=\\left(\\int_{\\mathbb{T}}|f|^{r}\\,\\mathrm{d} m\\right)^{1/r}$, with $\\lVert f\\rVert_{0}$ and $\\lVert f\\rVert_{\\infty}$ interpreted as the geometric mean and the supremum norm, respectively. Set $Q_n(z)=1+z^n$. We prove that, for every nonzero polynomial $p$ of degree $n$ whose zeros all lie on $\\mathbb{T}$, $$ \\frac{\\lVert p\\rVert_s}{\\lVert Q_n\\rVert_s} \\le \\frac{\\lVert p\\rVert_t}{\\lVert Q_n\\rVert_t}, \\qquad 0\\le s\\le t\\le\\infty. $$ This settles Baernstein's quasi-norm monotonicity conjecture. As corollaries, we obtain an $L^r$ extension of Visser's coefficient inequality, the sharp O'Hara--Rodriguez inequality and its higher-power analogues, the Erdős--Szekeres product bound $ \\left\\lVert\\prod_{j=1}^N(1-z^{s_j})\\right\\rVert_\\infty\\ge2\\sqrt N $ for all positive integers $s_1,\\ldots,s_N$ and Agler--McCarthy's entropy conjecture. We also provide a Lean 4 formalization of the main results.",
      "title": "Baernstein's quasi-norm monotonicity conjecture for polynomials with unimodular zero",
      "updated": "2026-10-01",
      "url": "https://arxiv.org/abs/2610.02009"
    },
    {
      "age_days": 2,
      "arxiv_id": "2610.05412",
      "authors": [
        "Chenghua Liu",
        "Boning Meng"
      ],
      "content_date": "2026-10-04",
      "freshness": "fresh",
      "id": "arxiv:2610.05412",
      "kind": "paper",
      "label": "adjacent",
      "matched_signals": [
        "lean_formal_proving_agents"
      ],
      "published": "2026-10-04",
      "score": 3.5,
      "source": "arxiv-ai4math-core",
      "summary": "We prove a complete complexity dichotomy for planar graph homomorphism counting with any fixed symmetric nonnegative matrix of arbitrary finite order, giving an explicit criterion for tractability. We also characterize exactly which fixed positive vertex weights preserve tractability, with both classifications extending from algebraic weights to fixed real weights in a prescribed exact representation. Our proof hinges on an entropy-based continuation argument: maximal logarithmic support identifies distance kernels as maximum-entropy completions, extending their positive definiteness throughout the parameter interval. This enables distance geometry to recover hidden product coordinates even when planar gadgets cannot distinguish colors; counting-hardness arguments then force the factors to be zero-field Boolean Ising interactions. The classification also yields complete tractability criteria for clock models, coupled Ising systems, and planar contractions of stoquastic imaginary-time kernels. All results have been formally verified in Lean 4.",
      "title": "A Dichotomy for Planar Graph Homomorphisms with Nonnegative Weights",
      "updated": "2026-10-04",
      "url": "https://arxiv.org/abs/2610.05412"
    },
    {
      "age_days": 3,
      "arxiv_id": "2610.04275",
      "authors": [
        "Ruoran Xu"
      ],
      "content_date": "2026-10-03",
      "freshness": "fresh",
      "id": "arxiv:2610.04275",
      "kind": "paper",
      "label": "adjacent",
      "matched_signals": [
        "lean_formal_proving_agents"
      ],
      "published": "2026-10-03",
      "score": 3.5,
      "source": "arxiv-ai4math-core",
      "summary": "Formal theorem proving increasingly combines learned semantic guidance with verified symbolic execution. The quality of the symbolic search substrate therefore determines how much useful mathematical structure can be accumulated, reused, and exposed under a finite inference budget. We introduce ViaLean, a Lean4 prover that organizes symbolic reasoning as bounded exploration of a persistent proof-state graph. Its search preserves coverage across complementary reasoning modes, merges semantically equivalent goals, retains verified intermediate structure, and observes short symbolic futures before committing to a transition. On the complete miniF2F test split, the model-free configuration solves 122/244 problems (50.0% pass@1). Historical and recent symbolic reference points range from Lean's earlier tidy search to modern grind and SMT-backed verification, showing that proof-space organization remains a substantial source of capability. The same persistent state also provides a natural interface for neural--symbolic agents: neural reasoning can operate over verified regions and intermediate objects while Lean continuously expands and validates the local proof space.",
      "title": "Symbolic Search Is Not Exhausted: Persistent Proof-Space Exploration in Lean4",
      "updated": "2026-10-03",
      "url": "https://arxiv.org/abs/2610.04275"
    },
    {
      "age_days": 4,
      "arxiv_id": "2610.02792",
      "authors": [
        "Sophia Simeng Han",
        "Yoshiki Takashima",
        "Anjiang Wei",
        "Zhaoyu Li",
        "Michael Genesereth"
      ],
      "content_date": "2026-10-02",
      "freshness": "fresh",
      "id": "arxiv:2610.02792",
      "kind": "paper",
      "label": "adjacent",
      "matched_signals": [
        "autoformalization"
      ],
      "published": "2026-10-02",
      "score": 3.5,
      "source": "arxiv-ai4math-core",
      "summary": "Legal systems are increasingly implemented through software, yet scalable methods for translating legal texts into accurate symbolic representations remain underdeveloped. We study this problem through tax law, where forms and filing instructions define large computational structures involving arithmetic, branching, recursion, and tabular reasoning. We propose Law&Order, a neuro-symbolic framework for automatically formalizing tax forms and instructions into executable symbolic programs. Our approach establishes two forms of correspondence between law and logic: structural correspondence, which aligns legal and symbolic components such as cells and schedules, and denotational correspondence, which requires symbolic components to implement the computations specified by their legal counterparts. We combine large language model synthesis with cell-level verification and iterative localized error repair using human-written OpenTaxSolver tax returns. We then evaluate the resulting formalizations on independently authored, held-out TaxCalcBench returns, that are never exposed during generation or repair. Although the most advanced LLM achieves only 66% accuracy, Law&Order achieves 100% cell-level and form-level accuracy on 51 held-out returns, demonstrating the effectiveness of combining LLM-based synthesis with symbolic verification for scalable and verifiable large-scale legal autoformalization compared with using an LLM alone.",
      "title": "Law And Order: Tax Law Autoformalization",
      "updated": "2026-10-02",
      "url": "https://arxiv.org/abs/2610.02792"
    },
    {
      "age_days": 4,
      "arxiv_id": "2610.02621",
      "authors": [
        "Kishor Bharti",
        "Tobias Haug",
        "Runzhou Tao",
        "Kevin Ye"
      ],
      "content_date": "2026-10-02",
      "freshness": "fresh",
      "id": "arxiv:2610.02621",
      "kind": "paper",
      "label": "adjacent",
      "matched_signals": [
        "verifier_guided_reasoning"
      ],
      "published": "2026-10-02",
      "score": 3.5,
      "source": "arxiv-ai4math-core",
      "summary": "We study a family of quantum subsystem codes defined on rectangular coordinate grids of any dimension, with even side lengths of at least four. The directly measured operators act along coordinate lines. For every member of the family, we derive the number of protected logical qubits and prove its dressed distance without a numerical search. To illustrate the resulting design choices, we impose a budget of at most 10,000 data qubits and determine the complete tradeoff between encoding rate and measurement weight within this family at distances 16, 32, and 64. At distance 16, one code protects 4,096 logical qubits among 10,000 data qubits using weight-ten measurements. At distances 32 and 64, codes with 7,776 and 9,216 data qubits protect 1,024 and 256 logical qubits, respectively, using weight-six measurements. We include and explain Lean code snippets throughout the paper to guide readers through the formal verification of our subsystem-code construction and its dressed and bare distances.",
      "title": "High-Rate Quantum Codes with Proven Distance and Low-Weight Measurements",
      "updated": "2026-10-02",
      "url": "https://arxiv.org/abs/2610.02621"
    },
    {
      "age_days": 1,
      "arxiv_id": "2610.06336",
      "authors": [
        "Jordi Daura Serrano",
        "Aron Gohr",
        "Marie-Amélie Lawn",
        "Travis Schedler"
      ],
      "content_date": "2026-10-05",
      "freshness": "fresh",
      "id": "arxiv:2610.06336",
      "kind": "paper",
      "label": "adjacent",
      "matched_signals": [
        "general_ai_math_reasoning"
      ],
      "published": "2026-10-05",
      "score": 1.9,
      "source": "arxiv-ai4math-core",
      "summary": "We prove the LeBrun--Salamon conjecture for positive quaternionic Kähler manifolds of quaternionic dimension n <= 14. To do so, we first show that the isometry group Isom(M) of any minimal counterexample M has rank at most one, hence dimension at most three. On the other hand, we consider a characteristic number which equals dim Isom(M) - (n+3) if n is even and dim Isom(M) - (n+1) if n is odd, and show that this index is a sum of positive quantities when n <= 14, yielding a contradiction. The toolset used to show positivity enlarges in stages. Projection curvature moments suffice through n=8; positive-semidefinite curvature moments extend the argument through n=10; and signed scalar orbital moments give the new inequalities needed for n=11,12. Scalar orbital moments alone fail at n=13; combining them with Hodge--Riemann positivity for squares of cohomology classes extends the proof to n=13,14. We give exact separating obstructions explaining the changes of method.",
      "title": "The Lebrun--Salamon conjecture is true in dimension up to 56",
      "updated": "2026-10-05",
      "url": "https://arxiv.org/abs/2610.06336"
    },
    {
      "age_days": 5,
      "arxiv_id": "2610.02159",
      "authors": [
        "Scott W. Viteri",
        "Laura Gomezjurado Gonzalez",
        "Clark Barrett"
      ],
      "content_date": "2026-10-01",
      "freshness": "fresh",
      "id": "arxiv:2610.02159",
      "kind": "paper",
      "label": "adjacent",
      "matched_signals": [
        "reasoning_rl_distillation"
      ],
      "published": "2026-10-01",
      "score": 1.9,
      "source": "arxiv-ai4math-core",
      "summary": "Intrinsic rewards are designed to guide exploration in reinforcement learning by assigning value to an agent's experience, for example through prediction error or learning progress. However, maximizing these rewards need not produce the most informative experience available. We propose a formal criterion for exploration that compares policies by the counterfactual information they acquire: how well their histories can substitute for experience under alternative policies. We construct a single, simple environment in which specified count-based, prediction-error, empowerment, and information-gain objectives have maximizing policies that are Pareto-suboptimal at acquiring counterfactual information. We explain these failures and establish conditions under which existing intrinsic rewards successfully encourage optimal exploration. We also construct an objective that assigns a higher value whenever exploration strictly improves under our criterion.",
      "title": "When Do Intrinsic Rewards Lead to Exploration?",
      "updated": "2026-10-01",
      "url": "https://arxiv.org/abs/2610.02159"
    },
    {
      "age_days": 0,
      "authors": [
        "Sebastian Graf"
      ],
      "content_date": "2026-10-06",
      "freshness": "fresh",
      "id": "github:leanprover/lean4:2369f26faa87",
      "kind": "github_update",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-10-06",
      "repo": "leanprover/lean4",
      "score": 0.8,
      "source": "lean4-github",
      "summary": "Recent commit on leanprover/lean4.",
      "title": "leanprover/lean4: refactor: index `Lean.Order.iSup` and `iInf` over `Sort` (#15517)",
      "updated": "2026-10-06",
      "url": "https://github.com/leanprover/lean4/commit/2369f26faa873e34e281057dbac329a57a356b2d"
    },
    {
      "age_days": 0,
      "authors": [
        "Sebastian Ullrich"
      ],
      "content_date": "2026-10-06",
      "freshness": "fresh",
      "id": "github:leanprover/lean4:6f79a3e117e4",
      "kind": "github_update",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-10-06",
      "repo": "leanprover/lean4",
      "score": 0.8,
      "source": "lean4-github",
      "summary": "Recent commit on leanprover/lean4.",
      "title": "leanprover/lean4: perf: share server extension entries with the main table outside the server (#15512)",
      "updated": "2026-10-06",
      "url": "https://github.com/leanprover/lean4/commit/6f79a3e117e4cffc425d7514df3ae9b512a37ea3"
    },
    {
      "age_days": 0,
      "authors": [
        "Henrik Böving"
      ],
      "content_date": "2026-10-06",
      "freshness": "fresh",
      "id": "github:leanprover/lean4:3dc0360c4af4",
      "kind": "github_update",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-10-06",
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      "published": "2026-10-05",
      "repo": "leanprover-community/mathlib4",
      "score": 0.8,
      "source": "mathlib4-github",
      "summary": "Recent commit on leanprover-community/mathlib4.",
      "title": "leanprover-community/mathlib4: chore(Algebra/Homology/Opposite): add implicit_reducible tags and remove set_options (#44272)",
      "updated": "2026-10-05",
      "url": "https://github.com/leanprover-community/mathlib4/commit/7414889cb3a5af9f4e5ce1157fc548d43d7dcc45"
    },
    {
      "age_days": 2,
      "authors": [
        "Leonardo de Moura"
      ],
      "content_date": "2026-10-04",
      "freshness": "fresh",
      "id": "github:leanprover/lean4:d2d056ac1f97",
      "kind": "github_update",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-10-04",
      "repo": "leanprover/lean4",
      "score": 0.8,
      "source": "lean4-github",
      "summary": "Recent commit on leanprover/lean4.",
      "title": "leanprover/lean4: fix: rewrite casts of numerals to numerals in the `Sym` arithmetic normalizer (#15476)",
      "updated": "2026-10-04",
      "url": "https://github.com/leanprover/lean4/commit/d2d056ac1f97efb01755545896d4ab7271bbb46d"
    },
    {
      "age_days": 2,
      "authors": [
        "Leonardo de Moura"
      ],
      "content_date": "2026-10-04",
      "freshness": "fresh",
      "id": "github:leanprover/lean4:e321b5de8ce3",
      "kind": "github_update",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-10-04",
      "repo": "leanprover/lean4",
      "score": 0.8,
      "source": "lean4-github",
      "summary": "Recent commit on leanprover/lean4.",
      "title": "leanprover/lean4: fix: accept only `denoteInt` spellings as numerals in `grind`'s field inverse rules (#15478)",
      "updated": "2026-10-04",
      "url": "https://github.com/leanprover/lean4/commit/e321b5de8ce37e05ebd1915ea4dcd5bd8a7cc5e7"
    },
    {
      "age_days": 1,
      "arxiv_id": "2610.06742",
      "authors": [
        "Patrick Hayden",
        "Alexander Maloney",
        "Jinzhao Wang",
        "Yuxiang Yang"
      ],
      "content_date": "2026-10-05",
      "freshness": "fresh",
      "id": "arxiv:2610.06742",
      "kind": "paper",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-10-05",
      "score": 0.5,
      "source": "arxiv-ai4math-core",
      "summary": "We study a variant of Schumacher compression. The task asks for the minimal memory cost for compressing many copies of a density matrix with known spectrum and unknown eigenbasis, without preserving its purification. For fixed dimension and distinct positive eigenvalues, we obtain the cost through its additive constant. Achievability uses a generalization of Werner's cloning map to $\\mathrm{GL}(d,\\mathbb C)$ irreducible representations, with finite trace-distance bounds controlled by highest-weight differences and row gaps. The matching converse follows from a quantitative form of Koashi--Imoto incompressibility for irreducible group orbits under a spectral-gap assumption. We also identify the relation between this quantum minimum description length and universal lossless coding overhead, and its connection to free entropy is explained in a companion letter. We provide a Lean certificate for our proofs.",
      "title": "Quantum minimum description of density matrices",
      "updated": "2026-10-05",
      "url": "https://arxiv.org/abs/2610.06742"
    },
    {
      "age_days": 1,
      "arxiv_id": "2610.06674",
      "authors": [
        "Srija Chakraborty"
      ],
      "content_date": "2026-10-05",
      "freshness": "fresh",
      "id": "arxiv:2610.06674",
      "kind": "paper",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-10-05",
      "score": 0.5,
      "source": "arxiv-ai4math-core",
      "summary": "Nighttime lights from NASA's Black Marble product suite capture thermal and light emission signals from anomalous events including fires, volcanic eruptions, and gas flaring. Existing detection approaches rely primarily on thermal bands, limiting sensitivity to weaker signals. We propose a novel machine learning framework that jointly models Black Marble M-band and Day/Night Band (DNB) signals to derive a generalized, spatio-temporally robust ensemble of anomaly detectors. The framework iteratively builds detectors that scale across regions, seasons, anomaly classes, and extends over land and ocean. Detection sets at varying confidence levels are derived based on relevant bands and detector agreement. The approach improves true detection rate while reducing spurious detections and results demonstrate strong generalizability with applications in natural hazard monitoring and energy extraction.",
      "title": "Detecting Nighttime Anomalies from NASA Black Marble Using a Generalized Spatio-Temporally Robust Framework of Machine Leaning Ensembles",
      "updated": "2026-10-05",
      "url": "https://arxiv.org/abs/2610.06674"
    },
    {
      "age_days": 1,
      "arxiv_id": "2610.06368",
      "authors": [
        "Jörg Endrullis",
        "Ievgen Ivanov",
        "Femke van Raamsdonk"
      ],
      "content_date": "2026-10-05",
      "freshness": "fresh",
      "id": "arxiv:2610.06368",
      "kind": "paper",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-10-05",
      "score": 0.5,
      "source": "arxiv-ai4math-core",
      "summary": "Confluence is a fundamental property of nondeterministic computations, arising from parallelism, concurrency, or freedom in the evaluation order. It guarantees that such a computation always yields the same result, regardless of the order in which steps are taken. The decreasing diagrams technique of van Oostrom is one of the most versatile methods for establishing confluence of transition systems (abstract rewriting systems). It reduces global confluence to local confluence: a system is confluent whenever its transitions admit a locally decreasing labeling. Essentially all classical confluence criteria arise as corollaries. A central question, posed by van Oostrom in 1993, asks whether the decreasing diagrams technique is complete: Does every confluent transition system admit a locally decreasing labeling? This is Problem 56 of the RTA List of Open Problems. A positive answer was known only for countable systems, and recently up to the first uncountable cardinal $\\aleph_1$. The general case remained open. We settle this thirty-three-year-old problem in full. We prove that every confluent transition system admits a locally decreasing labeling using only three labels. This bound is optimal, as two labels do not suffice even at the first uncountable cardinality. It follows that this single criterion can, in principle, certify the confluence of every confluent system, and hence of every confluent program. The entire development is machine-checked in the Isabelle/HOL and Lean proof assistants and relies only on classical logic and the axiom of choice.",
      "title": "Decreasing Diagrams are Complete for Confluence",
      "updated": "2026-10-05",
      "url": "https://arxiv.org/abs/2610.06368"
    },
    {
      "age_days": 2,
      "arxiv_id": "2610.05490",
      "authors": [
        "Leo Brunswic"
      ],
      "content_date": "2026-10-04",
      "freshness": "fresh",
      "id": "arxiv:2610.05490",
      "kind": "paper",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-10-04",
      "score": 0.5,
      "source": "arxiv-ai4math-core",
      "summary": "Generative flows sample from an unnormalized target by training a flow to be balanced, and the training loss is the signal a practitioner watches. We ask what that signal is worth: whether a small loss certifies an accurate sampler, whether the loss can be driven to zero, and how fast gradient descent does so. The loss decides the first. Losses that compare the two sides of the balance by their difference bound, in total variation, the error of the sampler the flow implies, with explicit constants that do not involve the policy; flow-matching losses that compare them through a ratio admit no such bound, already on a single cycle, whenever their generator is continuous at balance. On graphs, the backward policy decides the other two. Once it is frozen, balance becomes invariance under the backward chain, so that existence is free on finite graphs, and one constant --- the norm of that chain's Green operator, which plays the role of an inverse spectral gap --- fixes the order of the curvature of the loss around the balanced flow, from above and below, and sets a floor under the rate at which training converges near it. The mechanism is that gradient descent diffuses the flow along the backward policy. For the squared-logarithm generator of detailed and trajectory balance, training the balance loss on states converges globally on every finite path-connected graph, from every positive initialization. The constant can be infinite while backward trajectories are short on average, and exact flow matching can then fail. The bounds and rates are tested by exact computation on enumerable state spaces, and every theorem carries a certification status computed from a Lean~4 development.",
      "title": "Universality and Convergence of Generative Flows",
      "updated": "2026-10-04",
      "url": "https://arxiv.org/abs/2610.05490"
    },
    {
      "age_days": 2,
      "arxiv_id": "2610.05186",
      "authors": [
        "Jeff Calder",
        "Nadejda Drenska"
      ],
      "content_date": "2026-10-04",
      "freshness": "fresh",
      "id": "arxiv:2610.05186",
      "kind": "paper",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-10-04",
      "score": 0.5,
      "source": "arxiv-ai4math-core",
      "summary": "We prove that no single rank ordered adversary strategy is globally optimal for the prediction with expert advice problem with six or more experts, in both the geometric stopping and finite time horizon settings. The proof is based on establishing a leading order correction when one expert moves far ahead of the others. This allows us to connect optimal strategies between $n$ and $j<n$ experts and utilize recent results on the exact optimality set for the five expert problem.",
      "title": "On prediction from expert advice with more than five experts",
      "updated": "2026-10-04",
      "url": "https://arxiv.org/abs/2610.05186"
    },
    {
      "age_days": 3,
      "arxiv_id": "2610.04774",
      "authors": [
        "Matteo Nerini",
        "Xuekang Liu",
        "Bruno Clerckx"
      ],
      "content_date": "2026-10-03",
      "freshness": "fresh",
      "id": "arxiv:2610.04774",
      "kind": "paper",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-10-03",
      "score": 0.5,
      "source": "arxiv-ai4math-core",
      "summary": "Analog computing with microwave signals can perform linear transformations directly in the analog domain, as the signals propagate through a microwave network. A fundamental question is which transformations can be computed with a given set of microwave components. In our previous work, we answered this question for networks of hybrid couplers and phase shifters by deriving a necessary and sufficient condition on the transformations these networks can compute, and we showed that the discrete Fourier transform (DFT) satisfies it. In this paper, we take a first step toward the formalization of analog computing with microwaves in Lean, a programming language and proof assistant that is increasingly adopted in mathematics. We formalize the considered components, their series and parallel connections, and the class of networks they can implement. Then, we formally prove the necessary and sufficient condition characterizing these networks, as well as the implementability of the DFT of any size power of two. All proofs are checked by the Lean kernel, and the code is openly available at: https://github.com/matteonerini/formalizing-analog-computing.",
      "title": "Toward a Lean Formalization of Analog Computing with Microwaves",
      "updated": "2026-10-03",
      "url": "https://arxiv.org/abs/2610.04774"
    },
    {
      "age_days": 3,
      "arxiv_id": "2610.04511",
      "authors": [
        "Marianna Girlando",
        "Fabrizio Montesi"
      ],
      "content_date": "2026-10-03",
      "freshness": "fresh",
      "id": "arxiv:2610.04511",
      "kind": "paper",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-10-03",
      "score": 0.5,
      "source": "arxiv-ai4math-core",
      "summary": "Modal logic comprises a broad family of logics to reason about relational structures. The literature presents many such logics, displaying diverse operators, semantics, and applications. This plurality is reflected in a zoo of mechanised modal logics, which often duplicate syntax, semantics, metatheory, and reasoning infrastructure. We present a library-grade formalisation of modal logic in Lean, developed as part of CSLib (the Lean Computer Science Library) and designed around two complementary forms of reuse: vertical reuse, whereby specialised logics inherit from common abstractions, and horizontal reuse, whereby modal logic becomes a reasoning tool for independently formalised domains. Our development provides a generic framework for polyadic modal languages, reusable metatheory and proof automation, and derived interfaces for specialised modal logics. We exemplify our infrastructure by deriving basic modal logic, basic temporal logic, and Hennessy-Milner Logic; the latter subsumes and extends CSLib's previous implementation while reducing its logic-specific code by 67%. We further apply the same modal infrastructure to reasoning about mathematics (radicals of ideals), theory of programming languages (the type safety strategy for the simply typed $λ$-calculus), and concurrency theory (reasoning about processes in the Calculus of Communicating Systems). Our experience suggests that reuse, automation, and integration with the strong Lean ecosystem should be treated as first-class design concerns when formalising theories for shared libraries.",
      "title": "Library-Grade Modal Logic",
      "updated": "2026-10-03",
      "url": "https://arxiv.org/abs/2610.04511"
    },
    {
      "age_days": 3,
      "arxiv_id": "2610.04182",
      "authors": [
        "Levi Boxell",
        "Jacob Conway",
        "Nicolas Longuet-Marx"
      ],
      "content_date": "2026-10-03",
      "freshness": "fresh",
      "id": "arxiv:2610.04182",
      "kind": "paper",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-10-03",
      "score": 0.5,
      "source": "arxiv-ai4math-core",
      "summary": "What role do journalists play in determining the political slant of the news they produce? We develop and estimate a model where journalists and newspaper outlets contract over both slant and wages. The model implies a set of conditions under which we can consistently estimate the role of journalist preferences in driving the observed variation in slant across outlets by leveraging journalist transitions between outlets. To measure slant, we train a transformer-based, machine learning model using articles tweeted by politicians and apply it to a full-text database of 9+ million newspaper articles published in the US between 2013 and 2018. Under our model and identifying assumptions, our estimates (a) reject the hypothesis that journalists have zero ideological preferences over the content they produce and (b) imply that 10% of the variation in slant can be explained by journalists. When journalists move across outlets, their average slant shifts by 77% of the gap between their destination and origin outlet averages. In counterfactuals that reweight journalist party composition, moving from the observed left-leaning distribution to an even distribution of Democrats, Republicans, and non-partisans shifts average article slant in a more conservative direction, but only by 0.04 standard deviations in slant.",
      "title": "Journalist Ideology and the Production of News: Evidence from Movers",
      "updated": "2026-10-03",
      "url": "https://arxiv.org/abs/2610.04182"
    },
    {
      "age_days": 3,
      "arxiv_id": "2610.04666",
      "authors": [
        "Zijian Gong",
        "Zhaobin Lyu",
        "Jingjing Hu",
        "Dengfeng Li",
        "Shuoming An"
      ],
      "content_date": "2026-10-03",
      "freshness": "fresh",
      "id": "arxiv:2610.04666",
      "kind": "paper",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-10-03",
      "score": 0.5,
      "source": "arxiv-ai4math-core",
      "summary": "Quantum computers are claimed to produce samples ordinary classical computers cannot reproduce. The sharpest such claim known pits constant-depth quantum circuits, given one entangled state per run, against shallow classical circuits of few-input gates and limited randomness. No efficient test was known that certifies such a claim from classical samples alone. Here we show that what such a test must read is decided by the classical class's pattern of single-bit and pair correlations, not by a distance. The test rests on one inequality: any machine producing the samples sits no further from the target than the fraction it mislabels plus the deviation of its string half, the bits beyond the label, from uniform. A collapse theorem, machine-checked in the Lean~4 proof assistant, confines that second term, one setting of the sampler's random input bits at a time. Four natural checks provably fail; a fifth, reading those same correlations, catches a far-from-target construction they cannot reach, and is exact on the class of few-input gates. The label test is proven and sample-optimal in its own tolerance up to a logarithmic factor; the fifth check is sound, not only effective, on the bounded pinned-residue class, pairwise-uniform samplers whose residue, the string's weight modulo the prime, is pinned by few seeds and whose seed sharing is bounded. What binds an experiment is the $43$-qubit entangled resource state at fidelity near $0.99$, not the readout. What remains are two named open problems; a positive answer to the first would extend the guarantee to the full class.",
      "title": "Correlations decide a shallow-circuit advantage",
      "updated": "2026-10-03",
      "url": "https://arxiv.org/abs/2610.04666"
    },
    {
      "age_days": 4,
      "arxiv_id": "2610.02814",
      "authors": [
        "Naing Oo Lwin"
      ],
      "content_date": "2026-10-02",
      "freshness": "fresh",
      "id": "arxiv:2610.02814",
      "kind": "paper",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-10-02",
      "score": 0.5,
      "source": "arxiv-ai4math-core",
      "summary": "Keeping a Python program and its formal guarantees aligned is a continuing maintenance problem. Specifications must describe the implementation that actually runs, and updates must preserve the behavior on which existing callers depend. VeriPy brings these obligations into a common workflow for annotated Python components. Developers and agents express contracts, invariants, and proof hooks as Python comments, then refine checked auxiliary lemmas using source-located diagnostics. Direct encoders produce Dafny or Lean artifacts while retaining admitted executable bodies and explicit models of their dependencies. A relational product checks whether an updated component still admits old inputs and preserves returned values and modeled exceptions. The resulting workflow connects source-level proof development with functional verification and backward compatibility, recording the assumptions under which each guarantee applies. The code is available at https://github.com/astrio-labs/veripy.",
      "title": "VeriPy Source-Preserving Verification and Compatibility Checking for Python Components",
      "updated": "2026-10-02",
      "url": "https://arxiv.org/abs/2610.02814"
    },
    {
      "age_days": 4,
      "arxiv_id": "2610.02833",
      "authors": [
        "Deep Bhattacharjee",
        "Priyabrata Mandal",
        "Ushashi Bhattacharya"
      ],
      "content_date": "2026-10-02",
      "freshness": "fresh",
      "id": "arxiv:2610.02833",
      "kind": "paper",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-10-02",
      "score": 0.5,
      "source": "arxiv-ai4math-core",
      "summary": "We prove both Erdős-Ulam conjectures on monochromatic union-closed families. Every two-colouring of the subsets of a finite set contains a monochromatic union-closed family whose size grows faster than any fixed power of the size of the set, while suitable colourings admit no monochromatic union-closed family of exponential size. Both results hold for any number of colours and are verified in Lean.",
      "title": "Resolving Erdős-Ulam Monochromatic Union-Closed Family Conjectures",
      "updated": "2026-10-02",
      "url": "https://arxiv.org/abs/2610.02833"
    },
    {
      "age_days": 4,
      "arxiv_id": "2610.03214",
      "authors": [
        "Shuoming An",
        "Fusheng Yang"
      ],
      "content_date": "2026-10-02",
      "freshness": "fresh",
      "id": "arxiv:2610.03214",
      "kind": "paper",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-10-02",
      "score": 0.5,
      "source": "arxiv-ai4math-core",
      "summary": "A code is described by three numbers: its physical qubits, its logical qubits, and the smallest error it cannot detect, counted in qubits. The first two are linear algebra. The third, the distance, is an optimum over an exponentially large set, read off a solver whose answer carries no certificate. Whether a code's parameters can be checked uniformly across codes, with a certificate of proved size, is open. Here we show that they can, by moving the unit of work from the code to a \\emph{certificate}, a short object, either a list, a pairing or a symbolic instance, whose correctness the kernel of the \\Lean\\ proof assistant decides by computation. The kernel is the small core that checks each step. The certificate's size is itself a theorem: its enumeration form lists the vectors of weight below the distance $d$, $\\sum_{j<d}\\binom{n}{j}$ of them, a polynomial in the code's length $n$ at fixed distance $d$. No bound uniform over codes in both $n$ and $d$ is smaller. For product families the certificate is smaller still, the bound being proved once, symbolically. Deciding that a vector lies outside the row space of the check matrix, the set of sums of its rows, becomes one matrix-vector product and one inner product. On an eighteen-qubit toric code, a surface code closed into a torus, the whole-file check of that decision falls from 42\\,s to 9\\,s. Eleven code families and thirty-nine parameter sets follow, the widest at 1872 qubits. Nothing beyond the three standard axioms of \\Lean's logic is trusted, though the generator that writes a code's checks stops at twenty qubits, and the lower bound for the 144-qubit code is imported rather than proved here. A distance becomes checkable rather than believed, and the same move applies wherever else a computation ends in a solver.",
      "title": "Quantum code parameters, checkable by a certificate of provable size",
      "updated": "2026-10-02",
      "url": "https://arxiv.org/abs/2610.03214"
    },
    {
      "age_days": 4,
      "arxiv_id": "2610.03651",
      "authors": [
        "Sean Culatana",
        "Shang-En Huang",
        "Kang Li"
      ],
      "content_date": "2026-10-02",
      "freshness": "fresh",
      "id": "arxiv:2610.03651",
      "kind": "paper",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-10-02",
      "score": 0.5,
      "source": "arxiv-ai4math-core",
      "summary": "Dense-retrieval services must switch among embedding-prefix dimensions and index bit rates as latency, quality, and memory budgets change. Tuning a quantizer separately for each rate gives the best quality, but the retrieval tier then holds several code streams and quantizer states at once. We introduce Matryoshka Residual Vector Quantization (MRVQ), a post-hoc residual quantizer for frozen embeddings. Its maximum-rate code can be truncated two ways: dropping residual stages lowers the rate, and dropping embedding coordinates lowers the dimension. One resident artifact therefore serves every (dimension, rate) pair we evaluate. Across FiQA and NFCorpus, four embedding families, and {4, 8, 16}-byte codes, MRVQ is the lowest-RAM design we evaluate. It uses 17.8-22.0x less memory than three separately trained QINCo2 indices, and 1.89-2.02x less than a lean shared-model steelman. The saving is not free: per-rate QINCo2 is 0.026-0.107 nDCG@10 better on FiQA. But MRVQ beats PQ, OPQ, and AdANNS-OPQ at matched code size. We also evaluate a low-build-cost PCA-scalar design that attains quality comparable to RaBitQ and its extension while fitting 420x faster at the median. Finally, we report two negative results: QINCo2 collapses when trained at high rates, and a ranking-bound hypothesis misses its pre-specified acceptance criteria. MRVQ is therefore a low-memory operating point for elastic retrieval, not a universal quality winner.",
      "title": "MRVQ: One Resident Index for Dimension- and Rate-Elastic Vector Search",
      "updated": "2026-10-02",
      "url": "https://arxiv.org/abs/2610.03651"
    },
    {
      "age_days": 4,
      "arxiv_id": "2610.03364",
      "authors": [
        "Jens Palsberg",
        "Keli Huang",
        "Abdullah Almanei"
      ],
      "content_date": "2026-10-02",
      "freshness": "fresh",
      "id": "arxiv:2610.03364",
      "kind": "paper",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-10-02",
      "score": 0.5,
      "source": "arxiv-ai4math-core",
      "summary": "A well-known theorem states that the Clifford+T gate set is universal for quantum computing. The theorem combines approximation using a distance measure on unitary matrices with equality up to global phase. Previous proofs combine these two notions for specific distance measures, but do not identify the properties that govern how they work together. Which properties are sufficient to state and prove the theorem? We answer this question by defining projective distance measures using four axioms. We prove that the universality theorem holds for every distance measure satisfying these axioms. Thus, our formulation of the theorem is independent of any particular choice of distance measure. We show that the Hilbert-Schmidt distance is already a projective distance measure. We also develop a general construction that transforms a large class of distance measures into projective distance measures and apply it to obtain projective versions of the operator-norm distance, the Frobenius distance, and the trace distance. Finally, we formalize the proof of the universality theorem in Lean.",
      "title": "Distance-Independent Universality of Clifford+T",
      "updated": "2026-10-02",
      "url": "https://arxiv.org/abs/2610.03364"
    },
    {
      "age_days": 5,
      "arxiv_id": "2610.02500",
      "authors": [
        "João Araújo",
        "Jan Hůla",
        "Mikoláš Janota",
        "Edmond W. H. Lee",
        "Bartosz Naskręcki"
      ],
      "content_date": "2026-10-01",
      "freshness": "fresh",
      "id": "arxiv:2610.02500",
      "kind": "paper",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-10-01",
      "score": 0.5,
      "source": "arxiv-ai4math-core",
      "summary": "We introduce SemiBase, a project that computes and formally certifies finite identity bases for small semigroups. Deciding finite basability is undecidable for finite algebras and remains open for finite semigroups. The task requires a proof that a candidate basis is complete, or a proof that none exists, rather than a single first-order validity query. LLM-guided agents search for these proofs; a referee agent rebuilds them from source, and the Lean kernel checks the resulting corpus in a final audit. Humans choose targets and approve final outcomes. We certify every semigroup of order at most 6: all 1309 semigroups of order at most 5 and all 15973 of order 6, including proofs that the four known nonfinitely based semigroups have no finite basis. The bases for order 6 define 505 distinct varieties, whose inclusion order Vampire determines except for four pairs. The resulting catalogue is a machine-checked account of results scattered across the literature and a tested foundation for order 7.",
      "title": "Proving at Scale for Universal Algebra",
      "updated": "2026-10-01",
      "url": "https://arxiv.org/abs/2610.02500"
    },
    {
      "age_days": 5,
      "arxiv_id": "2610.02335",
      "authors": [
        "Fengyang Wang"
      ],
      "content_date": "2026-10-01",
      "freshness": "fresh",
      "id": "arxiv:2610.02335",
      "kind": "paper",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-10-01",
      "score": 0.5,
      "source": "arxiv-ai4math-core",
      "summary": "Computer-assisted proofs in dynamics establish results about nonlinear systems by rigorous numerical computation. Their correctness rests on a trusted base of interval-arithmetic libraries and analytic estimates checked by hand. We formalize in Lean~4 a framework for the radii polynomial method, which certifies an exact solution near a numerical approximation by verifying four norm bounds and the resulting polynomial inequality. Weighted coefficient algebras provide the common setting for polynomial equations and initial value problems in Taylor and Chebyshev series. Their universal properties construct the bounded operators and the evaluation maps, and the universal property of the free commutative algebra makes polynomial substitution commute with evaluation. Finite/tail reductions turn the four norm bounds into finite rational inequalities, which are checked in Lean. The radii theorem then yields an exact coefficient solution, and realization theorems carry it to a solution of the original equation. The worked examples are a square-root branch given by a convergent power series and polynomial initial value problems, among them the Lorenz system, for which the library proves existence, uniqueness within the trajectory ball, and analyticity of the function-level solution.",
      "title": "A Lean~4 Framework for the Radii Polynomial Method",
      "updated": "2026-10-01",
      "url": "https://arxiv.org/abs/2610.02335"
    }
  ],
  "lookback_days": 21,
  "schema": "ai4math-radar-run-v1",
  "timezone": "America/Los_Angeles"
}