{
  "counts": {
    "adjacent": 13,
    "core": 2,
    "errors": 0,
    "negative": 74,
    "total": 89
  },
  "date": "2026-10-05",
  "errors": [],
  "fresh_content_days": 21,
  "generated_at": "2026-10-05T22:08:43Z",
  "items": [
    {
      "age_days": 4,
      "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": 5,
      "arxiv_id": "2610.00760",
      "authors": [
        "Chenjun Guo",
        "Manooshree Patel",
        "Arnav Mehta",
        "Krishiv Kothari",
        "Thomas Lu",
        "Niels Voss",
        "Rayna Bhattacharyya",
        "Peter Donovan",
        "Bjoern Hartmann",
        "Gireeja Ranade"
      ],
      "content_date": "2026-09-30",
      "freshness": "fresh",
      "id": "arxiv:2610.00760",
      "kind": "paper",
      "label": "core",
      "matched_signals": [
        "lean_formal_proving_agents",
        "verifier_guided_reasoning"
      ],
      "published": "2026-09-30",
      "score": 9.5,
      "source": "arxiv-ai4math-core",
      "summary": "Large language models are increasingly used as collaborators on deductive-reasoning tasks, but their outputs can hallucinate or pull users away from intended reasoning. Formal proof assistants provide machine-checked verification, but have a steep learning curve and require more granular reasoning than human written proofs. We explore an interface that combines these strengths, allowing users to write and revise free-form natural-language proofs while a verified backend checks their reasoning and returns feedback at the user's granularity. We study this interface in the context of undergraduate mathematics education by developing LeanSide, a formally verified co-reasoning system, which auto-formalizes student reasoning into Lean and informalizes verifier output into understandable feedback. We conducted user studies through classroom deployment and analyzed which system properties helped students make progress and which caused them to get stuck. We use these findings to derive design implications for using a formally verified backend in human-AI co-reasoning systems.",
      "title": "LeanSide: A Formally Verified Co-Reasoning System for Natural-language Proofs",
      "updated": "2026-09-30",
      "url": "https://arxiv.org/abs/2610.00760"
    },
    {
      "age_days": 4,
      "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": 4,
      "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": 4,
      "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": 4,
      "arxiv_id": "2610.01240",
      "authors": [
        "Tom Verhoeff"
      ],
      "content_date": "2026-10-01",
      "freshness": "fresh",
      "id": "arxiv:2610.01240",
      "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 1965, D. H. Lehmer conjectured that the permutations of every multiset admit an imperfect Hamiltonian traversal by adjacent swaps: a walk in the neighbor-swap graph that visits every word, with some words visited twice in order to reach a neighbor and return. The question is posed as an unsolved research problem in Knuth's Art of Computer Programming. Verhoeff (2017) chose the stutter words, in which every domino is a double, as the words to be reached this way, and reformulated the conjecture as the Hamiltonicity of the graph $N(S)$ on the non-stutter words, with two exceptional families --- binary signatures with an odd multiplicity, and the permutations of $(2k,1,1)$ --- that admit a Hamiltonian path but no cycle. This article proves the reformulated conjecture, and with it Lehmer's conjecture. The key structure is a partition of the words into hypercubes: the swaps inside dominoes turn each class of words with the same domino contents into a hypercube, and the stutters are exactly the $0$-dimensional classes. When every multiplicity is even, Hamiltonian cycles of the hypercubes are glued along a spanning tree, with no finite check. The case of exactly one odd multiplicity reduces to the all-even case and to a theorem of Stachowiak (1992), the one inherited Hamiltonicity input, which also settles two or more odd multiplicities. The only finite ingredients are two explicit cycles, of 28 and 84 words. Every construction is implemented in Python and checked against brute-force graphs, and the proof is formalized in Lean 4 over Mathlib.",
      "title": "A proof of Lehmer's permutation conjecture for neighbor-swap graphs",
      "updated": "2026-10-01",
      "url": "https://arxiv.org/abs/2610.01240"
    },
    {
      "age_days": 3,
      "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": 3,
      "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": 4,
      "arxiv_id": "2610.02276",
      "authors": [
        "John Fairfax-Ball"
      ],
      "content_date": "2026-10-01",
      "freshness": "fresh",
      "id": "arxiv:2610.02276",
      "kind": "paper",
      "label": "adjacent",
      "matched_signals": [
        "verifier_guided_reasoning"
      ],
      "published": "2026-10-01",
      "score": 3.5,
      "source": "arxiv-ai4math-core",
      "summary": "Choose a uniformly random ordering of the vertices of a finite tree and run the usual greedy maximal-independent-set algorithm. We compare the resulting law on maximal independent sets with the uniform law. We prove that exact uniformity occurs only for the one-vertex tree and the single edge. The proof is structural: a diameter endpoint exposes a pendant star, and the remaining one-pendant-leaf case is resolved by a strict injection between exact permutation fibres obtained by swapping the pendant leaf with its support vertex. Exact uniformity is therefore rigid, but it can be approached closely. For an explicit mixed-spider family $T_{k,l}$ we count the maximal independent sets and compute the exact probability of every output. With $l=2^k-k$ the total-variation bias is positive and satisfies [ b(T_{k,2^k-k})=O!\\left(\\frac{\\sqrt{k}}{4^k}\\right) =O!\\left(\\frac{\\sqrt{\\log n_k}}{n_k^2}\\right), \\qquad n_k=2^k+k+1. ] The theorem package has also been formalised in Lean and registered with Palomar. These records document machine-checked formal verification and the checked axiom boundary; they are not peer review or a certificate of novelty.",
      "title": "Greedy Uniformity on Trees: Exact Obstruction and Near-Uniform Spiders",
      "updated": "2026-10-01",
      "url": "https://arxiv.org/abs/2610.02276"
    },
    {
      "age_days": 4,
      "arxiv_id": "2610.01527",
      "authors": [
        "Kabir Murjani",
        "Nisarg Patel"
      ],
      "content_date": "2026-10-01",
      "freshness": "fresh",
      "id": "arxiv:2610.01527",
      "kind": "paper",
      "label": "adjacent",
      "matched_signals": [
        "lean_formal_proving_agents"
      ],
      "published": "2026-10-01",
      "score": 3.5,
      "source": "arxiv-ai4math-core",
      "summary": "Non-Markovian environments are often modeled as Regular Decision Processes (RDPs), where dynamics depend on the interaction history through a finite automaton. Existing offline guarantees for RDPs rely on a distinguishability assumption on the behaviour policy but provide no means of verifying it. When the assumption is violated, distinct models may explain the data equally well. We study when data collected under a fixed behaviour policy can distinguish two candidate RDPs. We prove that the posterior odds between observationally equivalent candidates remain equal to the prior odds at every sample size, even when the policy visits every automaton state, and verify both results formally in Lean 4. We then characterize this equivalence exactly and derive PEC, an algorithm that decides it in time linear in the size of the product automaton. The distinguishability assumption of prior work fails on three of our four test environments, and the experiment identified by PEC restores it in each case.",
      "title": "Exact Distinguishability in Non-Markovian Decision Processes",
      "updated": "2026-10-01",
      "url": "https://arxiv.org/abs/2610.01527"
    },
    {
      "age_days": 4,
      "arxiv_id": "2610.01121",
      "authors": [
        "Youngwoo Kwon"
      ],
      "content_date": "2026-10-01",
      "freshness": "fresh",
      "id": "arxiv:2610.01121",
      "kind": "paper",
      "label": "adjacent",
      "matched_signals": [
        "verifier_guided_reasoning"
      ],
      "published": "2026-10-01",
      "score": 3.5,
      "source": "arxiv-ai4math-core",
      "summary": "Gibbs samplers, and more generally component-wise Markov chain Monte Carlo algorithms such as Metropolis-within-Gibbs, can be implemented using either random-scan or deterministic-scan updates. How much convergence can depend on this choice of scanning rule has been a longstanding question. We study this problem through $L^2$ spectral gaps, measuring computational cost in units of component updates. For a $d$-block Gibbs sampler, the cost of random scan is at most twice that of any fixed deterministic scan, while the reverse cost ratio is at most of order $d^2$. We extend these comparisons to general reversible component-wise updates under the global block-wise contraction condition. If $K_j$ denotes the update of block $j$ and $P_j$ its Gibbs counterpart, and $\\|K_j-P_j\\|\\leq λ_0<1$ for $j=1,\\ldots,d$, then the cost of random scan is at most $2/(1-λ_0)$ times that of deterministic scan, while the reverse cost ratio is of order at most $d^2/(1-λ_0)$. Examples show that the cost bounds for random scan relative to deterministic scan are asymptotically sharp and that the joint dependence on $d$ and $(1-λ_0)^{-1}$ in the reverse comparison cannot be improved uniformly. This work was assisted by generative AI, including for formal verification of mathematical results in Lean. The human author reviewed and verified the mathematical content and takes full responsibility for the results.",
      "title": "Cost Comparisons for Random and Deterministic Scans in Component-Wise Markov Chains",
      "updated": "2026-10-01",
      "url": "https://arxiv.org/abs/2610.01121"
    },
    {
      "age_days": 5,
      "arxiv_id": "2609.39544",
      "authors": [
        "Jules Viennot",
        "Guillaume Baudart",
        "Marc Lelarge"
      ],
      "content_date": "2026-09-30",
      "freshness": "fresh",
      "id": "arxiv:2609.39544",
      "kind": "paper",
      "label": "adjacent",
      "matched_signals": [
        "verifier_guided_reasoning"
      ],
      "published": "2026-09-30",
      "score": 3.5,
      "source": "arxiv-ai4math-core",
      "summary": "Recent achievements in AI-assisted mathematics require intensive interaction of agents with proof assistants to generate machine-checked proof certificates. Agents interact with proof assistants such as Rocq or Lean through an interface that controls what the agent receives from the prover and the cost of these interactions. Today, these interfaces are adapted from tools designed for humans and not optimized for agents. We propose an evolutionary method where a frontier model incrementally proposes new features and only keeps the ones that improve the overall performance of smaller models. We demonstrate the effectiveness of our method by growing, on a curated set of mathematical problems, \\rme, a new MCP server for the Rocq prover. On the held-out \\texttt{test} split of miniF2F-Rocq, an agent equipped with \\rme outperforms both the baseline that only exposes the Rocq compiler and an established MCP server, across four models from two families, in success rate, cost per solve, and time per solve. Although evolved for Rocq, the resulting server transfers to Lean, improving cost and time per solve on a subset of PutnamBench. We release \\rme and its port to Lean.",
      "title": "Growing an Agent/Prover Interface: Evolutionary Tool Design for Cost-Efficient Theorem Proving in Rocq and Lean",
      "updated": "2026-10-01",
      "url": "https://arxiv.org/abs/2609.39544"
    },
    {
      "age_days": 5,
      "arxiv_id": "2609.40062",
      "authors": [
        "Aparna Gupte",
        "Seyoon Ragavan",
        "Mark Zhandry"
      ],
      "content_date": "2026-09-30",
      "freshness": "fresh",
      "id": "arxiv:2609.40062",
      "kind": "paper",
      "label": "adjacent",
      "matched_signals": [
        "lean_formal_proving_agents"
      ],
      "published": "2026-09-30",
      "score": 3.5,
      "source": "arxiv-ai4math-core",
      "summary": "We establish a no-go theorem for a broad class of quantum algorithms for the dihedral coset problem (DCP). We consider the Fourier-sampling and subset-sum-measurement template proposed by Regev (SIAM Journal on Computing, 2004), which is one of the main approaches to solving DCP. Suppose that, after measuring the lower $n-1$ bits of the subset sum, the algorithm discards any $ω(\\log n)$ bits from each of the Fourier labels. Then we prove that the algorithm cannot succeed in solving DCP. This shows that any algorithm following this template must make extensive use of the Fourier labels, and thus serves as a useful guide for developing algorithms for DCP. As a main application, we show that the recent algorithm by Simon (IACR ePrint:2026/1591, August 11 2026) does not solve DCP. We show that after the subset-sum measurement, this algorithm can be implemented (up to exponentially-small error) using only the most-significant third of the Fourier labels, and is therefore subject to our general no-go theorem. To help with verifiability, we release Lean 4 code for our results.",
      "title": "A Fourier-Label Information-Loss Barrier for Dihedral Coset Algorithms",
      "updated": "2026-09-30",
      "url": "https://arxiv.org/abs/2609.40062"
    },
    {
      "age_days": 4,
      "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": 5,
      "arxiv_id": "2610.00413",
      "authors": [
        "Haruhisa Enomoto"
      ],
      "content_date": "2026-09-30",
      "freshness": "fresh",
      "id": "arxiv:2610.00413",
      "kind": "paper",
      "label": "adjacent",
      "matched_signals": [
        "general_ai_math_reasoning"
      ],
      "published": "2026-09-30",
      "score": 1.9,
      "source": "arxiv-ai4math-core",
      "summary": "Børve, Horiatakis, and Kalck defined the magnitude of the module category of a representation-finite algebra as the sum of the entries of the inverse of the matrix of Hom dimensions between indecomposable modules. They showed it is an Euler characteristic, the alternating count of vertices, arrows, and meshes of the Auslander--Reiten quiver, and conjectured that it is at least the number of simple modules, with equality precisely for special biserial algebras. We prove this conjecture over an algebraically closed field. For a representation-directed algebra, we relate the magnitudes of the module category, the subcategory of modules without a given simple composition factor, and the ideal quotient by it. By the theory of hammocks of Ringel and Vossieck, this quotient is equivalent to a category of representations of a finite poset, whose magnitude we compute. The general case reduces to the representation-directed case through algebras obtained by grading a standard form.",
      "title": "Magnitude of module categories and special biserial algebras",
      "updated": "2026-09-30",
      "url": "https://arxiv.org/abs/2610.00413"
    },
    {
      "age_days": 0,
      "authors": [
        "Henrik Böving"
      ],
      "content_date": "2026-10-05",
      "freshness": "fresh",
      "id": "github:leanprover/lean4:7537fb9348bf",
      "kind": "github_update",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-10-05",
      "repo": "leanprover/lean4",
      "score": 0.8,
      "source": "lean4-github",
      "summary": "Recent commit on leanprover/lean4.",
      "title": "leanprover/lean4: perf: make variable access in IR interpreter branch free (#15493)",
      "updated": "2026-10-05",
      "url": "https://github.com/leanprover/lean4/commit/7537fb9348bfd6ab0787bb636b576baeb9ed2e7b"
    },
    {
      "age_days": 0,
      "authors": [
        "Henrik Böving"
      ],
      "content_date": "2026-10-05",
      "freshness": "fresh",
      "id": "github:leanprover/lean4:05402572b139",
      "kind": "github_update",
      "label": "negative",
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      "source": "mathlib4-github",
      "summary": "Recent commit on leanprover-community/mathlib4.",
      "title": "leanprover-community/mathlib4: chore: move `QuasiMeasurePreserving` to the namespace of `MeasurePreserving` (#44142)",
      "updated": "2026-10-04",
      "url": "https://github.com/leanprover-community/mathlib4/commit/700eb6b6f6b5126e67b944a9744a1041a1f82548"
    },
    {
      "age_days": 1,
      "authors": [
        "Jovan Gerbscheid"
      ],
      "content_date": "2026-10-04",
      "freshness": "fresh",
      "id": "github:leanprover-community/mathlib4:98e02336d90b",
      "kind": "github_update",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-10-04",
      "repo": "leanprover-community/mathlib4",
      "score": 0.8,
      "source": "mathlib4-github",
      "summary": "Recent commit on leanprover-community/mathlib4.",
      "title": "leanprover-community/mathlib4: chore: adaptation for upcoming `on_goal` change (#44446)",
      "updated": "2026-10-04",
      "url": "https://github.com/leanprover-community/mathlib4/commit/98e02336d90b5495d098afdd27070f7e68b454d3"
    },
    {
      "age_days": 1,
      "authors": [
        "Snir Broshi"
      ],
      "content_date": "2026-10-04",
      "freshness": "fresh",
      "id": "github:leanprover-community/mathlib4:ca73e8b041d5",
      "kind": "github_update",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-10-04",
      "repo": "leanprover-community/mathlib4",
      "score": 0.8,
      "source": "mathlib4-github",
      "summary": "Recent commit on leanprover-community/mathlib4.",
      "title": "leanprover-community/mathlib4: chore(README.md): update the Lean link (#44482)",
      "updated": "2026-10-04",
      "url": "https://github.com/leanprover-community/mathlib4/commit/ca73e8b041d54ec36afc258dcd7856d06ce8bbb6"
    },
    {
      "age_days": 3,
      "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": 3,
      "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": 3,
      "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": 3,
      "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": 3,
      "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": 4,
      "arxiv_id": "2610.01728",
      "authors": [
        "Jakob Paul Zimmermann",
        "Moritz Grillo",
        "Andrei Balakin",
        "Georg Loho"
      ],
      "content_date": "2026-10-01",
      "freshness": "fresh",
      "id": "arxiv:2610.01728",
      "kind": "paper",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-10-01",
      "score": 0.5,
      "source": "arxiv-ai4math-core",
      "summary": "Understanding loss landscapes is central to explaining neural-network training, yet their structure remains only partially understood even in simple models. We study the Gaussian population loss of shallow, bias-free ReLU networks in the teacher--student setting. This provides a simple model for studying essential aspects such as feature learning and overparameterization. For teacher networks with positive output weights and planar features, we show that including a learned linear skip removes all spurious local minima with non-negative student output weights once the student network is at least as wide as the teacher network. In contrast, without the skip, we construct a fixed teacher network with positive output weights and only three hidden neurons in input dimension two whose spurious local minima persist at every student width at least three. Thus, a learned linear skip can remove spurious minima that persist under arbitrary overparameterization. Furthermore, we show that a positive output weight student network always learns the subspace spanned by the teacher features: student features at local minima with non-negative student output weights lie in the span of the teacher features. For ReLU networks in two dimensions, even heavily overparameterized student networks have effective width controlled by the teacher width: every critical point with positive student output weights has at most twice as many distinct student feature directions as teacher neurons. Finally, we transfer the benignity result to empirical minima over parameter balls of any prescribed radius, with the required sampling accuracy depending on that radius.",
      "title": "Removing spurious minima for planar features by skip connections",
      "updated": "2026-10-01",
      "url": "https://arxiv.org/abs/2610.01728"
    },
    {
      "age_days": 4,
      "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": 4,
      "arxiv_id": "2610.01639",
      "authors": [
        "Isaac Rudich",
        "Louis-Martin Rousseau"
      ],
      "content_date": "2026-10-01",
      "freshness": "fresh",
      "id": "arxiv:2610.01639",
      "kind": "paper",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-10-01",
      "score": 0.5,
      "source": "arxiv-ai4math-core",
      "summary": "Strassen showed that two 2x2 matrices can be multiplied with 7 multiplications instead of 8. Applied recursively, his algorithm multiplies two nxn matrices with O(n^2.807) multiplications, beating the naive O(n^3). The best known 3x3 recursive matrix multiplication algorithm uses 23 multiplications O(n^2.854). The best published lower bound of 21 (on algorithms with integer constants) leaves room for an algorithm with O(n^2.771) multiplications, and thus does not rule out the possibility of an algorithm that would beat Strassen's. We prove a lower bound of 22 multiplications for any 3x3 recursive algorithm with integer constants, proving that no such algorithm can do better than O(n^2.814) multiplications, and eliminating the possibility of a 3x3 algorithm that beats Strassen's 2x2 method. The proof builds on a recent decomposition method from Wang, who approached the problem by turning it into 496 subproblems. We provide exact solutions for 359 of them. The proof is in Lean; verification requires auditing only a few short files. The Lean formalization directly encodes statements about the limitations of recursive algorithms for matrix multiplication, as opposed to just a statement about the rank of the problem.",
      "title": "Lower Bound of 22 for 3x3 Matrix Multiplication over the Integers",
      "updated": "2026-10-01",
      "url": "https://arxiv.org/abs/2610.01639"
    },
    {
      "age_days": 4,
      "arxiv_id": "2610.00917",
      "authors": [
        "Yixuan Li",
        "Yiyun Zhou",
        "Yao Long Teng",
        "Fuchao Yang",
        "Yanchen Deng",
        "Zhiyi Lyu",
        "Xuyu Dong",
        "Feng Chen",
        "Bo An"
      ],
      "content_date": "2026-10-01",
      "freshness": "fresh",
      "id": "arxiv:2610.00917",
      "kind": "paper",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-10-01",
      "score": 0.5,
      "source": "arxiv-ai4math-core",
      "summary": "Choosing an agent system means choosing both a language model and the harness through which it acts. We ask whether a strong model, harness, or pairing stays strong when the setting changes. We evaluate 66 configurations: four configurable harnesses (OpenHands, DeepSeek Harness, PI, and openJiuwen) paired with five models on TUA-Bench, ALE-CLI, and Terminal-Bench 4, plus the native Codex-GPT and Claude Code-Claude pairings. Model rankings reverse across harnesses. On Terminal-Bench 4, Claude leads GPT by 7.94 points in OpenHands but trails it by 30.16 points in PI. For four of the five models, the best harness changes from one benchmark to another, yet some pairings hold: openJiuwen gives Kimi its highest score on all three benchmarks, by 5.61 to 11.11 points. A model's own vendor harness is not reliably its best, and higher cost does not reliably buy a higher score. On Terminal-Bench 4, GPT scores higher under PI than under DSH at less than a quarter of the cost per task. Matched trajectories suggest why fit varies. Models start almost all repairs themselves, so much depends on whether the harness hands failures back in a form the model can use. GPT does best with PI's lean scaffold, while Kimi, which often issues malformed tool calls, does best in openJiuwen. We argue that the model, the harness, and the task should be evaluated together, and we release the harness adapters, evaluation code, and all 6,204 scored trajectories at https://github.com/liyix/finding-the-right-fit and https://huggingface.co/datasets/yixuanli97/finding-the-right-fit.",
      "title": "Finding the Right Fit: Model-Harness Interactions across Agent Tasks",
      "updated": "2026-10-01",
      "url": "https://arxiv.org/abs/2610.00917"
    },
    {
      "age_days": 4,
      "arxiv_id": "2610.00885",
      "authors": [
        "Naing Oo Lwin"
      ],
      "content_date": "2026-10-01",
      "freshness": "fresh",
      "id": "arxiv:2610.00885",
      "kind": "paper",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-10-01",
      "score": 0.5,
      "source": "arxiv-ai4math-core",
      "summary": "Coding agents increasingly automate Lean proof development, but successful compilation alone does not establish that a candidate proves the intended statement under acceptable assumptions. We present FORALL-LEAN-AGENT, a frontend-agnostic framework for auditable reasoning in formal mathematics and software verification. The framework combines isolated workspaces, Lean tools, and fresh review with statement comparison, axiom audits, and independent proof checking where supported. Verification evidence and reviewer decisions are bound to the same candidate artifact, making acceptance traceable. We evaluate the framework on VeriSoftBench, PutnamBench, and both problems in the Lean Eval softwareverification track. On the 100-task VeriSoftBench subset, integration with FORALLLEAN-AGENT raises benchmark-rule success from 93 to 100 for GPT-5.6 Sol at low effort while reducing cost from $69 to $62. The PutnamBench evaluation accepts all 672 problems at an average of $4.72 each. These results show that agent harness design can improve correctness and efficiency while providing evidence beyond aggregate solve counts.",
      "title": "FORALL-LEAN-AGENT for Auditable Reasoning in Formal Mathematics and Software Verification",
      "updated": "2026-10-01",
      "url": "https://arxiv.org/abs/2610.00885"
    },
    {
      "age_days": 4,
      "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"
    },
    {
      "age_days": 5,
      "arxiv_id": "2609.39872",
      "authors": [
        "Bruno Le Floch"
      ],
      "content_date": "2026-09-30",
      "freshness": "fresh",
      "id": "arxiv:2609.39872",
      "kind": "paper",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-09-30",
      "score": 0.5,
      "source": "arxiv-ai4math-core",
      "summary": "We consider the 653 equational laws of order up to 4 for an associative binary operation, and all of their conjunctions. We determine that there are only 456 equivalence classes of such conjunctions (associative equational theories), and find all implications between them. The conjunction operation makes this set of theories into a semi-lattice. These results are formalized in Lean. This is a semigroup analogue of the Equational Theories Project, but extended to conjunctions of equations.",
      "title": "Lattice of 456 semigroup varieties from equations of order up to 4",
      "updated": "2026-09-30",
      "url": "https://arxiv.org/abs/2609.39872"
    },
    {
      "age_days": 5,
      "arxiv_id": "2610.00530",
      "authors": [
        "Chengxi Yang",
        "Tej Chajed",
        "Thomas Reps"
      ],
      "content_date": "2026-09-30",
      "freshness": "fresh",
      "id": "arxiv:2610.00530",
      "kind": "paper",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-09-30",
      "score": 0.5,
      "source": "arxiv-ai4math-core",
      "summary": "Modern incremental computation theories like DBSP have enabled efficient incrementalization of general recursive computations. To do so, they require a runtime Fixpoint Detection (FPD) mechanism to detect whether an iterative computation has reached the fixpoint and thus should terminate. However, we show that the commonly suggested \"FirstZero\" strategy is unsound even in naturally arising cases, and that exact FPD is impossible for arbitrary DBSP circuits with expressive primitive nodes. This issue reveals a fundamental gap between the mathematical specification and implementations of such theories. To fill this gap, using DBSP as a core calculus, we develop a formal theory of convergence detection. Within this theory, we define internal convergence (IntConv) as a declarative criterion corresponding to the internal-state-stability strategy used by practical implementations, and prove that IntConv is a sufficient condition for external convergence. We then define the state fixpoint (StFP) predicate and a sound and complete StFP detector. Combining the StFP detector with fixed-input and zero-output checks yields a sound and complete IntConv detector. Moreover, for a large class of useful circuits (programs) including Datalog queries, nested while queries, and their incrementally optimized versions, we show that IntConv is not only sound but also complete (meaning any convergence in the theory implies the convergence in our criterion). As a result, our theory provides semantic guarantees for convergence detection on all these circuits. Our results are formally verified in Lean, with the formalization available at https://github.com/Arcadia-Y/fixing-the-fixpoint/.",
      "title": "Fixing the Fixpoint: A Formal Theory of Convergence Detection for Incremental Recursive Computation",
      "updated": "2026-09-30",
      "url": "https://arxiv.org/abs/2610.00530"
    },
    {
      "age_days": 5,
      "arxiv_id": "2610.02252",
      "authors": [
        "Dingling Yao",
        "Kahaan Gandhi",
        "Valentin Duruisseaux",
        "Boris Bonev",
        "Francesco Locatello",
        "Anima Anandkumar"
      ],
      "content_date": "2026-09-30",
      "freshness": "fresh",
      "id": "arxiv:2610.02252",
      "kind": "paper",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-09-30",
      "score": 0.5,
      "source": "arxiv-ai4math-core",
      "summary": "Many scientific questions require reasoning about what was never observed: What if the conditions, interventions, or history had been different? Models can predict accurately on observed data yet fail on such what-if queries when correlated inputs are varied independently. A common remedy is to add controlled simulation data in which these factors are explicitly disentangled, but this requires access to a simulator, can be computationally expensive, and inherits the simulator's modeling assumptions. We introduce ReRoute, a framework for targeted scientific what-if prediction that combines factual data with partial mechanistic knowledge, without requiring controlled intervention data for adaptation. ReRoute fixes the queried input of a pretrained backbone to a reference value, reintroduces its variation through a known mechanistic pathway, and fine-tunes on the original factual data, while leaving downstream effects to the learned dynamics. We provide a causal identification result for this construction under explicit structural assumptions, with the core argument machine-checked in Lean. After showing that ReRoute achieves highly accurate counterfactual predictions in a controlled advection-diffusion system where exact responses are available, we turn to state-of-the-art climate emulation. On held-out coupled-climate interventions, ReRoute reduces aggregate climate error by 18.2-31.8% under severe CO$_2$ distribution shifts while preserving skill under standard conditions, at a small fraction of the cost of retraining on additional controlled simulations, without even accounting for the substantial expense of generating such data. Finally, on an emulator trained from historical ERA5 reanalysis, where no counterfactual reference exists, ReRoute preserves substantially more of the surface warming implied by the observed boundary conditions under a fixed-CO$_2$ counterfactual.",
      "title": "Counterfactual Predictions in Scientific Emulators Without Controlled Experiments",
      "updated": "2026-09-30",
      "url": "https://arxiv.org/abs/2610.02252"
    },
    {
      "age_days": 5,
      "arxiv_id": "2609.39633",
      "authors": [
        "John Fairfax-Ball"
      ],
      "content_date": "2026-09-30",
      "freshness": "fresh",
      "id": "arxiv:2609.39633",
      "kind": "paper",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-09-30",
      "score": 0.5,
      "source": "arxiv-ai4math-core",
      "summary": "We study a support-collapse version of graph pebbling on paths. A configuration is stackable if a sequence of legal pebbling moves can produce a nonzero configuration supported on a single vertex. On the path P_n, we choose a configuration uniformly from all weak compositions of total n times mu_n, where mu_n is a positive integer. We prove a two-sided fixed-offset transition for the logarithmic density. The transition is centred at sqrt(log_2 n) - (1/2) log_2 log_2 n + log_2(3e). For every fixed epsilon greater than zero, the stackability probability tends to zero when log_2 mu_n is eventually at most the centre minus epsilon, and tends to one when it is eventually at least the centre plus epsilon. No assertion is made at zero offset. The proof uses an exact recursive stackability score on trees, a one-dimensional path-message recurrence, binary-partition asymptotics for rare dyadic deficit excursions, a constant-cost regeneration argument, and an exact deep-message necessity theorem. Conditioning independent geometric occupancies on their sum returns the uniform fixed-total model. The finite deterministic necessity theorem and its exact fixed-total corollary are formalised in Lean and registered with Palomar; the full probabilistic asymptotic theorem is not part of that registration.",
      "title": "A Fixed-Offset Transition for Random Stackability on Paths",
      "updated": "2026-09-30",
      "url": "https://arxiv.org/abs/2609.39633"
    }
  ],
  "lookback_days": 21,
  "schema": "ai4math-radar-run-v1",
  "timezone": "America/Los_Angeles"
}