{ "counts": { "adjacent": 19, "core": 4, "errors": 0, "negative": 67, "total": 90 }, "date": "2026-09-29", "errors": [], "fresh_content_days": 21, "generated_at": "2026-09-29T20:10:18Z", "items": [ { "age_days": 1, "arxiv_id": "2609.34460", "authors": [ "Sareh Nabi", "Roland Vogl", "Marzieh Nabi" ], "content_date": "2026-09-28", "freshness": "fresh", "id": "arxiv:2609.34460", "kind": "paper", "label": "core", "matched_signals": [ "lean_formal_proving_agents", "mathlib_retrieval" ], "published": "2026-09-28", "score": 6.5, "source": "arxiv-ai4math-core", "summary": "Does structured mathematical knowledge help LLMs prove theorems in Lean 4? If so, for which models, and does the answer vary by problem? Formal libraries such as Mathlib encode 285,000+ verified theorems with syntactic dependencies, but the semantic layer mathematicians rely on for discovery (analogies, generalizations, cross-domain bridges) remains implicit. We introduce MathAgent, which builds this layer as a knowledge graph, MathKG, and uses it to augment LLM theorem provers. MathKG connects 364 Mathlib theorems and definitions by 9,434 typed semantic edges inferred via LLM-based relation extraction anchored to verified Mathlib declarations. We run a controlled ablation across four augmentation modes (no context, knowledge-graph context, Mathlib retrieval, both) and five models: Qwen3-8B/32B, their Lean-specialized derivatives Goedel-Prover-V2-8B/32B, and Claude Sonnet 4.6, on miniF2F, plus PutnamBench and MathOlympiadBench for Sonnet. Three findings emerge. (i) Specialization dominates augmentation: Lean fine-tuning adds 33-38 percentage points of solve rate in every mode, and a specialized 8B model beats a $4\\times$ larger general one by 29-35 points, while no augmentation mode improves solve rate by more than 3 points. (ii) Augmentation is capability-conditioned: knowledge-graph context helps small models but hurts large ones, with the specialized model gaining more relative to its general base at every scale. (iii) Yet the augmentation modes solve different problems: an oracle selecting the best mode per problem solves 6% to 58% more than the unaugmented prover, a complementarity effect that strengthens on harder problems (32% more on PutnamBench). These results motivate adaptive strategies that select augmentation by model capability and problem. Code, data, and artifacts are available at https://github.com/sarehnabi/mathagent", "title": "When Does Structured Knowledge Help Neural Theorem Proving?", "updated": "2026-09-28", "url": "https://arxiv.org/abs/2609.34460" }, { "age_days": 2, "arxiv_id": "2609.33066", "authors": [ "Volkan Dağlı", "Zerrin Dağlı", "Dağhan Dağlı" ], "content_date": "2026-09-27", "freshness": "fresh", "id": "arxiv:2609.33066", "kind": "paper", "label": "core", "matched_signals": [ "lean_formal_proving_agents", "verifier_guided_reasoning" ], "published": "2026-09-27", "score": 6.5, "source": "arxiv-ai4math-core", "summary": "Contemporary neural inference architectures rely on dense floating-point weight matrices stored in high-bandwidth memory (VRAM), incurring severe memory-wall bottlenecks and preventing native execution inside deterministic virtual machines like the Ethereum Virtual Machine (EVM). Verifying termination and arithmetic invariants for recursive dynamical systems over continuous domains is generally undecidable in the Blum-Shub-Smale model. Here, we present the formal verification and bare-metal empirical validation of WERR (Waves & Errors) and Phase III Orbital Error Dynamics (OED), a non-tensor decision paradigm that procedurally synthesizes non-linear decision boundaries on demand from a 24-byte coordinate seed $Θ= (c_x, c_y, \\text{zoom})$ along the boundary of the Mandelbrot set ($\\partial\\mathcal{M}$). By projecting the recurrence $z_{n+1} = z_n^2 + c$ onto the modular residue ring $\\mathbb{Z}/9\\mathbb{Z}$ and the fixed-point domain $\\mathbb{Q}_{16.16}$, we establish ten machine-verified theorems in Lean 4 (v4.34.1) with Mathlib4 and zero unproven conjectures (sorry): proving $\\mathcal{I}_3 = \\{0,3,6\\} \\subset \\mathbb{Z}/9\\mathbb{Z}$ ideal closure, universal fuel-bounded halting ($\\le 9$ and $\\le 12$ steps), absence of $\\mathbb{Q}_{16.16}$ square overflow below $2^{63}-1$, non-constant boundary escape sensitivity, and a parametric EVM gas bound ($\\le 22,557 \\le 24,000$ gas). Evaluated on a 40-core Dual Intel Xeon server, the vectorized 36-iteration CPU kernel processes 100,000 decisions in 6.49 s (15,397 decisions/s, 0 Bytes VRAM, 15.15x speedup), while a sigmoidal outlier gate suppresses 100.00% of adversarial spikes while preserving 89.60% of clean baseline signals.", "title": "Zero-Storage Procedural Neural Synthesis via Boundary Dynamics: Formal Verification in Lean 4 and Bare-Metal Gauntlet Validation", "updated": "2026-09-27", "url": "https://arxiv.org/abs/2609.33066" }, { "age_days": 3, "arxiv_id": "2609.32561", "authors": [ "Alejandro José Soto Franco", "Kobe Marshall-Stevens" ], "content_date": "2026-09-26", "freshness": "fresh", "id": "arxiv:2609.32561", "kind": "paper", "label": "core", "matched_signals": [ "lean_formal_proving_agents", "mathlib_retrieval" ], "published": "2026-09-26", "score": 6.5, "source": "arxiv-ai4math-core", "summary": "We formalise in Lean 4, on top of Mathlib, the solvability of the Dirichlet problem for second-order linear elliptic operators in divergence form. The machine-verified results, with no sorry in the development, include the Poincaré inequality, the existence of weak solutions by the Lax-Milgram theorem, Rellich-Kondrachov compactness, the Fredholm alternative, the spectral theorem, interior regularity estimates, and the Sobolev embedding theorem. From these results we obtain a formalisation of classical solvability for sufficiently regular coefficients and data. Our Lean library includes a self-contained theory of Sobolev spaces developed independently of existing formalisations. Throughout the paper we associate each prose statement with the named machine-checked Lean declaration that discharges it.", "title": "Formalising Linear Elliptic PDE Theory in Lean 4", "updated": "2026-09-26", "url": "https://arxiv.org/abs/2609.32561" }, { "age_days": 4, "arxiv_id": "2609.31903", "authors": [ "Yidi Qi", "Melanie Weber" ], "content_date": "2026-09-25", "freshness": "fresh", "id": "arxiv:2609.31903", "kind": "paper", "label": "core", "matched_signals": [ "autoformalization", "lean_formal_proving_agents" ], "published": "2026-09-25", "score": 6.5, "source": "arxiv-ai4math-core", "summary": "AI agents can now formalize entire textbooks and major theorems in proof assistants such as Lean, but current efforts are typically centralized: a single team runs all agents and bears the full computational cost. We introduce Choir, an open protocol for distributed formalization. Choir decomposes a project into tasks that can be completed by independent contributors, each running their own agent with their own LLM subscription, while coordinating entirely through the project's GitHub repository. To support open participation, every contribution is checked by a deterministic gate before merge. Choir supports Lean 4, Isabelle, and Rocq, and is open source and modular, allowing projects to replace individual components or extend the protocol.", "title": "Choir: An Open Protocol for Distributed Multi-Agent Autoformalization", "updated": "2026-09-25", "url": "https://arxiv.org/abs/2609.31903" }, { "age_days": 3, "arxiv_id": "2609.32201", "authors": [ "Hantao Yu", "Sandy Han", "Udaya Ghai", "Ferhat Erata", "Joe Lilien", "Aman Goel", "Ali Torkamani" ], "content_date": "2026-09-26", "freshness": "fresh", "id": "arxiv:2609.32201", "kind": "paper", "label": "adjacent", "matched_signals": [ "autoformalization", "reasoning_rl_distillation" ], "published": "2026-09-26", "score": 4.9, "source": "arxiv-ai4math-core", "summary": "On-Policy Context Distillation (OPCD) has recently emerged as a powerful technique for transferring context to student models and for self-improvement. In OPCD, the teacher is conditioned on privileged information, and the goal is to minimize the Kullback-Leibler (KL) divergence between the privileged teacher and the student, evaluated on student-generated tokens. Many existing studies show that using instance-specific gold answers or gold demonstrations as the default privilege can hurt training performance, especially out-of-distribution (OOD). In this work, we instead design general instructions that target common student mistakes observed on the training samples, and show that such simple instructions can outperform gold as the OPCD privilege. In autoformalization tasks, using a matched formatting instruction as the privilege could outperform gold in OOD accuracy by a large margin. In 7 out of 8 experiments using ProverQA, ProofWriter, and ProntoQA as datasets, and Qwen3-Thinking and Olmo3-Thinking families as models, matched instruction privileges outperform gold in OOD by 4 to 17 points, while remaining on par with gold in-domain. Each instruction is only a few sentences (and thus contains much less information compared to all instance-specific gold) and is applied uniformly to every training sample. These results indicate that a general instruction, which applies equally to source and target domain examples, can be substantially more transferable than instance-specific gold in OPCD while maintaining in-domain performance.", "title": "Instruct, Not Answer: Using Instruction Privileges in On-Policy Context Distillation", "updated": "2026-09-26", "url": "https://arxiv.org/abs/2609.32201" }, { "age_days": 1, "arxiv_id": "2609.34927", "authors": [ "Alexandre Zua Caldeira" ], "content_date": "2026-09-28", "freshness": "fresh", "id": "arxiv:2609.34927", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-09-28", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "We prove that the state space of every well-formed session type, quotiented by strongly connected components, forms a bounded lattice; n-ary parallel composition yields product lattices. Two consequences follow: duality preserves the lattice up to isomorphism, and Gay-Hole width subtyping corresponds to lattice embedding for non-recursive types. We validate this on 108 benchmark protocols across networking, databases, distributed systems, AI, and fault tolerance: all form lattices, 93 distributive, 15 non-distributive. Mechanised in Lean 4 with two independently developed tool implementations.", "title": "Session Type State Spaces Form Lattices", "updated": "2026-09-28", "url": "https://arxiv.org/abs/2609.34927" }, { "age_days": 1, "arxiv_id": "2609.35762", "authors": [ "Jaewook J. Suh", "TaeHo Yoon", "Edward Duc Hien Nguyen", "Bicheng Ying", "Shiqian Ma" ], "content_date": "2026-09-28", "freshness": "fresh", "id": "arxiv:2609.35762", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-09-28", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "This paper presents Peppy, an AI-assisted workflow for discovering tight, analytic convergence proofs for first-order optimization algorithms. Generic approaches to using LLMs to conduct mathematical research target an unspecified, broad spectrum of problems and sometimes use the Lean 4 proof assistant for formalization. On the other hand, Peppy leverages domain-specific knowledge more heavily and is thereby capable of constructing the proofs in a more structured manner, which allow a minimal and accessible verification through SymPy. We experimentally demonstrate through examples that Peppy provides a rigorous, practical, and reproducible paradigm for AI-assisted theorem synthesis in optimization. We further highlight its capability of closing several open problems on tight convergence analysis of first-order optimization algorithms, including conjectures for Nesterov's FGM. Overall, Peppy is designed to turn the art of optimization algorithm analysis into a science.", "title": "Peppy: An AI-Assisted Workflow for Tight Convergence Analysis of Optimization Algorithms", "updated": "2026-09-28", "url": "https://arxiv.org/abs/2609.35762" }, { "age_days": 1, "arxiv_id": "2609.35187", "authors": [ "Xinyu Wang" ], "content_date": "2026-09-28", "freshness": "fresh", "id": "arxiv:2609.35187", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-09-28", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "Let $f$ be a Steinhaus random multiplicative function and let $g$ be a polynomial of degree $d$. Write $S_N=N^{-1/2}\\sum_{n\\le N}f(n)\\,e(g(n))$. We prove a quantitative dichotomy for the moments of $S_N$: for each integer $s\\ge 2$, either $\\mathbb{E}|S_N|^{2s}$ is close to the Gaussian moment $s!$, or the coefficients of $g$ can be approximated by rationals with a small denominator. In particular, if the coefficients of $g$ satisfy a Diophantine condition, then $S_N$ converges in law to a complex normal distribution with mean $0$ and variance $1$. Lean 4 code for the proofs is provided, for convenience of verification.", "title": "Moments of random multiplicative functions with polynomial coefficients", "updated": "2026-09-28", "url": "https://arxiv.org/abs/2609.35187" }, { "age_days": 1, "arxiv_id": "2609.35062", "authors": [ "Shuo Li Liu" ], "content_date": "2026-09-28", "freshness": "fresh", "id": "arxiv:2609.35062", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-09-28", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "We give a finite simplex theorem. When the number of states equals the number of source signals and the source posterior likelihood-ratio vectors form a simplex, Blackwell dominance is characterized by a universal barycentric-coordinate condition. We also give an explicit countable-state, countable-signal diagnostic-monitor theorem with a closed-form universal criterion and a closed-form garbling; tensorization yields dominance at every sample size. Finally, we provide a three-state, two-signal counterexample to the sufficiency conjectured by Mu, Pomatto, Strack, and Tamuz for their many-state moment-generating-function conditions for large-sample Blackwell dominance. The pair satisfies strict comparisons on both parameter domains, all ordered Kullback-Leibler inequalities, bounded likelihood ratios and pairwise genericity, but dominance fails at every positive sample size. The finite theorem, obstruction, and continuum inequalities are verified in Lean 4.30 with Mathlib; the countable diagnostic proof is given analytically.", "title": "Blackwell Boundaries", "updated": "2026-09-28", "url": "https://arxiv.org/abs/2609.35062" }, { "age_days": 2, "arxiv_id": "2609.33043", "authors": [ "Kristian Muri Knausgård" ], "content_date": "2026-09-27", "freshness": "fresh", "id": "arxiv:2609.33043", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-09-27", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "The lower asymptotic proportion of distinct nontrivial zeros of the Riemann zeta function, counted with multiplicity, is at least $0.8369928814\\ldots$. The previous bound was $0.83625\\ldots$. As in the proof of that bound, an unconditional version of Montgomery's pair-correlation theorem gives an asymptotic energy estimate. The new ingredient is a short matrix inequality with a free clipping parameter. It strengthens the lower bound for this energy in terms of the number of distinct zeros. The gain is a nonnegative correction from overlaps between different nearby zeros on the critical line, which is retained even when some of these zeros are double. The matrix inequality, the threshold lemma, the block dichotomy, the counting assembly and the exact arithmetic are proved formally in Lean 4. The constant relies on a computer-assisted local inequality from recent work that has not yet been refereed. That computation was re-run independently, and every imported input is listed. This paper is primarily an experiment in AI-assisted mathematical research (Section 4).", "title": "More than 83.69% of the zeros of the Riemann zeta function are distinct", "updated": "2026-09-27", "url": "https://arxiv.org/abs/2609.33043" }, { "age_days": 2, "arxiv_id": "2609.33033", "authors": [ "Weinan Wang" ], "content_date": "2026-09-27", "freshness": "fresh", "id": "arxiv:2609.33033", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-09-27", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "We formalize in Lean 4 the Galerkin construction of global Leray--Hopf weak solutions for the unforced two-dimensional Navier--Stokes equations on arbitrary bounded open domains with no-slip boundary conditions. For every initial datum in the solenoidal \\(L^2\\) velocity space, the solution has a weakly continuous velocity path, satisfies the weak equation locally in time, and obeys the energy inequality at every time. For rectangles, we also formalize finite-horizon solutions with continuous forcing in the dual energy space. The development includes divergence-free graph spaces, compact spectral coordinates, transport cancellation, Ladyzhenskaya estimates, and simultaneous space--time compactness. The abstract Hilbert-space results apply to evolution equations with a compact energy-to-velocity embedding and a skew trilinear nonlinearity.", "title": "Formalization of the Galerkin Construction for the Two-Dimensional Navier--Stokes Equations in Lean", "updated": "2026-09-27", "url": "https://arxiv.org/abs/2609.33033" }, { "age_days": 3, "arxiv_id": "2609.32140", "authors": [ "Shisheng Li" ], "content_date": "2026-09-26", "freshness": "fresh", "id": "arxiv:2609.32140", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-09-26", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "We give an elementary proof that every positive rational number $a/b$ with $b$ squarefree is a finite sum of distinct unit fractions $1/n$, where each $n$ is a product of two distinct primes (Erdős Problem #306). After a reduction to small targets, we take a single complete bipartite graph between the primes in $(y^2,2y^2]$, together with $2$ and the primes of $b$, and a tuned initial segment of the primes in $(y^8,y^9]$, and show that some subgraph has reciprocal sum congruent to $a/b$ modulo $1$; the small total mass then forces equality. Writing the number of such subgraphs as a finite Fourier sum, we sort the frequencies into three cases using a table indexed by the two sides of the graph. The small integer frequencies give a positive main term, and all other frequencies are negligible by a divisor-counting argument and a no-wrap-around form of the Chinese remainder theorem. The only inputs about primes are Chebyshev-type bounds. The circle-method framework comes from Tang's Lean development, which gave the first proof; our construction removes its anchor-synchronisation step. The proof has been formalised in Lean 4, apart from a cited inequality of Ramanujan. This work is a human-AI collaboration: AI tools contributed substantially to the construction, the experiments and the writing.", "title": "Unit fractions with semiprime denominators: an elementary proof of Erdős Problem #306", "updated": "2026-09-26", "url": "https://arxiv.org/abs/2609.32140" }, { "age_days": 3, "arxiv_id": "2609.32251", "authors": [ "Junyi Gong", "Zijie Qiu", "Ben Zhong Tang" ], "content_date": "2026-09-26", "freshness": "fresh", "id": "arxiv:2609.32251", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-09-26", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "Chemistry's phenomenological theories multiply with its observables: the same physical reality is repeatedly named in different measurement contexts, and the relations among theories have never had an auditable carrier. Here we present PhotoLean, a machine-checked genealogy of seventeen photochemical and photophysical theories formalized in Lean 4 over a shared kernel. Every pair of theories carries either a verified relation (equivalence, implication, composition with registered premises, definitional certificate) or a registered reason for its absence, covering all 136 pairs. Three adjudications resolve inherited conflations with if-and-only-if boundaries and witnesses: the symmetry factor $β$ equals one-half exactly when reactant and product force constants coincide; Kasha's rule and the Kasha-Vavilov rule agree only under closed quantification with a loss channel; and intensity-only Stern-Volmer data cannot distinguish static from dynamic quenching, whereas the lifetime channel discriminates exactly. As proposition generation shifts to machines, such machine-checked relational ledgers offer a replicable audit layer for theory itself.", "title": "The Same and Not the Same: A Machine-Checked Genealogy of Photochemical Theory", "updated": "2026-09-26", "url": "https://arxiv.org/abs/2609.32251" }, { "age_days": 3, "arxiv_id": "2609.32908", "authors": [ "Aarnav Choudhary" ], "content_date": "2026-09-26", "freshness": "fresh", "id": "arxiv:2609.32908", "kind": "paper", "label": "adjacent", "matched_signals": [ "verifier_guided_reasoning" ], "published": "2026-09-26", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "Neural theorem provers must both propose tactics and decide which valid successor states to explore. We study whether one-step Lean transitions provide a self-supervised signal for branch ordering. A JEPA-style model predicts latent successor representations and scores only kernel-validated, nonterminal successors generated by a fixed pretrained ByT5 proposer. JEPA achieves higher Top-1 than matched InfoNCE on a same-theorem ranking diagnostic(50.18% versus 31.55%), but averages 282.3 of 987 solved theorems across three seeds versus 308 for proposer ordering, while requiring more tactic checks. In this setting, accurate one-step transition ranking is therefore insufficient as a long-horizon search value. The controlled evaluation separates representation from proposal quality and treats kernel-checked proof completion as the primary endpoint.", "title": "Predicting the Next State Is Not Enough: JEPA Representations for Lean Theorem Proving", "updated": "2026-09-26", "url": "https://arxiv.org/abs/2609.32908" }, { "age_days": 3, "arxiv_id": "2609.32491", "authors": [ "Jorryt de Jong", "Stefan Moraca", "Ettore Cesari", "Lasha Abzianidze" ], "content_date": "2026-09-26", "freshness": "fresh", "id": "arxiv:2609.32491", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-09-26", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "Large Language Models (LLMs) are highly capable of natural language reasoning and appear to store a great deal of lexical knowledge, but it is still unclear how much of this knowledge they actually use when reasoning, and whether they use it in the right way. On the other hand, logic-based Natural Language Inference (NLI) systems provide transparent and formally grounded reasoning, but they need to be supplied with rich lexical knowledge to prove inferences beyond purely logical ones. In this paper, we evaluate whether LLMs can identify all lexical knowledge needed to solve NLI problems and how much this knowledge contributes to proof search in a logic-based NLI system. Our research focuses exclusively on structured lexical entailments (e.g., chinchilla$\\sqsubseteq$small animal) as a proxy for structured explanations for NLI problems with an entailment label. First, we curate a dataset for a new task of explaining sentential entailments with a set of lexical entailments. The dataset is used to intrinsically evaluate LLMs on generating structured lexical explanations. Then, we use NLI as an extrinsic evaluation in a simple neuro-symbolic setting, assessing whether LLMs can supply sufficient lexical relations to LangPro, a natural-logic theorem prover for natural language. The results show that the proposed task remains challenging even for hosted proprietary LLMs, and that their contribution to theorem proving is moderate: generated relations are often only partially sound and may be tailored to the specific NLI problem rather than representing generally valid lexical knowledge.", "title": "Explaining Textual Entailment with Lexical Entailments: Using LLMs to Supply Lexical Relations for Formal Proofs", "updated": "2026-09-26", "url": "https://arxiv.org/abs/2609.32491" }, { "age_days": 3, "arxiv_id": "2609.32995", "authors": [ "Tianlong Nan", "Garud Iyengar", "Christian Kroer", "Shuvomoy Das Gupta" ], "content_date": "2026-09-26", "freshness": "fresh", "id": "arxiv:2609.32995", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-09-26", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "Alternating gradient descent-ascent (AltGDA) is a simple and practically effective method for solving finite two-player zero-sum matrix games. However, the theory of AltGDA remains limited: existing results either apply only to unconstrained settings or require restrictive assumptions on the equilibrium in constrained settings. We show that AltGDA converges globally at an $O(1/T)$ ergodic rate in every finite two-player zero-sum matrix game. Unlike prior results, our guarantee holds for every initialization and every horizon $T$: the uniform averages of the AltGDA iterates satisfy an $O(1/T)$ duality-gap bound. Our proof is inspired by numerical results obtained using a novel performance estimation programming (PEP) framework for Lyapunov function search over compact convex sets. Additionally, we provide simple counterexamples showing that the last-iterate duality gap of AltGDA does not converge to zero. This justifies why averaging of iterates is indeed necessary to achieve an $O(1/T)$ rate. We have formalized and machine-checked our global ergodic convergence result in Lean 4.", "title": "AltGDA Achieves Global $O(1/T)$ Ergodic Convergence in Matrix Games", "updated": "2026-09-26", "url": "https://arxiv.org/abs/2609.32995" }, { "age_days": 3, "arxiv_id": "2609.32115", "authors": [ "Zhaoxi Chen", "Daniel Raggi" ], "content_date": "2026-09-26", "freshness": "fresh", "id": "arxiv:2609.32115", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-09-26", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "Rewriting in proof assistants spans a strict hierarchy. Equational rewriting substitutes based on extensional equality; subequational rewriting substitutes based on homogeneous relations such as $\\leq$ or $\\subseteq$, justified by monotonicity lemmas; and generalized rewriting relates different operations across different types, justified by transfer rules. All three are specializations of one schema, the function relator $(R \\Rightarrow S)\\,f\\,g$. Lean 4's rw and Mathlib's grw/gcongr implement the first two levels, but the third has had no Lean implementation. We present one: a transfer tactic family over a tagged rule database, benchmarked using Mathlib's own test suites, of which 112 of 178 ported tests close through the transfer engine. An evaluation of the design's three motivating hypotheses returns a mixed result: generality holds, authoring effort wins in its amortized form (averaged per use), but on dependency footprint, transfer only ties a mature library and loses on coercions. This shows that transfer's value is concentrated on transferring into domains where few theorems exist.", "title": "A Transfer Tactic for Lean", "updated": "2026-09-26", "url": "https://arxiv.org/abs/2609.32115" }, { "age_days": 4, "arxiv_id": "2609.31811", "authors": [ "John Fairfax-Ball" ], "content_date": "2026-09-25", "freshness": "fresh", "id": "arxiv:2609.31811", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-09-25", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "The stacking number of a graph is the least integer t >= 2 such that every configuration of t pebbles can be transformed by pebbling moves into a configuration supported on one vertex. We prove that, for every finite tree T with at least two vertices, this number equals the rooted distance-and-degree estimator conjectured by Csernák and Soukup. The proof uses an exact recursive characterization of stackability at a prescribed vertex, an explicit zero-score obstruction, and a weighted cancellation argument for arbitrary nonstackable configurations. The complete theorem is formalized in Lean 4; the formal result has also passed Palomar mechanical verification and is publicly registered as PALOMAR-2026-09-25-000010.", "title": "The stacking number of a tree", "updated": "2026-09-25", "url": "https://arxiv.org/abs/2609.31811" }, { "age_days": 4, "arxiv_id": "2609.31557", "authors": [ "Yuchen Yang", "Xinan Dai", "Wenhao Deng", "Yingdong Shi", "Feng Xu", "Tailin Wu" ], "content_date": "2026-09-25", "freshness": "fresh", "id": "arxiv:2609.31557", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-09-25", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "Chambolle-Dossal acceleration guarantees $F(x_n)-F^*=o(n^{-2})$ for every fixed smooth convex loss with a minimizer. We show that this qualitative improvement admits no universal quantitative rate. For every damping parameter $α>3$ and positive nondecreasing gain $G(n)\\to\\infty$, we construct a fixed one-dimensional smooth convex loss whose exact CD orbit satisfies $\\sup_{n\\ge1} n^2G(n)\\bigl(F(x_n)-F^*\\bigr)=\\infty$. Thus no divergent gain improves the $n^{-2}$ scale for all fixed losses, even with instance-dependent constants. The construction prescribes queried gradients and realizes infinitely many slow blocks within one smooth convex objective. Under local $p$-power growth with $p>2$ and sufficiently strong damping, we also construct a fixed loss whose exact CD orbit satisfies $F(x_n)-F^*\\sim Dn^{-2p/(p-2)}$, $D>0$, establishing the sharpness of the known convergence rate. Both main results are formally verified in Lean 4.", "title": "No Universal Remainder Rate for Chambolle-Dossal Acceleration", "updated": "2026-09-25", "url": "https://arxiv.org/abs/2609.31557" }, { "age_days": 0, "authors": [ "Michael Rothgang" ], "content_date": "2026-09-29", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:ed72f1faaeee", "kind": "github_update", "label": "adjacent", "matched_signals": [ "general_ai_math_reasoning" ], "published": "2026-09-29", "repo": "leanprover-community/mathlib4", "score": 2.2, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: fix(Archive/IMO/Imo2019Q4): fix flexible simp hiding as norm_num (#42868)", "updated": "2026-09-29", "url": "https://github.com/leanprover-community/mathlib4/commit/ed72f1faaeee9bea5805932ae309af92264cd8d1" }, { "age_days": 1, "arxiv_id": "2609.35632", "authors": [ "Tejasvi Singh Tomar" ], "content_date": "2026-09-28", "freshness": "fresh", "id": "arxiv:2609.35632", "kind": "paper", "label": "adjacent", "matched_signals": [ "general_ai_math_reasoning" ], "published": "2026-09-28", "score": 1.9, "source": "arxiv-ai4math-core", "summary": "We prove that for every choice of four positive masses and every cyclic ordering of the bodies there is exactly one strictly convex planar central configuration with that ordering, up to similarity. This answers Problem 10 in the list of Albouy, Cabral and Santos, which Santoprete calls the Simó-Yoccoz conjecture. The main step is a uniform lower bound for the Hessian at convex central configurations: it is at least one quarter of its radial part, which vanishes only on translations and rotations. Dziobek's relations turn the indefinite part of the Hessian into a negative multiple of a square, and a Cauchy-Schwarz argument bounds this term by the radial part times the trace of an explicit $2\\times 2$ matrix in which the masses do not appear. We prove that this trace is less than 3/4 by interval arithmetic on the three-dimensional set of normalized convex central configurations, in the coordinates of Corbera, Cors and Roberts. Uniqueness then follows by a covering argument from the case of four equal masses. The computation has been repeated with a second, independently written program. The Hessian bound and the uniqueness theorem, including the computation, have been formalized in the Lean proof assistant and checked by its kernel, using only the standard axioms. As consequences, every degenerate four-body central configuration is concave, the convex central configuration depends analytically on the masses, and the known symmetry theorems for kites, isosceles trapezoids and rhombi follow in a few lines. The analytic dependence and the symmetry theorems are also formalized.", "title": "Uniqueness of four-body convex central configurations for all masses", "updated": "2026-09-28", "url": "https://arxiv.org/abs/2609.35632" }, { "age_days": 1, "arxiv_id": "2609.34960", "authors": [ "Feiming Wang", "Daibo Li", "Kun Yuan" ], "content_date": "2026-09-28", "freshness": "fresh", "id": "arxiv:2609.34960", "kind": "paper", "label": "adjacent", "matched_signals": [ "tool_use_agents" ], "published": "2026-09-28", "score": 1.9, "source": "arxiv-ai4math-core", "summary": "Formalizing research-level stochastic optimization in Lean requires both an algorithm model and domain theory connecting foundational libraries to convergence proofs. Revising a model to restore provability can change the mathematical claim. We introduce ProofLoom, a fully automated LLM-agent system for Proof-Obligation-Driven Theory Construction. Given a published algorithm, target theorem, and source proof, ProofLoom autonomously constructs the Lean model and supporting theory. Open proof obligations drive the development of definitions, interfaces, lemmas, and proof plans. Signature contracts record evidence and obligations for model revisions; an independent Judge rejects unsupported assumptions and weakened conclusions. Planner expands the published argument into intermediate claims, and Audit checks whether the Lean proof follows it. Across tasks, SOptLib accumulates verified mathematics and construction experience: reusable results are extracted, generalized, and verified, while modeling decisions and failed proof routes are recorded. Later tasks retrieve these results and records and contribute new developments, forming a cycle of construction, accumulation, and reuse. On fifteen textbook and research-paper tasks, ProofLoom obtains mean human ratings of 6.3/7 and 6.4/7, compared with 4.9/7 and 5.0/7 for the strongest of six baselines. Across 33 developments, it produces 490,693 lines of algorithm-local Lean code with no sorry. The formalizations also expose 28 incorrect formulas, proof gaps, and algorithm-analysis mismatches in published sources across 22 developments, each with checked evidence.", "title": "ProofLoom: Proof-Obligation-Driven Theory Construction for Autoformalizing Research-Level Stochastic Optimization", "updated": "2026-09-28", "url": "https://arxiv.org/abs/2609.34960" }, { "age_days": 2, "arxiv_id": "2609.33842", "authors": [ "Ziyang Qin", "Yuan Liao", "Ayush Khaitan", "Bennett Chow" ], "content_date": "2026-09-27", "freshness": "fresh", "id": "arxiv:2609.33842", "kind": "paper", "label": "adjacent", "matched_signals": [ "general_ai_math_reasoning" ], "published": "2026-09-27", "score": 1.9, "source": "arxiv-ai4math-core", "summary": "We formalize the smooth three-dimensional Poincaré conjecture, together with the Moise smoothing theorem, yielding the topological three-dimensional Poincaré conjecture. The smooth proof follows the Hamilton--Perelman route through Ricci flow with surgery and finite-time extinction.", "title": "A Lean Formalization of the Hamilton--Perelman Proof of the Three-Dimensional Poincaré Conjecture", "updated": "2026-09-27", "url": "https://arxiv.org/abs/2609.33842" }, { "age_days": 0, "authors": [ "Leonardo de Moura" ], "content_date": "2026-09-29", "freshness": "fresh", "id": "github:leanprover/lean4:a332ad100000", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-29", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: test: re-enable the `grind_bitvec2` benchmark (#15408)", "updated": "2026-09-29", "url": "https://github.com/leanprover/lean4/commit/a332ad1000004723f3fedf2990e775762dbe6ecd" }, { "age_days": 0, "authors": [ "Hongyu Ouyang" ], "content_date": "2026-09-29", "freshness": "fresh", "id": "github:leanprover/lean4:62e2afd1a5a7", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-29", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: perf: memoize the search path root of each package in `findOLean` (#15291)", "updated": "2026-09-29", "url": "https://github.com/leanprover/lean4/commit/62e2afd1a5a79562d37cabc44fb1619b7e0556d6" }, { "age_days": 0, "authors": [ "Leonardo de Moura" ], "content_date": "2026-09-29", "freshness": "fresh", "id": "github:leanprover/lean4:633a200e4aac", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-29", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: fix: sharing violation in the `Sym.Arith` normalizer (#15405)", "updated": "2026-09-29", "url": "https://github.com/leanprover/lean4/commit/633a200e4aac7eba2b6f85719eb6e3679a7a2320" }, { "age_days": 0, "authors": [ "Sebastian Graf" ], "content_date": "2026-09-29", "freshness": "fresh", "id": "github:leanprover/lean4:c266b91ed310", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-29", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: fix: introduce `__`-prefixed binders as implementation details in `Sym.intros` (#15322)", "updated": "2026-09-29", "url": "https://github.com/leanprover/lean4/commit/c266b91ed31065c5ebfa6000329f1f1d9a6e35fc" }, { "age_days": 0, "authors": [ "Henrik Böving" ], "content_date": "2026-09-29", "freshness": "fresh", "id": "github:leanprover/lean4:6dae63ba38d3", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-29", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: fix: avoid sharing bug in bv_decide (#15403)", "updated": "2026-09-29", "url": "https://github.com/leanprover/lean4/commit/6dae63ba38d3c288288e34cb3dfda27c1a2c79a9" }, { "age_days": 0, "authors": [ "Henrik Böving" ], "content_date": "2026-09-29", "freshness": "fresh", "id": "github:leanprover/lean4:089d938f77b5", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-29", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: feat: make codegen for ctorIdx O(1) (#15407)", "updated": "2026-09-29", "url": "https://github.com/leanprover/lean4/commit/089d938f77b5249b8eaf945de42f533c5e617313" }, { "age_days": 0, "authors": [ "Sebastian Graf" ], "content_date": "2026-09-29", "freshness": "fresh", "id": "github:leanprover/lean4:67a862927484", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-29", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: feat: lattice lemmas and `PredTrans` rules for `do` notation (#15387)", "updated": "2026-09-29", "url": "https://github.com/leanprover/lean4/commit/67a8629274847c29155324086f6c0f49ba20ec8d" }, { "age_days": 0, "authors": [ "Leonardo de Moura" ], "content_date": "2026-09-29", "freshness": "fresh", "id": "github:leanprover/lean4:58d47647ae50", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-29", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: feat: control-flow reduction in the `Sym.simp`-based `grind` normalizer (#15409)", "updated": "2026-09-29", "url": "https://github.com/leanprover/lean4/commit/58d47647ae5036505c6452f36e2ea871121c67a9" }, { "age_days": 0, "authors": [ "Henrik Böving" ], "content_date": "2026-09-29", "freshness": "fresh", "id": "github:leanprover/lean4:4ac30d495e49", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-29", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: feat: add support for incremental (partially verified) bitblasting (#15404)", "updated": "2026-09-29", "url": "https://github.com/leanprover/lean4/commit/4ac30d495e4917a6de26ceae7163abc440938a6a" }, { "age_days": 0, "authors": [ "Sebastian Ullrich" ], "content_date": "2026-09-29", "freshness": "fresh", "id": "github:leanprover/lean4:8d338845e5de", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-29", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: chore: disable broken `grind_bitvec2` benchmark temporarily (#15397)", "updated": "2026-09-29", "url": "https://github.com/leanprover/lean4/commit/8d338845e5de0a87977768f0d0bd7536620d3245" }, { "age_days": 0, "authors": [ "JX-Mo" ], "content_date": "2026-09-29", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:696364883ced", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-29", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: refactor(RepresentationTheory): record the concrete binary biproduct (#41678)", "updated": "2026-09-29", "url": "https://github.com/leanprover-community/mathlib4/commit/696364883ced7a851c9365688be38f405f352f90" }, { "age_days": 0, "authors": [ "Sebastien Gouezel" ], "content_date": "2026-09-29", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:cba67dece737", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-29", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: feat: introduce an IsNormableSpace class (#42983)", "updated": "2026-09-29", "url": "https://github.com/leanprover-community/mathlib4/commit/cba67dece737ecab8cd5a3e96c2bfa81b0890111" }, { "age_days": 0, "authors": [ "Snir Broshi" ], "content_date": "2026-09-29", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:0f564b73d28d", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-29", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: feat(Topology/GDelta/Basic): a set is meagre iff it's a countable union of nowhere dense sets (#43666)", "updated": "2026-09-29", "url": "https://github.com/leanprover-community/mathlib4/commit/0f564b73d28debce6da438a1be67a7deb7bc7d49" }, { "age_days": 0, "authors": [ "Ruiz Chen" ], "content_date": "2026-09-29", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:0a55f542b23e", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-29", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: feat(Topology/Covering): fundamental group of the circle is ℤ (#41856)", "updated": "2026-09-29", "url": "https://github.com/leanprover-community/mathlib4/commit/0a55f542b23e6c402abb5a77a78a4b53cb316510" }, { "age_days": 0, "authors": [ "Snir Broshi" ], "content_date": "2026-09-29", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:cdc909433e1a", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-29", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: feat(Topology/Baire/Lemmas): a space is Baire iff countable unions of closed nowhere dense sets have empty interior (#43665)", "updated": "2026-09-29", "url": "https://github.com/leanprover-community/mathlib4/commit/cdc909433e1a32fdad86ec012629ccc3a7a9d75d" }, { "age_days": 0, "authors": [ "Vlad Tsyrklevich" ], "content_date": "2026-09-29", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:d88ba3d5b96c", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-29", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: feat(SimpleGraph): the cycle graph and complete graph are Hamiltonian (#34799)", "updated": "2026-09-29", "url": "https://github.com/leanprover-community/mathlib4/commit/d88ba3d5b96cd6c389ac80f21a1fc07e3c346ad3" }, { "age_days": 0, "authors": [ "Xavier Roblot" ], "content_date": "2026-09-29", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:1cf8efbf580f", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-29", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: feat(RingTheory/RamificationInertia): ramification index and inertia degree of the zero ideal (#44281)", "updated": "2026-09-29", "url": "https://github.com/leanprover-community/mathlib4/commit/1cf8efbf580fbeed2d989c0d96ee21267e85252e" }, { "age_days": 0, "authors": [ "vaca22" ], "content_date": "2026-09-29", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:bf620d19e942", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-29", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: feat(RingTheory/ClassGroup): add ClassGroup.map and define mulEquiv via it (#41732)", "updated": "2026-09-29", "url": "https://github.com/leanprover-community/mathlib4/commit/bf620d19e9426227ad155d1a2624ef3ba161cd14" }, { "age_days": 0, "authors": [ "JX-Mo" ], "content_date": "2026-09-29", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:5e0436985028", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-29", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: feat(RepresentationTheory): add two simp lemmas (#43649)", "updated": "2026-09-29", "url": "https://github.com/leanprover-community/mathlib4/commit/5e043698502894993991089b2dd893867c0db5c2" }, { "age_days": 0, "authors": [ "Evgenia Karunus" ], "content_date": "2026-09-29", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:17be17fef3d8", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-29", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: feat(MeasureTheory/Integral/Bochner/Basic): add enorm_integral_norm_eq_lintegral_enorm (#43451)", "updated": "2026-09-29", "url": "https://github.com/leanprover-community/mathlib4/commit/17be17fef3d8e386bef090fcc37e9857825fc459" }, { "age_days": 0, "authors": [ "√2" ], "content_date": "2026-09-29", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:efe587b509bc", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-29", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: feat(LinearAlgebra/Matrix): the transpose of a totally nonnegative matrix is totally nonnegative (#43004)", "updated": "2026-09-29", "url": "https://github.com/leanprover-community/mathlib4/commit/efe587b509bc9bf831550fd92c66e81a3ed0ca8e" }, { "age_days": 0, "authors": [ "Snir Broshi" ], "content_date": "2026-09-29", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:e9930a2bc844", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-29", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: feat(Combinatorics/SimpleGraph/Paths): more path/circuit/cycle lemmas (#41433)", "updated": "2026-09-29", "url": "https://github.com/leanprover-community/mathlib4/commit/e9930a2bc844f15fc7ca8ff10be0abf5e6bb1ecb" }, { "age_days": 0, "authors": [ "Dagur Asgeirsson" ], "content_date": "2026-09-29", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:58ceb233cd2d", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-29", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: feat(CategoryTheory): the bicategory of profunctors (#39619)", "updated": "2026-09-29", "url": "https://github.com/leanprover-community/mathlib4/commit/58ceb233cd2d0eb3ec48572ea244f8673faf160c" }, { "age_days": 0, "authors": [ "Jireh Loreaux" ], "content_date": "2026-09-29", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:5b655492878e", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-29", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: feat(Analysis/Calculus): derivatives of parametric circle integrals (#42968)", "updated": "2026-09-29", "url": "https://github.com/leanprover-community/mathlib4/commit/5b655492878ebeef603e621859dbf83cd91104f1" }, { "age_days": 0, "authors": [ "Weiyi Wang" ], "content_date": "2026-09-29", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:3b531c35df33", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-29", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: feat(Algebra/Pi): extending a Pi.(mul)Single (#44258)", "updated": "2026-09-29", "url": 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"2026-09-29", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: feat(Algebra/Order/BigOperators/Group/Finset): add `Finset.map_prod_le_prod` (#44154)", "updated": "2026-09-29", "url": "https://github.com/leanprover-community/mathlib4/commit/e83c642143ddbbda27b6b4647957f0328b4a73a0" }, { "age_days": 0, "authors": [ "Snir Broshi" ], "content_date": "2026-09-29", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:0f64d30a9a45", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-29", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: doc(Topology/Algebra/InfiniteSum/UniformOn): fix docstring typo (#44163)", "updated": "2026-09-29", "url": 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"repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: chore: rename remaining `not_mem` declarations to `notMem` (#44296)", "updated": "2026-09-29", "url": "https://github.com/leanprover-community/mathlib4/commit/1b5006ca00888f8aaafa8939b28787d423e44bba" }, { "age_days": 0, "authors": [ "Sebastien Gouezel" ], "content_date": "2026-09-29", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:fd38abae3e47", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-29", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: chore: fix an implicit reducible diamond in nilpotent groups (#44126)", "updated": "2026-09-29", "url": "https://github.com/leanprover-community/mathlib4/commit/fd38abae3e470fda2d010ebe2dddc9fd28d242b3" }, { "age_days": 0, "authors": [ "Moritz Doll" ], "content_date": "2026-09-29", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:c73361e06476", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-29", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: chore: add `prodComm_trans_prodComm` for various cases (#43923)", "updated": "2026-09-29", "url": "https://github.com/leanprover-community/mathlib4/commit/c73361e06476dc2efdd449ebe9fd8c06d485e05c" }, { "age_days": 0, "authors": [ "JX-Mo" ], "content_date": "2026-09-29", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:4349c598a16a", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-29", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: chore(RepresentationTheory/Equiv): generalize to Type* (#44073)", "updated": "2026-09-29", "url": "https://github.com/leanprover-community/mathlib4/commit/4349c598a16a549b487865f3e8575c5702205746" }, { "age_days": 0, "authors": [ "Bhavik Mehta" ], "content_date": "2026-09-29", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:d0f8fbcbfcab", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-29", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: chore(Order/BooleanAlgebra/Basic): tag sdiff disjoint lemmas with grind (#43794)", "updated": "2026-09-29", "url": "https://github.com/leanprover-community/mathlib4/commit/d0f8fbcbfcab26855f0184049a51c3731191a47f" }, { "age_days": 0, "authors": [ "Marcelo Lynch" ], "content_date": "2026-09-29", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:ae6f1eeba630", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-29", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: chore(MathlibTest): temporarily disable two slow library suggestion tests (#44318)", "updated": "2026-09-29", "url": "https://github.com/leanprover-community/mathlib4/commit/ae6f1eeba63041889d82d9ed777b263435ca3424" }, { "age_days": 0, "authors": [ "Bhavik Mehta" ], "content_date": "2026-09-29", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:bc1fe1ff24de", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-29", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: chore(Data/Finset/BooleanAlgebra): tag univ_nonempty/nontrivial/eq_empty_iff with grind (#43808)", "updated": "2026-09-29", "url": "https://github.com/leanprover-community/mathlib4/commit/bc1fe1ff24de7683bfd563ac1bd0a4eded60c9b7" }, { "age_days": 0, "authors": [ "Yury G. 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leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: chore(Analysis/Normed/Algebra/UnitizationL1): add missing `simp` attributes (#44085)", "updated": "2026-09-29", "url": "https://github.com/leanprover-community/mathlib4/commit/ed07dcc1fb88123dc8ce507edf2fdfa72861880d" }, { "age_days": 0, "authors": [ "TJHeeringa" ], "content_date": "2026-09-29", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:0493595ecd5d", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-29", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: chore(*): rename (anti)lipschitz to (anti)lipschitzWith (#44262)", "updated": "2026-09-29", "url": "https://github.com/leanprover-community/mathlib4/commit/0493595ecd5da8ef14742febab99728f98e948e8" }, { "age_days": 1, "authors": [ "Leonardo de Moura" ], "content_date": "2026-09-28", "freshness": "fresh", "id": "github:leanprover/lean4:8dd5d448291e", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-28", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: fix: sharing violation in `Sym.dsimp` binder telescopes (#15389)", "updated": "2026-09-28", "url": "https://github.com/leanprover/lean4/commit/8dd5d448291eb0ca0180c8ae1f9d248b2c8a2a7b" }, { "age_days": 1, "authors": [ "Leonardo de Moura" ], "content_date": "2026-09-28", "freshness": "fresh", "id": "github:leanprover/lean4:076b0b706600", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-28", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: fix: canonicalize cast prefixes in the `Sym.Arith` normalizer (#15381)", "updated": "2026-09-28", "url": "https://github.com/leanprover/lean4/commit/076b0b706600d1b93ee11be105227fc42cd9f501" }, { "age_days": 1, "authors": [ "Leonardo de Moura" ], "content_date": "2026-09-28", "freshness": "fresh", "id": "github:leanprover/lean4:16df84b16f8e", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-28", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: fix: `grind` normalizer never pushed negation into implications and universal quantifiers (#15378)", "updated": "2026-09-28", "url": "https://github.com/leanprover/lean4/commit/16df84b16f8e90e4f1a16b97d6d144aee48fdf22" }, { "age_days": 1, "authors": [ "Leonardo de Moura" ], "content_date": "2026-09-28", "freshness": "fresh", "id": "github:leanprover/lean4:fabf217ac972", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-28", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: feat: decide trivially unsatisfiable and valid `Nat` relations in `Sym.simp` (#15380)", "updated": "2026-09-28", "url": "https://github.com/leanprover/lean4/commit/fabf217ac9727b42d40765fbcdad555d35805a87" }, { "age_days": 1, "authors": [ "Leonardo de Moura" ], "content_date": "2026-09-28", "freshness": "fresh", "id": "github:leanprover/lean4:b45f25031de9", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-28", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: feat: add `lhsOnly` mode to the `Sym.Arith` relation normalizer (#15376)", "updated": "2026-09-28", "url": "https://github.com/leanprover/lean4/commit/b45f25031de905288cc2ede6c5898cdc92b88517" }, { "age_days": 1, "authors": [ "Leonardo de Moura" ], 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divisibility in the `Sym.Arith` normalizer (#15382)", "updated": "2026-09-28", "url": "https://github.com/leanprover/lean4/commit/9f66b6cfc9c057b20bc545f4541a423ab342b161" }, { "age_days": 1, "authors": [ "Leonardo de Moura" ], "content_date": "2026-09-28", "freshness": "fresh", "id": "github:leanprover/lean4:c46507e3ea83", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-28", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: doc: explain the stop condition of the `[grind hom]` rewriter (#15379)", "updated": "2026-09-28", "url": "https://github.com/leanprover/lean4/commit/c46507e3ea83834737508bc4828199be7720280a" }, { "age_days": 1, "authors": [ "Lean stage0 autoupdater" ], "content_date": "2026-09-28", "freshness": "fresh", "id": "github:leanprover/lean4:56b48d816210", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-28", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: chore: update stage0", "updated": "2026-09-28", "url": "https://github.com/leanprover/lean4/commit/56b48d816210bc0c1b646fc76b83a43ce43d9607" }, { "age_days": 1, "authors": [ "Leonardo de Moura" ], "content_date": "2026-09-28", "freshness": "fresh", "id": "github:leanprover/lean4:86941be18ef3", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-28", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: chore: drop the E-matching copies of `[grind hom]` image rules (#15377)", "updated": "2026-09-28", "url": "https://github.com/leanprover/lean4/commit/86941be18ef336b765f61e4faca3207342f70b96" }, { "age_days": 2, "authors": [ "Sebastian Ullrich" ], "content_date": "2026-09-27", "freshness": "fresh", "id": "github:leanprover/lean4:6d9787cda00f", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-27", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: test: keep `Init` out of the `lake check` test projects (#15326)", "updated": "2026-09-27", "url": "https://github.com/leanprover/lean4/commit/6d9787cda00f5ae1c0aeca349d3072716fa7524a" }, { "age_days": 2, "authors": [ "Leonardo de Moura" ], "content_date": "2026-09-27", "freshness": "fresh", "id": "github:leanprover/lean4:aee6d42280de", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-27", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: fix: normalize `-0 : Int` to `0` (#15363)", "updated": "2026-09-27", "url": "https://github.com/leanprover/lean4/commit/aee6d42280defdf84888844e1ac16f3fc804ee7c" }, { "age_days": 2, "authors": [ "Leonardo de Moura" ], "content_date": "2026-09-27", "freshness": "fresh", "id": "github:leanprover/lean4:059fb8ae4da6", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-27", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: fix: make `grind` understand the `Bool`-valued signed comparisons `BitVec.slt` and `BitVec.sle` (#15370)", "updated": "2026-09-27", "url": "https://github.com/leanprover/lean4/commit/059fb8ae4da638fa0ecd806e9cbbdf1676c702ef" }, { "age_days": 2, "authors": [ "Leonardo de Moura" ], "content_date": "2026-09-27", "freshness": "fresh", "id": "github:leanprover/lean4:e39e715e5018", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-27", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: fix: `Sym.DSimp.zetaDelta` should unfold let-bound variables in head position (#15362)", "updated": "2026-09-27", "url": "https://github.com/leanprover/lean4/commit/e39e715e5018b4b923ea9bf59271cdb45fc745bd" }, { "age_days": 2, "authors": [ "Leonardo de Moura" ], "content_date": "2026-09-27", "freshness": "fresh", "id": "github:leanprover/lean4:1e92f02237ce", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-27", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: feat: support the unsigned value of `sshiftRight` in `grind`'s bit-vector homomorphism rules (#15365)", "updated": "2026-09-27", "url": "https://github.com/leanprover/lean4/commit/1e92f02237ce2f4910cfe5ed9c6b6f8dbf15575e" }, { "age_days": 2, "authors": [ "Leonardo de Moura" ], "content_date": "2026-09-27", "freshness": "fresh", "id": "github:leanprover/lean4:02c16a9da34a", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-27", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: feat: support signed `~~~` in `grind`'s homomorphism rules (#15369)", "updated": "2026-09-27", "url": "https://github.com/leanprover/lean4/commit/02c16a9da34a44ac6371c7e7d28d3799197bb103" }, { "age_days": 2, "authors": [ "Leonardo de Moura" ], "content_date": "2026-09-27", "freshness": "fresh", "id": "github:leanprover/lean4:47f4cf80fe10", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-27", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: feat: support signed `<<<` by a variable amount in `grind`'s bit-vector homomorphism rules (#15367)", "updated": "2026-09-27", "url": "https://github.com/leanprover/lean4/commit/47f4cf80fe1088c42b53fd7b647d040a721321e2" }, { "age_days": 2, "authors": [ "Leonardo de Moura" ], "content_date": "2026-09-27", "freshness": "fresh", "id": "github:leanprover/lean4:32ac62681efe", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-27", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: feat: support `rotateLeft` and `rotateRight` in `grind`'s bit-vector homomorphism rules (#15364)", "updated": "2026-09-27", "url": "https://github.com/leanprover/lean4/commit/32ac62681efed8ec10fd1eabea3f11c5a94762ee" }, { "age_days": 2, "authors": [ "Leonardo de Moura" ], "content_date": "2026-09-27", "freshness": "fresh", "id": "github:leanprover/lean4:368dc838819f", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-27", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: feat: make `simp [f]` and `dsimp [f]` in `sym =>` mode behave like the default simplifier (#15366)", "updated": "2026-09-27", "url": "https://github.com/leanprover/lean4/commit/368dc838819f6a8e6537e5c854312493dde9de62" }, { "age_days": 2, "authors": [ "Leonardo de Moura" ], "content_date": "2026-09-27", "freshness": "fresh", "id": "github:leanprover/lean4:ddbe8182954b", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-27", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: feat: add `grind_norm` for comparing the `simp`- and `Sym.simp`-based `grind` normalizers (#15375)", "updated": "2026-09-27", "url": "https://github.com/leanprover/lean4/commit/ddbe8182954ba0b92011f28c28e5b89f25404790" }, { "age_days": 1, "arxiv_id": "2609.34889", "authors": [ "Isabela Drǎmnesc", "Tudor Jebelean", "Sorin Stratulat" ], "content_date": "2026-09-28", "freshness": "fresh", "id": "arxiv:2609.34889", "kind": "paper", "label": "negative", "matched_signals": [], "published": "2026-09-28", "score": 0.5, "source": "arxiv-ai4math-core", "summary": "This is a case study on a specific version of the Patience Sort algorithm in which we illustrate the evolution of it from an intuitive but inefficient nested recursion into a more complex but more efficient tail recursion, together with its formal implementation and certification in Theorema and Rocq (formerly Coq). We identify some general principles of algorithm transformation, and we develop the necessary background theory and proof methods. As a significant distinctive aspect, the approach in Theorema uses multisets, which simplifies the whole process and makes it more intuitive. The certification process reveals significant differences between the Theorema and Rocq approaches, about which we provide a comparative analysis with respect to algorithm definition, proof development, and proof effort. This analysis offers insights into how algorithm design influences the complexity and structure of formal proofs and demonstrates how non-trivial algorithms can be effectively verified across different formal frameworks.", "title": "Certification of Bilateral Patience Sort in Theorema and Rocq", "updated": "2026-09-28", "url": "https://arxiv.org/abs/2609.34889" }, { "age_days": 1, "arxiv_id": "2609.35719", "authors": [ "Sergey Gusev", "David E. Bernal Neira" ], "content_date": "2026-09-28", "freshness": "fresh", "id": "arxiv:2609.35719", "kind": "paper", "label": "negative", "matched_signals": [], "published": "2026-09-28", "score": 0.5, "source": "arxiv-ai4math-core", "summary": "Artificial intelligence (AI) agents, language models connected to tools and run in a loop, can now carry out long, multi-step tasks with little supervision. We gave swarms of off-the-shelf coding agents a short statement of scope, from a narrow topic to a whole field, access to the literature and to computing tools, and one standing instruction: make real, correct, useful progress, and do not stop. We supplied no scientific ideas. Within weeks, the agents produced a large body of research notes, paper-length drafts, and formal proofs in five areas of optimization theory and physical science, and proposed untested laboratory experiments in a sixth. We do not claim that all of it is correct or new, but it is not noise: in what we have checked so far, we found no major scientific error, and several results are proved in a proof assistant. The agents produced results faster than we could review them; we estimate that a full review would take us months. Together with two widely discussed 2026 results in mathematics obtained with swarms, our runs suggest that agents can already do a large part of routine theoretical research, at least in areas that we experimented with. This raises questions we cannot yet answer: how to trust results when review, not production, is the scarce resource; what credit and publication counts mean when the human input is a prompt, and why institutions would pay researchers rather than buy computing time; and how people can learn a field, add to what agents do, and stay in control of research they cannot keep up with. Research institutions are not ready: models improve faster than institutions change, so they should decide now how to respond as capabilities increase. We offer tentative positions, release the agents' unedited output as of 25 September 2026, and invite readers to repeat the experiment in their own fields.", "title": "AI Agent Swarms as Researchers: Progress, Challenges, and Open Questions", "updated": "2026-09-28", "url": "https://arxiv.org/abs/2609.35719" }, { "age_days": 1, "arxiv_id": "2609.35229", "authors": [ "Christian Hägg", "Anders Mörtberg" ], "content_date": "2026-09-28", "freshness": "fresh", "id": "arxiv:2609.35229", "kind": "paper", "label": "negative", "matched_signals": [], "published": "2026-09-28", "score": 0.5, "source": "arxiv-ai4math-core", "summary": "We settle in the negative the longstanding question whether every Bézout domain is an elementary divisor domain by constructing a Bézout domain over which an explicit $2\\times 2$ matrix has no Smith normal form. The obstruction is topological and is detected by the Möbius line bundle.", "title": "A Bézout domain that is not an elementary divisor domain", "updated": "2026-09-28", "url": "https://arxiv.org/abs/2609.35229" }, { "age_days": 2, "arxiv_id": "2609.33421", "authors": [ "Benoit Cloitre", "Haobo Ma", "Wenlin Zhang" ], "content_date": "2026-09-27", "freshness": "fresh", "id": "arxiv:2609.33421", "kind": "paper", "label": "negative", "matched_signals": [], "published": "2026-09-27", "score": 0.5, "source": "arxiv-ai4math-core", "summary": "We study the sequence $a(0)=0$, $a(1)=1$ and $a(n)=n-a(n-a(n-a(n-1)))$ for $n\\ge 2$, listed as A076502 in the On-Line Encyclopedia of Integer Sequences. We identify $a(n)$ as a two-position shift in the greedy Padovan numeration system, with a finite-state correction. The proof constructs an addition automaton from an exact integer-carry invariant and certifies its completeness by finite-language inclusion; a synchronized automaton then verifies the nested recurrence. We establish bounded discrepancy from the line of slope $c$, where $c^3-c^2+2c-1=0$, and show that the exact set of offsets from $\\lfloor cn\\rfloor$ is $\\{-1,0,1,2\\}$. We construct an explicit 26-letter non-erasing morphic presentation of the first-difference word, prove that its least balance constant is 4, and give an effective procedure for enclosing the global discrepancy extrema to arbitrary accuracy. We formalize the recurrence identification, six-decimal discrepancy bound, exact offset set, concrete morphic identity, least balance constant, and an effective extrema algorithm in Lean. Separate exact computations refine the numerical enclosures.", "title": "A Padovan-automatic description of a nested recurrence", "updated": "2026-09-27", "url": "https://arxiv.org/abs/2609.33421" }, { "age_days": 3, "arxiv_id": "2609.32552", "authors": [ "Antonio Attili", "Ludovico Nista", "Tommaso Baffetti", "Geveen Arumapperuma", "Sofiane Al Kassar", "Lukas Berger", "Christoph D. K. Schumann", "Temistocle Grenga", "Heinz Pitsch" ], "content_date": "2026-09-26", "freshness": "fresh", "id": "arxiv:2609.32552", "kind": "paper", "label": "negative", "matched_signals": [], "published": "2026-09-26", "score": 0.5, "source": "arxiv-ai4math-core", "summary": "Convolutional Neural Networks (CNNs) are used as analytical tools to investigate the relationship between the progress variable field and the local chemical source term in lean premixed hydrogen flames. Rather than employing machine learning for modelling, CNNs are leveraged to analyse the physical information contained in spatially resolved fields from direct numerical simulations. CNNs are particularly well suited for this task because they exploit spatial correlations and multi-scale structures, allowing an assessment of how spatial organisation influences the source term. The analysis demonstrates that the progress variable based on water, $C_{\\rm H_2O}$, contains all the information required to accurately parametrise the chemical source term whereas $C_{\\rm H_2}$ alone does not. The inclusion of the mixture fraction $Z$ improves the accuracy of the latter but provides no significant additional information to the CNN when the spatial field of $C_{\\rm H2O}$ is used as input. The same behaviour is observed in both a laminar thermodiffusively unstable flame and a turbulent slot-jet hydrogen flame, indicating that $C_{\\rm H_2O}$ is a robust single variable for parametrisation. A complementary scale analysis in the laminar case shows that spatial features extending over at least two laminar flame thicknesses are required to reconstruct the source term accurately, thereby identifying the characteristic scale in the progress variable field that carries this information. These results demonstrate how machine learning can uncover physical dependencies that remain hidden to classical statistical analyses.", "title": "Uncovering flame physics with machine learning: application to the reaction rate in hydrogen flames", "updated": "2026-09-26", "url": "https://arxiv.org/abs/2609.32552" }, { "age_days": 3, "arxiv_id": "2609.32950", "authors": [ "Nikita Lebedev" ], "content_date": "2026-09-26", "freshness": "fresh", "id": "arxiv:2609.32950", "kind": "paper", "label": "negative", "matched_signals": [], "published": "2026-09-26", "score": 0.5, "source": "arxiv-ai4math-core", "summary": "Santos asked when intersections of dilated Hadamard simplices with cubes are integral, smooth, or normal, in a prescribed affine lattice. We construct a nonintegral example in dimension eleven and prove that no smaller-dimensional example exists. We characterize smoothness completely and show that every smooth member of this family is normal. An explicit example in dimension fifteen shows that integrality alone does not imply normality. For Sylvester simplices of order at least sixteen, we establish a sharp uniform integrality bound and construct counterexamples immediately below it. We also obtain sufficient normality bounds for general Hadamard simplices and stronger bounds for the Sylvester family. The proofs use integer decomposition for boxes with separated corner cuts and rounding under three signed slab constraints. All numbered results have formal counterparts verified in Lean.", "title": "Integrality, smoothness and normality bounds for cube-truncated Hadamard simplices", "updated": "2026-09-26", "url": "https://arxiv.org/abs/2609.32950" }, { "age_days": 3, "arxiv_id": "2609.32931", "authors": [ "Zhipeng Deng" ], "content_date": "2026-09-26", "freshness": "fresh", "id": "arxiv:2609.32931", "kind": "paper", "label": "negative", "matched_signals": [], "published": "2026-09-26", "score": 0.5, "source": "arxiv-ai4math-core", "summary": "Lebesgue's universal covering problem asks for the minimum area convex planar region capable of containing a congruent copy of every planar set of diameter at most one. In this paper, we develop an exact dual formulation of this problem through a minimum diameter analogue of Bellman's lost-in-a-forest problem. For a compact convex forest $F$, we define the critical escape diameter $D(F)$ as the minimum diameter of a closed path whose trace cannot be placed, under any rigid motion, entirely in the interior of $F$. We prove attainment of this minimum and establish a diameter cover/escape duality showing that the normalized body $D(F)^{-1}F$ is a Lebesgue universal cover. Consequently, the Lebesgue universal covering constant admits the exact representation \\[ \\mathcal L=\\inf_F\\frac{\\operatorname{Area}(F)}{D(F)^2}, \\] where the infimum ranges over compact convex planar bodies with nonempty interior. By reversing the rigid motion, we further characterize escape as intersection of a fixed path with every oppositely transformed boundary of $F$, and derive equivalent continuous curve, convex body, and support function optimization formulations. To make the infinite dimensional problem computationally tractable while retaining rigorous control of approximation error, we discretize the compact configuration space by an $η_m$-net and formulate the resulting problem as a minimum diameter traveling salesman problem with neighborhoods; for polygonal forests, an exact mixed-integer second-order cone formulation is obtained. We prove the quantitative certification, which yields convergent, rigorously certified universal cover bounds. The framework replaces finite tests of prescribed constant width shapes by a unified optimization over the full configuration space and extends naturally to other congruent and translative universal cover problems.", "title": "Closed escape path of smallest diameter in forest: dual formulation of Lebesgue's universal covering problem", "updated": "2026-09-26", "url": "https://arxiv.org/abs/2609.32931" }, { "age_days": 4, "arxiv_id": "2609.31334", "authors": [ "Lars Becker", "María Inés de Frutos-Fernández", "Leo Diedering", "Floris van Doorn", "Sébastien Gouëzel", "Evgenia Karunus", "Edward van de Meent", "Pietro Monticone", "Jasper Mulder-Sohn", "Jim Portegies", "Joris Roos", "Michael Rothgang", "James Sundstrom", "Jeremy Tan" ], "content_date": "2026-09-25", "freshness": "fresh", "id": "arxiv:2609.31334", "kind": "paper", "label": "negative", "matched_signals": [], "published": "2026-09-25", "score": 0.5, "source": "arxiv-ai4math-core", "summary": "We present the formalization of Carleson's theorem in the proof assistant Lean. This paper describes the mathematical content, organization of the project, the blueprint, and the main design decisions behind the formalization. It is the result of a large collaborative effort, written and developed in public.", "title": "Formalizing Carleson's Theorem in Lean", "updated": "2026-09-25", "url": "https://arxiv.org/abs/2609.31334" } ], "lookback_days": 21, "schema": "ai4math-radar-run-v1", "timezone": "America/Los_Angeles" }