{ "counts": { "adjacent": 14, "core": 11, "errors": 0, "negative": 69, "total": 94 }, "date": "2026-06-20", "errors": [], "generated_at": "2026-06-20T15:36:58Z", "items": [ { "arxiv_id": "2606.19399", "authors": [ "Manish Acharya", "Zhenyu Liao", "Yueke Zhang", "Kevin Leach", "Yu Huang", "Yifan Zhang" ], "id": "arxiv:2606.19399", "kind": "paper", "label": "core", "matched_signals": [ "lean_formal_proving_agents", "verifier_guided_reasoning" ], "published": "2026-06-17", "score": 12.5, "source": "arxiv-ai4math-core", "summary": "LLM-based formal provers often collapse rich verifier signals (syntax errors, type mismatches, partial goal progress) into a binary pass/fail bit. We present VERITAS, a zero-shot framework that routes every verifier signal back into proof search through a two-phase protocol: Best-of-N sampling first, then a critic-guided MCTS pass that ingests Phase 1 failures as explicit negative examples. The protocol preserves every theorem solved by its own Phase 1 sweep, so Phase 2's additional solves are attributable to feedback-driven exploration. VERITAS reaches 40.6% on miniF2F (vs. an independently run Best-of-5 at 36.9%, Portfolio 26.2%) and 7.3% on VERITAS-CombiBench, a 55-theorem combinatorics benchmark we release on which Best-of-5 (1.8%) falls below Portfolio (3.6%), exposing that unguided sampling hurts when correct lemma names must be recovered iteratively from verifier feedback. Artifacts are available on GitHub.", "title": "VERITAS: Verifier-Guided Proof Search for Zero-Shot Formal Theorem Proving", "updated": "2026-06-17", "url": "https://arxiv.org/abs/2606.19399" }, { "arxiv_id": "2403.13310", "authors": [ "Guoxiong Gao", "Jiedong Jiang", "Haocheng Ju", "Bin Dong", "Zihan Qin" ], "id": "manual:semantic-scholar-mathlib4-search", "kind": "paper", "label": "core", "matched_signals": [ "mathlib_retrieval", "seed_author:Guoxiong Gao", "seed_author:Haocheng Ju", "seed_author:Jiedong Jiang" ], "observed_date": "2026-06-20", "published": "2025-02-04", "score": 9.5, "source": "semantic-scholar-library", "source_app": "semantic_scholar", "summary": "Semantic Scholar library seed for mathlib retrieval and premise search. This is infrastructure-level signal for theorem-proving agents.", "title": "A Semantic Search Engine for Mathlib4", "updated": "2026-06-20", "url": "http://arxiv.org/abs/2403.13310" }, { "arxiv_id": "", "authors": [], "id": "manual:scholar-inbox-distilling-lean-feedback", "kind": "paper", "label": "core", "matched_signals": [ "verifier_guided_reasoning" ], "observed_date": "2026-06-20", "published": "", "score": 9.0, "source": "scholar-inbox-manual", "source_app": "scholar_inbox", "summary": "Scholar Inbox positive seed for verifier-guided learning from Lean feedback. Useful for proof repair loops and training signal design.", "title": "Distilling LLM Feedback for Lean Theorem Proving", "updated": "2026-06-20", "url": "https://www.semanticscholar.org/search?q=Distilling%20LLM%20Feedback%20for%20Lean%20Theorem%20Proving" }, { "arxiv_id": "2606.20068", "authors": [ "Minsu Kim", "Se-Young Yun" ], "id": "arxiv:2606.20068", "kind": "paper", "label": "core", "matched_signals": [ "reasoning_rl_distillation", "verifier_guided_reasoning" ], "published": "2026-06-18", "score": 7.9, "source": "arxiv-ai4math-core", "summary": "While reinforcement learning from verifiable rewards (RLVR) typically has relied on a single binary verification signal, symbolic proof assistants in formal reasoning offer rich, fine-grained structured feedback. This gap between structured processes and unstructured rewards highlights the importance of feedback that is both dense and sound. In this work, we demonstrate that the Lean proof assistant itself can serve as a symbolic process oracle, supplying both outcome-level and fine-grained tactic-level verified feedback during training. Proof attempts are parsed into tactic sequences, and Lean's elaboration marks both locally sound steps and the earliest failing step, yielding dense, verifier-grounded credit signals rooted in type theory. We incorporate these structured rewards into a GRPO-style reinforcement learning objective with first-error propagation and first-token credit methods that balances outcome- and process-level advantages. Experiments with STP-Lean and DeepSeek-Prover-V1.5 show that tactic-level supervision outperforms outcome-only baselines in most settings, delivering improvements on benchmarks such as MiniF2F and ProofNet. Beyond empirical gains, our study highlights a broader perspective: symbolic proof assistants are not only verifiers at evaluation time, but can also act as process-level reward oracles during training. This opens a path toward reinforcement learning frameworks that combine the scalability of language models with the reliability of symbolic verification for formal reasoning.", "title": "Process-Verified Reinforcement Learning for Theorem Proving via Lean", "updated": "2026-06-18", "url": "https://arxiv.org/abs/2606.20068" }, { "arxiv_id": "2606.15972", "authors": [ "Ji Feng", "Zhouxing Shi" ], "id": "arxiv:2606.15972", "kind": "paper", "label": "core", "matched_signals": [ "autoformalization", "general_ai_math_reasoning", "lean_formal_proving_agents" ], "published": "2026-06-14", "score": 7.9, "source": "arxiv-ai4math-core", "summary": "With large language models (LLMs) increasingly applied to mathematical reasoning, formal proof assistants such as Lean can be leveraged to verify reasoning outputs with machine-checkable rigor, enabling use cases such as answer selection in test-time scaling with K sampled candidate answers. However, employing Lean requires that LLM outputs, originally in natural language, first be formalized. Existing Lean-based answer-selection work uses an autoformalization model to generate a formal statement in Lean for each candidate answer independently, incurring a significant computational cost. We propose BASE, a base-and-edit pipeline that formalizes a single base candidate per problem and derives the remaining K-1 statements by editing the answer expression in place. To facilitate this, we train a rewriter model LEANSCRIBE to localize the answer in the base formalization and generate a reusable edit function for the other K-1 candidates. BASE simultaneously improves selection accuracy and reduces formalization cost - a Pareto improvement that holds on all 12 (dataset, solver) configurations across four benchmarks and three solvers, cutting autoformalizer calls by about 5x at K=8, with the reduction expected to become larger as K grows. Code is available at https://github.com/ucr-rai/base-and-edit.", "title": "Formalize Once, Edit the Rest: Efficient Lean-Based Answer Selection for Math Reasoning", "updated": "2026-06-14", "url": "https://arxiv.org/abs/2606.15972" }, { "arxiv_id": "", "authors": [ "Jui-Hui Chung" ], "id": "manual:x-juihuichung-goedel-architect", "kind": "post", "label": "core", "matched_signals": [ "lean_formal_proving_agents", "seed_author:Jui-Hui Chung" ], "observed_date": "2026-06-20", "published": "2026-06-08", "score": 7.2, "source": "x-manual", "source_app": "x", "summary": "X post surfaced during tuning about Goedel-Architect, a Lean 4 formal theorem proving agent built around blueprint generation and refinement.", "title": "Jui-Hui Chung: Goedel-Architect launch thread", "updated": "2026-06-20", "url": "https://x.com/juihuichung/status/2064023094197707161" }, { "arxiv_id": "2606.18557", "authors": [ "Patrick Cooper", "Alvaro Velasquez" ], "id": "arxiv:2606.18557", "kind": "paper", "label": "core", "matched_signals": [ "lean_formal_proving_agents", "verifier_guided_reasoning" ], "published": "2026-06-17", "score": 6.5, "source": "arxiv-ai4math-core", "summary": "A rule-based logic solver resolves every instance in our benchmark in under 50 microseconds with 100% accuracy; the best frontier language model reaches 65% at best and drops to 23.5% under rendering-robust evaluation (worst case over four surface renderings). We introduce DeFAb (Defeasible Abduction Benchmark), a dataset and generation pipeline that converts four decades of publicly funded knowledge bases into formally grounded instances for defeasible abduction: constructing hypotheses that explain anomalies by overriding defaults while preserving unrelated expectations. Because every hypothesis must pass polynomial-time checks for valid derivation, conservativity, and minimality, DeFAb makes logical rigor the instrument for measuring creativity and theoretical reasoning, scoring the disciplined construction of theory revisions rather than fluent but theory-destroying prose. The pipeline pairs taxonomic hierarchies (OpenCyc, YAGO, Wikidata) with behavioral property graphs (ConceptNet, UMLS) to produce 372,648+ instances across 33.75M materialized rules from 18 sources, in three levels with polynomial-time verifiable gold standards. Four frontier models do not reliably internalize defeasible reasoning: rendering-robust Level 2 accuracy is 7.8-23.5%; chain-of-thought variance (~36 pp) exceeds any inter-model gap; and a matched contamination control isolates a +19.4 pp Level 3 gap. We further release DeFAb-Hard (a 235-instance Level 3 difficulty variant; best model 53.3% vs 100% symbolic) and CONJURE (a kernel-verified transformative-creativity variant of 560 Lean 4/Mathlib instances whose gold answers are definitions the proof kernel did not previously contain, judge-free verifier; a pilot finds zero novel concepts). The same verifier doubles as an exact reward for preference optimization (DPO, RLVR/GRPO). Released under MIT at https://huggingface.co/datasets/PatrickAllenCooper/DeFAb.", "title": "DeFAb: A Verifiable Benchmark for Defeasible Abduction in Foundation Models", "updated": "2026-06-17", "url": "https://arxiv.org/abs/2606.18557" }, { "arxiv_id": "2606.16679", "authors": [ "Dhyey Dharmendrakumar Mavani", "Nathan Pflueger" ], "id": "arxiv:2606.16679", "kind": "paper", "label": "core", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-06-15", "score": 6.5, "source": "arxiv-ai4math-core", "summary": "The Riemann--Roch theorem for graphs, due to Baker and Norine, is a foundational result establishing a powerful analogy between finite graphs and algebraic curves. We describe a complete formal proof of this theorem implemented in the Lean 4 theorem prover. Our formalization includes the existence and uniqueness of q-reduced divisors, a modified form of Dhar's burning algorithm, the bijection between acyclic orientations with unique source and maximal superstable configurations, and Clifford's theorem. We also include several challenges for future formalization.", "title": "Formalizing chip-firing and Riemann--Roch for graphs in Lean 4", "updated": "2026-06-15", "url": "https://arxiv.org/abs/2606.16679" }, { "arxiv_id": "2606.16541", "authors": [ "Noor Islam S. Mohammad", "Tamim Sheikh" ], "id": "arxiv:2606.16541", "kind": "paper", "label": "core", "matched_signals": [ "autoformalization", "lean_formal_proving_agents" ], "published": "2026-06-15", "score": 6.5, "source": "arxiv-ai4math-core", "summary": "Autoformalization, translating natural-language mathematics into formal proof assistants, is bottlenecked not by translation fluency but by \\emph{faithfulness}: a formal statement can typecheck and be provable, yet still encode a different theorem than the source intended. We introduce \\emph{Bidirectional Provability Fingerprinting} (\\bpf{}), a framework that certifies faithfulness by characterizing each candidate through its forward and backward consequence neighborhoods in the ambient theory and matching these against probes derived from the natural-language statement. We further introduce four novel components: (i) \\emph{Counterfactual Probe Generation} (\\cpg{}), a contrastive procedure that synthesizes probes targeting specific drift directions; (ii) the \\emph{Equivalence Spectrum}, a continuous faithfulness score that replaces brittle binary verdicts; (iii) \\emph{Adaptive Probe Budget Allocation} (\\apba{}), an information-theoretic budget router; and (iv) \\emph{Faithfulness-Guided Decoding} (\\fgd{}), which uses \\bpf{} signals as a reward during autoformalization. We prove a \\emph{drift detection theorem} and a \\emph{PAC-faithfulness} result establishing that the equivalence class of a natural language statement is learnable from $\\mathcal{O}(\\log(1/δ)/\\varepsilon)$ probes under mild assumptions. We release \\driftbench{}, a benchmark of $2{,}183$ NL/Lean~4 pairs with controlled drift labels across six subfields of mathlib4. \\bpf{}\\,+\\,\\cpg{} detects $89.6\\%$ of drifted formalizations at a $3.0\\%$ false-positive rate-against $41.2\\%$ for typecheck and $63.3\\%$ for LLM-judge baselines, and \\fgd{} reduces the rate at which a state-of-the-art autoformalizer emits drifted statements by $47\\%$. https://pmlrbd.github.io/BPF/", "title": "The Faithfulness Gap: Certifying Semantic Equivalence Between Natural-Language and Formal Mathematical Statements", "updated": "2026-06-15", "url": "https://arxiv.org/abs/2606.16541" }, { "arxiv_id": "2606.16144", "authors": [ "Xiaohui Bei", "Jiajun Ma", "Zhan Jing", "Hongfei Fu", "Zhihao Gavin Tang" ], "id": "arxiv:2606.16144", "kind": "paper", "label": "core", "matched_signals": [ "lean_formal_proving_agents", "mathlib_retrieval" ], "published": "2026-06-15", "score": 6.5, "source": "arxiv-ai4math-core", "summary": "Mathematical formalization uses interactive theorem provers to turn informal mathematical statements into machine-checkable artifacts. The success of mathlib, a large collaborative library for Lean, illustrates the potential of this approach. Recent progress in AI-assisted programming and theorem proving is also making large-scale formalization more practical. This paper presents EconCSLib, an early Lean 4 library for computational economics, as both infrastructure and a case study for AI-assisted formalization. The library aims to provide reusable definitions and theorems for game theory, mechanism design, social choice, and related areas. Beyond verified proofs of existing results, the library also aims to host machine-checked open problems and formalization of modern research papers. We discuss the design principles behind the library, the lessons learned from its development, and future directions for AI-assisted formalization in computational economics.", "title": "EconCSLib: A Lean Library for Computational Economics and AI-Assisted Research", "updated": "2026-06-15", "url": "https://arxiv.org/abs/2606.16144" }, { "arxiv_id": "2512.17260", "authors": [ "Huajian Xin", "Zhicheng Jiang", "Allan Jie", "Xiaoran Jin", "Xing Jin" ], "id": "manual:semantic-scholar-seed-prover-1-5", "kind": "paper", "label": "core", "matched_signals": [ "lean_formal_proving_agents", "seed_author:Allan Jie", "seed_author:Huajian Xin", "seed_author:Zhicheng Jiang" ], "observed_date": "2026-06-20", "published": "2025-12-19", "score": 6.5, "source": "semantic-scholar-library", "source_app": "semantic_scholar", "summary": "Semantic Scholar folder seed for experience-driven formal theorem proving. Watch for methods that turn failed proof attempts into useful training data.", "title": "Seed-Prover 1.5: Mastering Undergraduate-Level Theorem Proving via Learning from Experience", "updated": "2026-06-20", "url": "http://arxiv.org/abs/2512.17260" }, { "arxiv_id": "2606.19315", "authors": [ "Ruida Wang", "Rui Pan", "Pengcheng Wang", "Shizhe Diao", "Tong Zhang" ], "id": "arxiv:2606.19315", "kind": "paper", "label": "adjacent", "matched_signals": [ "general_ai_math_reasoning", "lean_formal_proving_agents" ], "published": "2026-06-17", "score": 4.9, "source": "arxiv-ai4math-core", "summary": "Enhancing the formal math reasoning capabilities of Large Language Models (LLMs) has become a key focus in both mathematical and computer science communities in recent years. While significant progress has been made in using state-of-the-art Auto-Regressive (AR) LLMs for formal theorem proving, these models suffer from inherent limitations. Their next-token prediction generation methods may yield suboptimal performance due to the challenges of long-range coherence and the compounding of errors over long sequences. Recent advancements in diffusion LLMs (dLLMs), which generate text through iterative denoising of a multi-token block, offer a promising alternative. However, the application of dLLMs to formal mathematics, where maintaining long-range coherence is critical, remains largely understudied. To address the challenges above, we propose **Diffusion-Proof**, to the best of our knowledge, the first framework to train and apply dLLMs for formal theorem proving. Our frameworks contain training and inference methods for two models. The first one is *dLLM-Prover-7B*, which performs whole-proof writing with long-range coherent tactic usage. The second one is *dLLM-Corrector-7B*, which is a novel large block diffusion-based correction model. It leverages the in-filling capabilities of dLLMs to perform local proof correction using bi-directional information. Extensive experiments demonstrate that **Diffusion-Proof** relatively significantly outperforms the AR LLM baseline trained under the same dataset. **Diffusion-Proof** achieves an absolute improvement of **1.61%** on ProofNet-Test and **6.14%** on MiniF2F-Test benchmarks compare to the baseline. Notably, **Diffusion-Proof** successfully resolves one IMO problem that more advanced thinking model DeepSeek-Prover-V2-7B could not solve, showcasing the unique advantage of dLLMs in formal theorem proving.", "title": "Diffusion-Proof: Recipe for Formal Theorem Proving Beyond Auto-Regressive Generation", "updated": "2026-06-17", "url": "https://arxiv.org/abs/2606.19315" }, { "arxiv_id": "2306.15626", "authors": [ "Kaiyu Yang", "Aidan M. Swope", "Alex Gu", "Rohan Chalamala", "Peiyang Song", "Shuyuan Yu", "Saad Godil", "Ryan Prenger", "Anima Anandkumar" ], "id": "manual:scholar-inbox-leandojo", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents", "seed_author:Kaiyu Yang" ], "observed_date": "2026-06-20", "published": "2023-06-27", "score": 4.7, "source": "scholar-inbox-manual", "source_app": "scholar_inbox", "summary": "Scholar Inbox surfaced this as a core retrieval-augmented Lean theorem proving baseline. It is relevant to premise retrieval, proof search, and agent evaluation.", "title": "LeanDojo: Theorem Proving with Retrieval-Augmented Language Models", "updated": "2026-06-20", "url": "https://arxiv.org/abs/2306.15626" }, { "arxiv_id": "2606.20358", "authors": [ "Fubin Yan", "Kenneth W. Shum" ], "id": "arxiv:2606.20358", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-06-18", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "The extended complex plane is a fundamental object in complex analysis, hyperbolic geometry, and mathematical physics. Its geometry is governed by Möbius transformations, with the cross ratio serving as a central invariant. We present a formalization of these concepts in the Lean4 theorem prover. The extended complex plane is represented using Mathlib's Option type over $\\mathbb{C}$, where the additional element represents the point at infinity. On this foundation, we define Möbius transformations, their action on the extended complex plane, and the cross ratio. We formalize several basic properties of Möbius transformations, including their group structure, and identify them with a projective general linear group. We also prove the uniqueness of a Möbius transformation mapping any three distinct points to any other three distinct points, and the invariance of the cross ratio. All proofs are machine-checked in Lean 4. The complete development comprises approximately 6,000 lines of Lean code, including about 40 definitions and 150 lemmas and theorems. This work provides a verified foundation for future formalizations of conformal geometry, hyperbolic models, modular forms, and applications in mathematical physics.", "title": "Formalizing Extended Complex Numbers, Mobius Transformations, and Cross Ratio in Lean 4", "updated": "2026-06-18", "url": "https://arxiv.org/abs/2606.20358" }, { "arxiv_id": "2606.20121", "authors": [ "Ghilain Bergeron", "Vincent Trélat" ], "id": "arxiv:2606.20121", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-06-18", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "BARReL is a Lean 4 library bridging Atelier B, an industrial tool for the B method, and the Lean proof assistant by enabling users to conduct their formal B developments -- up to machine refinement and implementation -- interactively inside Lean, while retaining standard B syntax. B partial operators are carefully encoded by generating explicit well-definedness conditions, leveraging Lean's dependent types to enforce a well-definedness discipline by construction. That is, proof obligations and proof steps cannot silently rely on ill-typed or ill-defined instantiations. BARReL also features basic automation to try to discharge such well-definedness conditions automatically. The implementation is written entirely using Lean meta-programming and is designed to be modular: extending the supported B fragment typically requires only adding new syntax and encoding clauses. We illustrate the approach on a small but representative case study, and argue that BARReL can act as a stepping stone towards a strongly reliable Atelier B toolchain grounded in the Lean proof assistant.", "title": "BARReL: a modern backend for Atelier B in Lean", "updated": "2026-06-18", "url": "https://arxiv.org/abs/2606.20121" }, { "arxiv_id": "2606.19936", "authors": [ "Leni Aniva", "Claire Wang" ], "id": "arxiv:2606.19936", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-06-18", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "Music theory obeys a rich set of mathematical rules and symmetries. These symmetries follow mathematical structure which can be verified and expressioned in the precise language of a proof assistant. In this paper, we present Prismriver, a formalization of music theory in Lean 4. By formalizing music theory in Lean 4, we open the door to verifiable algorithmic composition and accompaniment generation. We also enable the analysis of monadic analysis of structures in music.", "title": "Prismriver: Formalization of Music Theory and Algorithmic Composition in Lean 4", "updated": "2026-06-18", "url": "https://arxiv.org/abs/2606.19936" }, { "arxiv_id": "2606.19761", "authors": [ "Lars Warren Ericson" ], "id": "arxiv:2606.19761", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-06-18", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "We present a machine-checked completeness theorem, in Lean 4, for the hybrid logic $L(\\forall)$: propositional modal logic with nominals, the satisfaction-style binder $\\forall$, and the box modality. (Machine-checked completeness for basic hybrid logic, without binders, was pioneered by Asta Halkjær From in Isabelle/HOL.) We build on Alex Oltean's 2023 Lean 4 formalization, which mechanized the syntax, semantics, Hilbert-style proof system, and soundness following Blackburn's Hybrid Completeness (1998), but left completeness unfinished. Finishing it requires manufacturing fresh names at two structurally different points, and our central finding is that they call for two different tools. (1) The root witnessed maximal consistent set, built by an extended Lindenbaum construction, needs at each step a nominal fresh for the whole set; the right tool is structural freshness: extend the language so an infinite supply of nominals is reserved by construction. We survey the design space (Oltean's odd/even encoding inside $\\mathbb{N}$, the disjoint-sum $N \\oplus \\mathbb{N}$ parameterization suggested by Bud Mishra, and From's synthetic-completeness frameworks) and explain the encoding we adopt. (2) The witnessed $\\Diamond$-successor of a maximal consistent set cannot be obtained this way: its canonical box-reduct provably mentions every nominal, so no reserved name is fresh. Here the right tool is one Oltean chose but left incomplete: an existence-lemma Henkin construction drawing each witness from the predecessor's witnessedness through a fresh state variable; we complete it with a data-carrying witness accumulator and a compactness argument. The theorem $Γ\\models \\varphi \\to Γ\\vdash \\varphi$ is fully formalized: the development is sorry-free, and #print axioms reports only propext, Classical.choice, and Quot.sound. We port the development to Lean v4.30.0 / mathlib v4.30.0.", "title": "Finishing Oltean's Completeness Proof in Lean 4 for Hybrid Logic $L(\\forall)$", "updated": "2026-06-18", "url": "https://arxiv.org/abs/2606.19761" }, { "arxiv_id": "2606.18462", "authors": [ "Alper Ferudun" ], "id": "arxiv:2606.18462", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-06-16", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "Dirac asked in 1970 whether for every k >= 4 there is a k-vertex-critical graph without critical edges; Jensen settled all k >= 5, and only k=4 remains open. Following Skottova and Steiner, call a graph G a (4,1)-graph if chi(G)=4, chi(G-v)=3 for every vertex v, and chi(G-e)=4 for every edge e; they proved delta(G) >= 6 and lambda(G) >= 6 for every (4,1)-graph and asked whether a 6-regular (4,1)-graph exists. We prove three results about this 6-regular case. Theorem A (computational): there is no 6-regular 4-vertex-critical graph on n <= 15 vertices, except for a unique graph (up to isomorphism) on n=13, whose 13 critical edges form a Hamilton cycle; hence any 6-regular (4,1)-graph has at least 16 vertices. Theorem B: in a 6-regular (4,1)-graph every 6-edge-cut is either the edge star of a vertex or has both shores of size at least 15; consequently every 6-regular (4,1)-graph on at most 29 vertices is super-6-edge-connected. Theorem C (all sizes): no shore of a nontrivial 6-edge-cut in a 6-regular (4,1)-graph induces a bipartite graph; more generally, a shore whose deficiency is concentrated on two vertices forces them to receive equal colours in every proper 3-colouring. The proof of Theorem B rests on an exact classification of the 3x3 cut matrices of 6-edge-cuts in (4,1)-graphs (exactly 21 matrices, five types up to row/column permutations) together with a boundary-shortfall lemma; the unique near-miss is K_{3,3,3} minus a rainbow 3-matching. Several supporting lemmas are machine-checked in Lean 4/Mathlib.", "title": "Exact 6-cut rigidity and small-order superconnectivity for the 6-regular case of Dirac's k=4 problem", "updated": "2026-06-16", "url": "https://arxiv.org/abs/2606.18462" }, { "arxiv_id": "2606.16893", "authors": [ "Aarne Ranta" ], "id": "arxiv:2606.16893", "kind": "paper", "label": "adjacent", "matched_signals": [ "autoformalization" ], "published": "2026-06-15", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "Symbolic informalization enables a reliable conversion of formal mathematics to natural language. It has the potential to make machine-checked content human-readable without loss of precision. In a traditional proof system usage, symbolic informalization generalizes the limited mechanisms of syntactic sugar into the ordinary language of mathematics. In a setting where proofs are constructed by artificial intelligence and autoformalization, symbolic informalization can explain what precisely has been constructed. This paper outlines the project Informath, which aims to show how symbolic informalization can produce fluent text with a reasonable development effort and address multiple formal and natural languages. Informath is based on an interlingual architecture, where Dedukti works as a hub between different proof systems (Agda, Lean, Rocq) and Grammatical Framework (GF) takes care of linguistic correctness and variation in different natural languages.", "title": "Symbolic Informalization: Fluent, Productive, Multilingual", "updated": "2026-06-15", "url": "https://arxiv.org/abs/2606.16893" }, { "arxiv_id": "2606.16688", "authors": [ "Andreas Florath" ], "id": "arxiv:2606.16688", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-06-15", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "We prove the exact octonary covering-code value $K_8(4, 2) = 23$ in Lean 4. The upper bound is given by an explicit 23-word radius-two code in $(Fin\\:8)^4$ , checked over all $8^4$ ambient words. The lower bound excludes covers with at most 22 words. A fiber-counting and missing-pair argument first rules out covers with at most 21 words. In the remaining 22-word case, the proof reduces a hypothetical cover to six missing-pair graphs coming from the coordinate-pair projections. Fiber-counting arguments constrain these graphs, and two Lean-checked Linear RAT (LRAT) refutations of stored conjunctive-normal-form (CNF) instances force a common 3 + 3 + 2 block structure. This structure is incompatible with a 22-word cover: the two three-symbol components already force 18 codewords, while the remaining two-symbol component would require a binary strength-two array of length four with at most four rows, which is impossible. The result is packaged as a proof-carrying Lean artifact: the explicit upper bound, structural lower bound, CNF instances, and LRAT refutations are checked inside Lean, with no external SAT solver used during proof replay.", "title": "A Lean-Certified Proof of $K_8(4, 2) = 23$", "updated": "2026-06-15", "url": "https://arxiv.org/abs/2606.16688" }, { "arxiv_id": "2606.16289", "authors": [ "Serhii Zabolotnii" ], "id": "arxiv:2606.16289", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-06-15", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "We give a characteristic-function formulation of Kunchenko's stochastic-polynomial construction for settings in which raw moments may fail to exist. In the finite-variance trigonometric case, the coefficients of the Kunchenko normal system are expressed through the characteristic function and its derivative. In the moment-free case, empirical characteristic functions on a fixed finite frequency grid define a bounded discrepancy geometry that remains meaningful for Cauchy, symmetric stable, and other heavy-tailed laws. We prove well-definedness and finite-grid almost sure consistency of this empirical characteristic-function geometry. We introduce the associated minimum-CF-distance estimator and establish its identifiability, strong consistency, and asymptotic normality on a fixed grid, with a covariance built from bounded trigonometric moments that stays finite even for Cauchy and stable laws; refining the grid increases the optimal-weight information monotonically to the Fisher information, so the estimator is asymptotically efficient in the dense-grid limit. We also relate bounded sine scores to weak stochastic-polynomial estimating equations. A small Lean 4 / Mathlib supplement checks selected deterministic identities underlying the bounded-score construction; convergence arguments and statistical interpretation remain outside the formalization.", "title": "Moment-Free Kunchenko Stochastic Polynomials via Empirical Characteristic Function", "updated": "2026-06-15", "url": "https://arxiv.org/abs/2606.16289" }, { "arxiv_id": "2606.16134", "authors": [ "Lars Warren Ericson" ], "id": "arxiv:2606.16134", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-06-15", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "We describe a Lean~4 formalization revisiting NYU Courant Technical Report TR1995-711 on the average-case complexity of Multilevel Syllogistic (MLS). The development encodes Reischuk--Schindelhauer average-case classes, an axiomatic MLS/EMLS semantics layer, a partial Ferro--Omodeo--Schwartz decision procedure with proved soundness and partial completeness on a membership-free fragment, serialization and step budgets, and conditional NP-average completeness and non-AvP hardness corollaries modulo explicitly documented structural axioms. Full Lean sources are inlined in the appendix modules.", "title": "Revisiting average case complexity of multilevel syllogistic: From the 1995 Courant Technical Report to Lean 4 Formalization", "updated": "2026-06-15", "url": "https://arxiv.org/abs/2606.16134" }, { "arxiv_id": "2606.15860", "authors": [ "Ryosuke Mizuno" ], "id": "arxiv:2606.15860", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-06-14", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "Let $N(Δ,D)$ denote the maximum number of vertices in a simple undirected connected graph with maximum degree at most $Δ$ and diameter at most $D$. Determining $N(Δ,D)$ is known as the degree/diameter problem. Although the problem has been studied for many years, the exact value of $N(Δ,D)$ is unknown for most pairs $(Δ,D)$. In this paper we construct explicit graphs, using a construction discovered through interaction with ChatGPT via its standard web interface, showing that $N(12,5)\\ge 34{,}992$ and $N(16,5)\\ge 147{,}456$. These improve the corresponding recorded lower bounds 29,621 and 132,496. The search was conducted without an external orchestration layer around ChatGPT: no custom agent framework, automated evaluator-driven search loop, problem-specific search engine, or formal proof assistant was set up in advance by the author. As far as the visible transcript shows, the author did not prompt the model with the concrete components or candidate families of the construction. After presenting the mathematical result, we describe the discovery process on the basis of the visible transcript. We focus on the meta-level interventions made during the approximately six-day search, and we identify the stage at which the abstraction underlying the construction first appeared.", "title": "New lower bounds for the degree/diameter problem via interaction with a browser-accessible LLM", "updated": "2026-06-14", "url": "https://arxiv.org/abs/2606.15860" }, { "arxiv_id": "2606.15520", "authors": [ "Lars Warren Ericson" ], "id": "arxiv:2606.15520", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-06-14", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "We describe a Lean 4 formalization of the algorithms and domain types from NYU Computer Science Technical Report \\#232, \\emph{An ICON Package for Experimenting with Euclidean Domains} (Ericson, 1986). The original system implemented Lipson's catalog of procedures over integers, rationals, modular rings, polynomial rings, and truncated power series via a custom runtime dispatch mechanism in Icon. The present work separates three concerns: mathematical definitions grounded in Mathlib's \\texttt{EuclideanDomain} hierarchy, computable mirrors suitable for evaluation and regression testing, and report-formatting infrastructure that reproduces the 1986 benchmark output line-for-line. All fourteen application algorithms from Section 3 of the report are defined and typecheck without \\texttt{sorry}; those grounded in Mathlib -- chiefly integer gcd and extended Euclid -- additionally carry machine-checked proofs. We classify each procedure by its epistemic status relative to Mathlib, enumerate the coherence obligations between the proof and computable layers, and state precisely what is theorem-backed versus regression-trusted. The formalization makes explicit the verification boundary that the 1986 package crossed only informally.", "title": "A Lean 4 Formalization of Euclidean Domain Algorithms from a 1986 Icon Experimentation Package", "updated": "2026-06-14", "url": "https://arxiv.org/abs/2606.15520" }, { "arxiv_id": "2601.14027", "authors": [ "Junqi Liu", "Marco Dos Santos", "Zekai Zhu", "Jiawei Liu", "Ran Wang", "Jia Li", "Wenda Li" ], "id": "manual:semantic-scholar-numina-lean-agent", "kind": "paper", "label": "adjacent", "matched_signals": [ "seed_author:Junqi Liu", "seed_author:Marco Dos Santos", "seed_author:Zekai Zhu" ], "observed_date": "2026-06-20", "published": "2026-01-20", "score": 3.5, "source": "semantic-scholar-library", "source_app": "semantic_scholar", "summary": "Semantic Scholar folder seed for agentic formal mathematics. Relevant to open Lean proof-agent workflows and reusable evaluation setups.", "title": "Numina-Lean-Agent: An Open and General Agentic Reasoning System for Formal Mathematics", "updated": "2026-06-20", "url": "http://arxiv.org/abs/2601.14027" }, { "authors": [ "Sebastian Ullrich" ], "id": "github:leanprover/lean4:755918567941", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-20", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: chore: CI: fix `macos-arm-ci` label spelling in restart-on-label (#14132)", "updated": "2026-06-20", "url": "https://github.com/leanprover/lean4/commit/75591856794182559e5a3713bdf8aa00c52fc7ca" }, { "authors": [ "Sebastian Ullrich" ], "id": "github:leanprover/lean4:437d4e58643f", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-20", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: test: reuse snapshots of most common headers across elab pile (#14077)", "updated": "2026-06-20", "url": "https://github.com/leanprover/lean4/commit/437d4e58643fc2eda231f2a7b9fe81945b16b43f" }, { "authors": [ "leiko" ], "id": "github:leanprover/lean4:62f25a54bea5", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-20", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: fix: lake: correct `Package.remoteUrl?` logic (#14130)", "updated": "2026-06-20", "url": "https://github.com/leanprover/lean4/commit/62f25a54bea55b68f2f2948b26441626f2c4a7da" }, { "authors": [ "mathlib-splicebot[bot]" ], "id": "github:leanprover-community/mathlib4:2dfe37a6fa59", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-20", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: chore(FieldTheory/IntermediateField/Basic): automated extraction from #38864 (#40844)", "updated": "2026-06-20", "url": "https://github.com/leanprover-community/mathlib4/commit/2dfe37a6fa59521018b61dc988495a84dd47dd30" }, { "authors": [ "mathlib-splicebot[bot]" ], "id": "github:leanprover-community/mathlib4:843d7890de00", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-20", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: chore(Algebra/Group/Subgroup/Basic): automated extraction from #38864 (#40842)", "updated": "2026-06-20", "url": "https://github.com/leanprover-community/mathlib4/commit/843d7890de006fe2cb1d2974f5bafcf1b0e3fad8" }, { "authors": [ "Thomas Browning" ], "id": "github:leanprover-community/mathlib4:fbfd7f57b6e2", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-20", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: refactor(RingTheory/Localization/FractionRing): remove bottom ring and field from `IsFractionRing.mulSemiringAction` (#40804)", "updated": "2026-06-20", "url": "https://github.com/leanprover-community/mathlib4/commit/fbfd7f57b6e24d35f345edee2d888a3e6b5f4cc4" }, { "authors": [ "Yi.Yuan" ], "id": "github:leanprover-community/mathlib4:8690e4fcb159", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-20", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: chore(RingTheory/Polynomial/Basic): squeeze terminal `simp`s (#40833)", "updated": "2026-06-20", "url": "https://github.com/leanprover-community/mathlib4/commit/8690e4fcb159b28ef4b3afd5c68404c9bc0a8992" }, { "authors": [ "Snir Broshi" ], "id": "github:leanprover-community/mathlib4:887d94632e78", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-20", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: doc(1000-yaml): we don't have Cauchy's integral theorem (#40335)", "updated": "2026-06-20", "url": "https://github.com/leanprover-community/mathlib4/commit/887d94632e78e6e41701cf57c3e7206c6c2b1e5b" }, { "authors": [ "Jireh Loreaux" ], "id": "github:leanprover-community/mathlib4:6eb4a1ce07e0", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-20", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: feat: in a dense order, `𝓝[<] a` has `fun x ↦ Ico x a` as a basis (#40523)", "updated": "2026-06-20", "url": "https://github.com/leanprover-community/mathlib4/commit/6eb4a1ce07e071ac0f8365e6999a3135b45d8ae4" }, { "authors": [ "Jireh Loreaux" ], "id": "github:leanprover-community/mathlib4:d3d49a109bea", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-20", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: feat: conditions for commuting with a unitary element (#40516)", "updated": "2026-06-20", "url": "https://github.com/leanprover-community/mathlib4/commit/d3d49a109bea04e0f3b9bf33b43d2ad80ec33c09" }, { "authors": [ "Henrik Böving" ], "id": "github:leanprover/lean4:cb5e33763601", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-19", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: fix: `lean_task_imp.m_canceled` needs to be accessed using atomics (#14127)", "updated": "2026-06-19", "url": "https://github.com/leanprover/lean4/commit/cb5e33763601d9460805607bfb7ffc5fc69119d7" }, { "authors": [ "Lean stage0 autoupdater" ], "id": "github:leanprover/lean4:74d7aef14788", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-19", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: chore: update stage0", "updated": "2026-06-19", "url": "https://github.com/leanprover/lean4/commit/74d7aef14788aed7b0e289a401119b9d13ab9067" }, { "authors": [ "Sebastian Ullrich" ], "id": "github:leanprover/lean4:0a0e533d2124", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-19", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: fix: avoid compiler panic when deriving computable `Inhabited` in a `noncomputable section` (#14125)", "updated": "2026-06-19", "url": "https://github.com/leanprover/lean4/commit/0a0e533d21242224de3a3d0fb876bd6f3b195b58" }, { "authors": [ "Kim Morrison" ], "id": "github:leanprover/lean4:9dd91738cef0", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-19", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: feat: split `implicit` transparency into `implicit` and `instances` (#13637)", "updated": "2026-06-19", "url": "https://github.com/leanprover/lean4/commit/9dd91738cef0bc19abf86bf4089179cc782ef1de" }, { "authors": [ "Garmelon" ], "id": "github:leanprover/lean4:f0c3607bec71", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-19", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: chore: update release scripts (#14126)", "updated": "2026-06-19", "url": "https://github.com/leanprover/lean4/commit/f0c3607bec71470a83a7362d53e4925c393cfb25" }, { "authors": [ "Sebastian Ullrich" ], "id": "github:leanprover/lean4:072fd7f782fa", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-19", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: perf: replace the object compactor's worklist with recursion (#14122)", "updated": "2026-06-19", "url": "https://github.com/leanprover/lean4/commit/072fd7f782fa004c9a62309a3f57cf60bcb408f5" }, { "authors": [ "Sebastian Graf" ], "id": "github:leanprover/lean4:1df6f6996307", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-19", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: feat: extract `WP` superclass from `WPMonad` to support deep embeddings (#14080)", "updated": "2026-06-19", "url": "https://github.com/leanprover/lean4/commit/1df6f699630747c449151cd8f05eb7da60e29864" }, { "authors": [ "Paul Reichert" ], "id": "github:leanprover/lean4:f32420668cf7", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-19", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: chore: skip fetching blobs in the \"push tag\" step of the PR release (#14123)", "updated": "2026-06-19", "url": "https://github.com/leanprover/lean4/commit/f32420668cf7018545c7d93fe138eaebbe03fb89" }, { "authors": [ "Sebastian Ullrich" ], "id": "github:leanprover/lean4:0af1abc8e800", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-19", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: chore: CI: avoid fsanitize OOMs (#14100)", "updated": "2026-06-19", "url": "https://github.com/leanprover/lean4/commit/0af1abc8e800438228e640297f7362cdac3d9645" }, { "authors": [ "Sebastian Ullrich" ], "id": "github:leanprover/lean4:9c236a98d4aa", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-19", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: fix: `getModuleEntries (level := .server)` should work even outside server mode in combination with `import all` (#14120)", "updated": "2026-06-19", "url": "https://github.com/leanprover/lean4/commit/9c236a98d4aa5d694c4f039c5b2a5e3fef3ad11a" }, { "authors": [ "Sebastian Graf" ], "id": "github:leanprover/lean4:d53f283b35bc", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-19", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: fix: support dependent discriminant telescopes in mvcgen match splitting (#14119)", "updated": "2026-06-19", "url": "https://github.com/leanprover/lean4/commit/d53f283b35bcce4c77956d015aa6e7f871549302" }, { "authors": [ "Eric Wieser" ], "id": "github:leanprover/lean4:fa69f382002b", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-19", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: refactor: use `std::optional` (#14094)", "updated": "2026-06-19", "url": "https://github.com/leanprover/lean4/commit/fa69f382002b8cbb92b574900fcb84b34d584ddd" }, { "authors": [ "Eric Wieser" ], "id": "github:leanprover-community/mathlib4:005f0aa67b69", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-19", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: chore: remove an unused import (#40825)", "updated": "2026-06-19", "url": "https://github.com/leanprover-community/mathlib4/commit/005f0aa67b6922eb1a8f5209fd0e707aeb867945" }, { "authors": [ "Snir Broshi" ], "id": "github:leanprover-community/mathlib4:7ba890ea1765", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-19", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: feat: add `Homeomorph.chartedSpace` (#39107)", "updated": "2026-06-19", "url": "https://github.com/leanprover-community/mathlib4/commit/7ba890ea1765d6395f6236529f9e4e8cae006bb0" }, { "authors": [ "Eric Wieser" ], "id": "github:leanprover-community/mathlib4:5392b8aa32a2", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-19", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: chore: move RingQuot results (#40823)", "updated": "2026-06-19", "url": "https://github.com/leanprover-community/mathlib4/commit/5392b8aa32a2d5f6d24ab5c9e76ea4fd1c593538" }, { "authors": [ "Eric Wieser" ], "id": "github:leanprover-community/mathlib4:1220aa6023f6", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-19", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: refactor: make `{Con,AddCon,RingCon}.congr` match `Quotient.congr` (#40819)", "updated": "2026-06-19", "url": "https://github.com/leanprover-community/mathlib4/commit/1220aa6023f69a3283decf8e204984a6075cd9bd" }, { "authors": [ "Michael Rothgang" ], "id": "github:leanprover-community/mathlib4:3685731497f3", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-19", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: feat: add `contMDiffWithinAt_iff_of_mem_maximalAtlas'` (#40810)", "updated": "2026-06-19", "url": "https://github.com/leanprover-community/mathlib4/commit/3685731497f316bb1b3ecbfa8282eca31494b20b" }, { "authors": [ "Leonid Ryvkin" ], "id": "github:leanprover-community/mathlib4:413a92168ed3", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-19", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: feat: Lie-Rinehart subalgebras introduced (#39850)", "updated": "2026-06-19", "url": "https://github.com/leanprover-community/mathlib4/commit/413a92168ed3397410753fb2cd9373eb0d28824a" }, { "authors": [ "teorth" ], "id": "github:leanprover-community/mathlib4:2216b5b1ea90", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-19", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: feat(NumberTheory/PrimeCounting): primesBelow/primesLE as filters of Ioo/Ioc (#40654)", "updated": "2026-06-19", "url": "https://github.com/leanprover-community/mathlib4/commit/2216b5b1ea909cc6bcd2c3b45516c1ece827135f" }, { "authors": [ "Thomas Browning" ], "id": "github:leanprover-community/mathlib4:e1f7efcd7be2", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-19", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: feat(RingTheory/Ideal/Maps): coheight of ideal under surjective ring homomorphism (#40780)", "updated": "2026-06-19", "url": "https://github.com/leanprover-community/mathlib4/commit/e1f7efcd7be2541b98efd51874fe6498299f6bab" }, { "authors": [ "Thomas Browning" ], "id": "github:leanprover-community/mathlib4:c367137fa1e2", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-19", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: refactor(RingTheory/QuasiFinite/Basic): golf proof of `eq_of_le_of_under_eq` (#40772)", "updated": "2026-06-19", "url": "https://github.com/leanprover-community/mathlib4/commit/c367137fa1e20d856c0d58e0efa86a62c87121fb" }, { "authors": [ "Thomas Browning" ], "id": "github:leanprover-community/mathlib4:4cf5258e7fca", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-19", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: feat(RingTheory/Ideal/Over): add instances for `LiesOver` in towers (#40437)", "updated": "2026-06-19", "url": "https://github.com/leanprover-community/mathlib4/commit/4cf5258e7fcaee29f7b3abb025add73bdea63f41" }, { "authors": [ "Thomas Browning" ], "id": "github:leanprover-community/mathlib4:2e6e057c0087", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-19", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: feat(RingTheory/Ideal/Pointwise): add `IsScalarTower` instance for pointwise action on ideals (#40386)", "updated": "2026-06-19", "url": "https://github.com/leanprover-community/mathlib4/commit/2e6e057c008759e50eda6b03746d6309d59d7b65" }, { "authors": [ "Thomas Browning" ], "id": "github:leanprover-community/mathlib4:1b82673ebcbc", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-19", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: feat(RingTheory/Ideal/Over): add `Ideal.smul_under` (#40385)", "updated": "2026-06-19", "url": "https://github.com/leanprover-community/mathlib4/commit/1b82673ebcbcca37455a100743b2c2f56b61bda6" }, { "authors": [ "Thomas Browning" ], "id": "github:leanprover-community/mathlib4:d846133bc965", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-19", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: chore(Topology/Algebra/Category/ProfiniteGrp/*): to_additivize some declarations (#39973)", "updated": "2026-06-19", "url": "https://github.com/leanprover-community/mathlib4/commit/d846133bc96504cc64e86ff8010b310a38d09849" }, { "authors": [ "Thomas Browning" ], "id": "github:leanprover-community/mathlib4:c9ba3100f3ae", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-19", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: chore(FieldTheory/Galois/IsGaloisGroup): generalize `IsGaloisGroup.of_ringEquiv` to surjective ring homs (#40805)", "updated": "2026-06-19", "url": "https://github.com/leanprover-community/mathlib4/commit/c9ba3100f3ae30fffc9f6d2bbfe033dcd44ce577" }, { "authors": [ "Jorge Luis Mayoral" ], "id": "github:leanprover-community/mathlib4:e72c92ce6faa", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-19", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: feat(FinitelyPresentedGroup): free product of finitely presented is finitely presented (#40332)", "updated": "2026-06-19", "url": "https://github.com/leanprover-community/mathlib4/commit/e72c92ce6faaaa9e1c19532a0a7b3cefd18006c6" }, { "authors": [ "Whysoserioushah" ], "id": "github:leanprover-community/mathlib4:4084fffec05c", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-19", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: feat(Representation/Continuous): show TopRep is a linear category (#40739)", "updated": "2026-06-19", "url": "https://github.com/leanprover-community/mathlib4/commit/4084fffec05cf6a84e0097e3c3e4083b92e04fa4" }, { "authors": [ "Julia Markus Himmel" ], "id": "github:leanprover-community/mathlib4:10212ec219a3", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-19", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: chore(CI): allow PR title to start with 'E2' (#40806)", "updated": "2026-06-19", "url": "https://github.com/leanprover-community/mathlib4/commit/10212ec219a35b6f8603f9dd72751931be2a6167" }, { "authors": [ "Michael Rothgang" ], "id": "github:leanprover-community/mathlib4:0c7d9c10847c", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-19", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: chore(Geometry/Manifold/ContMDiff/Defs): use `variable` more (#40798)", "updated": "2026-06-19", "url": "https://github.com/leanprover-community/mathlib4/commit/0c7d9c10847cd4a97cf50bd1a2f034acf5e9a9b0" }, { "authors": [ "Hannah Scholz" ], "id": "github:leanprover-community/mathlib4:01469fac233f", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-19", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: feat(Geometry/Manifold/Notation): add (d)elaborators for `UniqueMDiffOn` and `UniqueMDiffWithinAt` (#40748)", "updated": "2026-06-19", "url": "https://github.com/leanprover-community/mathlib4/commit/01469fac233f7969277e6f6c5e53f4d1bd018e75" }, { "authors": [ "Anatole Dedecker" ], "id": "github:leanprover-community/mathlib4:63c85da758e5", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-19", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: chore: rename Pi.algHom to AlgHom.pi (#40044)", "updated": "2026-06-19", "url": "https://github.com/leanprover-community/mathlib4/commit/63c85da758e583e0be796f43c133ae7529486a40" }, { "authors": [ "Mac Malone" ], "id": "github:leanprover-community/mathlib4:555b02bc956f", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-19", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: chore: prerequisite options for the Lake cache (#40784)", "updated": "2026-06-19", "url": "https://github.com/leanprover-community/mathlib4/commit/555b02bc956f8a17a8dc95eb1131efda57d8b837" }, { "authors": [ "Thomas Browning" ], "id": "github:leanprover-community/mathlib4:9158e203ca67", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-19", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: chore(Order/Northcott): use `TendstoCofinite` in `Northcott.lean` (#40778)", "updated": "2026-06-19", "url": "https://github.com/leanprover-community/mathlib4/commit/9158e203ca676281c73e395d6d295ad2552a3599" }, { "authors": [ "Whysoserioushah" ], "id": "github:leanprover-community/mathlib4:7c309b0e29db", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-19", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: feat(Projectivization/PSL/Stabilizer): Add stabilizer lemmas (#39999)", "updated": "2026-06-19", "url": "https://github.com/leanprover-community/mathlib4/commit/7c309b0e29db78e07335916aa82148910b5689eb" }, { "authors": [ "Kim Morrison" ], "id": "github:leanprover/lean4:bc5f89f4abe8", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-18", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: feat: tag `min`/`max` definitions with `@[lia]` (#14107)", "updated": "2026-06-18", "url": "https://github.com/leanprover/lean4/commit/bc5f89f4abe82ee105ce7922a83e286fd7a67774" }, { "authors": [ "Sebastian Ullrich" ], "id": "github:leanprover/lean4:2be4e45262b6", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-18", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: chore: bench build/server-mode open in `import Lean` (#14111)", "updated": "2026-06-18", "url": "https://github.com/leanprover/lean4/commit/2be4e45262b6773a99df6be93a7d7435d61bf33b" }, { "authors": [ "Julia Markus Himmel" ], "id": "github:leanprover/lean4:44232a0ea72a", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-18", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: feat: fix and expose `Float.ofScientific` (#14110)", "updated": "2026-06-18", "url": "https://github.com/leanprover/lean4/commit/44232a0ea72a86568ac9867fe0d2dcefc3dfd28a" }, { "authors": [ "Sebastian Graf" ], "id": "github:leanprover/lean4:8e073a29a6d0", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-18", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: refactor: reorder Triple to program-first (#14103)", "updated": "2026-06-18", "url": "https://github.com/leanprover/lean4/commit/8e073a29a6d0029980c8ee2486330307817df159" }, { "authors": [ "Sebastian Graf" ], "id": "github:leanprover/lean4:9ddaae5b4e5c", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-18", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: feat: add `wait_for_expected_type%` term elaborator (#14112)", "updated": "2026-06-18", "url": "https://github.com/leanprover/lean4/commit/9ddaae5b4e5c47c159450d8bb58ac4a2940d648d" }, { "authors": [ "Henrik Böving" ], "id": "github:leanprover/lean4:d24bd898d641", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-18", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: fix: make `get_task_state_core` thread safe (#14108)", "updated": "2026-06-18", "url": "https://github.com/leanprover/lean4/commit/d24bd898d6417ddcefda2135e3f070e454e3a82c" }, { "authors": [ "Sebastian Ullrich" ], "id": "github:leanprover/lean4:518dc662d70f", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-18", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: fix: do not max-share `IO.Ref` in compactor (#14106)", "updated": "2026-06-18", "url": "https://github.com/leanprover/lean4/commit/518dc662d70fe6c566a0c935157690ee8ae53558" }, { "authors": [ "Sebastian Graf" ], "id": "github:leanprover/lean4:04e7ed3e81fa", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-18", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: refactor: locate the spec program structurally in selectProg (#14105)", "updated": "2026-06-18", "url": "https://github.com/leanprover/lean4/commit/04e7ed3e81faa185024eab8f906836de1d8cb22f" }, { "authors": [ "Kim Morrison" ], "id": "github:leanprover/lean4:c2f84454df66", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-18", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: feat: add builtin `@[lia]` attribute supplying a lemma set to the `lia` tactic (#14098)", "updated": "2026-06-18", "url": "https://github.com/leanprover/lean4/commit/c2f84454df661afccf3a8641d6061f8269838f5d" }, { "authors": [ "Vladimir Gladshtein" ], "id": "github:leanprover/lean4:84fc958c1608", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-18", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: feat: add a shorthand for `while`-loop invariants (#14102)", "updated": "2026-06-18", "url": "https://github.com/leanprover/lean4/commit/84fc958c16084b092932067ecf2d816802025cdf" }, { "authors": [ "Sebastian Ullrich" ], "id": "github:leanprover/lean4:5781828fa124", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-18", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: chore: add `trace_timings` script for test suite profiling (#14104)", "updated": "2026-06-18", "url": "https://github.com/leanprover/lean4/commit/5781828fa124314d32846bc0a3433cf5c48fada1" }, { "authors": [ "Vladimir Gladshtein" ], "id": "github:leanprover/lean4:7821657e0630", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-06-18", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: fix: normalise `⊤` in `mvcgen'` (#14099)", "updated": "2026-06-18", "url": "https://github.com/leanprover/lean4/commit/7821657e0630f8e0dd487658de0459202fd55670" }, { "arxiv_id": "2606.20439", "authors": [ "Evan Chen", "Ken Ono", "Richard E. Schwartz", "Dinesh S. Thakur" ], "id": "arxiv:2606.20439", "kind": "paper", "label": "negative", "matched_signals": [], "published": "2026-06-18", "score": 0.5, "source": "arxiv-ai4math-core", "summary": "Start with four digits, arrange them in both descending and ascending order, subtract, and repeat. This simple process is known as the Kaprekar routine, famous in base ten for sending every nonconstant four-digit string to $6174$. We show that in every odd base $B>3$, the four-digit Kaprekar map has an unexpectedly rigid structure. After at most three iterations, every nonconstant orbit enters an explicit triangular region $\\mathcal{T}_B$, and on this region the map is conjugate to projective doubling: \\[ \\{[r],[s]\\}\\longmapsto \\{[2r],[2s]\\}. \\] This gives a complete finite description of all nonconstant terminal cycles, including an explicit formula for their lengths and counts. In particular, the longest terminal cycle has length at most $(B-1)/2$, and equality can occur only when $B$ is prime. For primes $p>5$, equality occurs precisely when the least positive $m$ with $2^m\\equiv\\pm1\\pmod p$ is $m=(p-1)/2$. The results proved here were first formulated by Schwartz and Thakur. As a test case for AI-assisted formal mathematics, AxiomProver produced Lean/mathlib formalizations of these results.", "title": "Four-digit Kaprekar dynamics in odd bases", "updated": "2026-06-18", "url": "https://arxiv.org/abs/2606.20439" }, { "arxiv_id": "2606.19093", "authors": [ "Ewan Pinnington", "Peter Lean", "Mihai Alexe", "Eulalie Boucher", "Simon Lang", "Patrick Laloyaux", "Gert Mertes", "Tomas Kral", "Patricia de Rosnay", "Matthew Chantry", "Anthony McNally" ], "id": "arxiv:2606.19093", "kind": "paper", "label": "negative", "matched_signals": [], "published": "2026-06-17", "score": 0.5, "source": "arxiv-ai4math-core", "summary": "We introduce the Artificial Intelligence Forecasting System for Direct Observation Prediction (AIFS-DOP). AIFS-DOP is trained on a 40-year harmonized dataset of gridded observations, without using numerical weather prediction (NWP) reanalysis or model data. The resulting model is competitive with ECMWF's Integrated Forecasting System (IFS) when scored on a one year period of forecasts across 2021/2022. This progress on Direct Observation Prediction represents the first time that a data-driven model, trained solely on observations, is competitive with the IFS at medium ranges for several key upper-air and surface headline scores, when verified against observation data.", "title": "AIFS-DOP: End-to-End Medium-Range Weather Prediction from Observations Alone with Machine Learning", "updated": "2026-06-17", "url": "https://arxiv.org/abs/2606.19093" }, { "arxiv_id": "2606.18342", "authors": [ "Zhiyuan Ji", "Yang Sun", "Mauro Giavalisco", "Yongda Zhu", "George H. Rieke", "Christina C. Williams", "Michael V. Maseda", "Jianwei Lyu", "Marcia Rieke", "Sandro Tacchella" ], "id": "arxiv:2606.18342", "kind": "paper", "label": "negative", "matched_signals": [], "published": "2026-06-16", "score": 0.5, "source": "arxiv-ai4math-core", "summary": "Little Red Dots (LRDs) appear extremely compact at rest-frame optical wavelengths, yet many show extended rest-frame UV morphology revealing more complex internal structure. We present a combined analysis of VLT/MUSE rest-frame UV integral-field spectroscopy and continuum-subtracted [O III], H$β$, and H$α$+[N II] emission-line maps from JWST/NIRCam imaging at sub-kpc resolution for LRD-204851 at $z=5.482$ in GOODS-S. We find that LRD-204851 hosts a remarkably thin, bipolar, elongated structure passing through the optical continuum centroid and extending several kpc on either side, traced by both the UV continuum and the rest-frame optical emission lines, with a bright [O III] clump-like structure $\\sim$2 kpc to the south-east of the centroid. The MUSE observations reveal a double-peaked Ly$α$ profile, with a broad and bright near-systemic red peak and a relatively faint peak blueshifted by $\\sim$430 km s$^{-1}$, accompanied by a tentative N V $λ1238$ detection at similar velocity. In narrow-band imaging extracted from the MUSE IFU cube, both the blue Ly$α$ peak and the tentative N V emission lean toward this same south-eastern direction. Independently, radiative-transfer modeling of the integrated Ly$α$ profile favors a biconical low-column-density cavity in a dense, slowly expanding neutral envelope, in support of the bipolar geometry traced by the line maps. Together, these results suggest that the elongated emission of LRD-204851 is connected to radiation and/or gas flow from its central engine through a low-column-density channel with a small opening angle that may trace either a slow outflow or a quasi-static ionization cone. LRD-204851 is one of the first LRDs where the central engine's impact on its host galaxy is potentially directly observable on kpc scales.", "title": "Compact Core, Extended Reach: A Bipolar kpc-Scale Elongation in a Little Red Dot at $z \\approx 5.5$", "updated": "2026-06-16", "url": "https://arxiv.org/abs/2606.18342" }, { "arxiv_id": "2606.17581", "authors": [ "Xiyu Zhai", "Xinyi Chen", "Yiping Wang", "Runlong Zhou", "Liao Zhang", "Simon S. Du" ], "id": "arxiv:2606.17581", "kind": "paper", "label": "negative", "matched_signals": [], "published": "2026-06-16", "score": 0.5, "source": "arxiv-ai4math-core", "summary": "We present a dependent-type-based prover designed around the way LLMs (and humans) tend to write mathematics, complementing existing systems such as Lean and Rocq. Its core design choices are a surface that imitates mathematical natural language and a rule-driven automation layer that closes the routine steps a textbook would omit, so that an accepted proof can be re-emitted as a checked Lean file. Early experiments suggest that, even without any prover-specific training data, LLMs can learn to use it effectively on the miniF2F benchmark. Lean output excerpts: https://github.com/xiyuzhai-husky-lang/visored/", "title": "Visored: A Controlled-Natural-Language Prover for LLM-Generated Mathematics", "updated": "2026-06-16", "url": "https://arxiv.org/abs/2606.17581" }, { "arxiv_id": "2606.17269", "authors": [ "Carlos Eduardo Sanoja" ], "id": "arxiv:2606.17269", "kind": "paper", "label": "negative", "matched_signals": [], "published": "2026-06-15", "score": 0.5, "source": "arxiv-ai4math-core", "summary": "In skill-constrained production-inventory systems, the qualified human capacity available tomorrow depends on training decisions made today: production requires certified workers, certifications decay unless maintained, and training consumes the same scarce worker hours that production needs now. We study a closed-loop skill-constrained model predictive controller that, at every shift, solves a finite-horizon mixed-integer program over production, inventory, backlog, and training, with binary predicted certification, hard production eligibility, and an interpretable terminal value that prices certified-capacity gaps at the horizon boundary; only the first-period action is applied before replanning. On synthetic, seed-controlled SkillChain-Gym scenarios - announced and surprise new-skill shocks, demand shocks, absenteeism, forecast- and availability-quality modes, capacity-boundary and training-rate sweeps, and negative controls - we evaluate the controller against production-only and maintenance-only ablations, static cross-training insurance plans, and a strong reactive heuristic, under an ex-ante locked configuration and paired statistics. The result is regime dependence, not superiority: no policy class dominates. Predictive control helps when skill or labor bottlenecks are forecastable early enough for training to complete; lean static insurance remains hard to beat under surprise shocks, near the demand-capacity boundary, and wherever pre-shock slack makes insurance cheap. Attribution ablations separate certification maintenance, re-acquisition of lapsed certifications, and greenfield skill acquisition. Forecastability, not adaptivity per se, decides when predictive control pays.", "title": "Skill-Constrained Model Predictive Control for Resilient Manufacturing Supply Chains", "updated": "2026-06-15", "url": "https://arxiv.org/abs/2606.17269" }, { "arxiv_id": "2606.17266", "authors": [ "Carlos Eduardo Sanoja" ], "id": "arxiv:2606.17266", "kind": "paper", "label": "negative", "matched_signals": [], "published": "2026-06-15", "score": 0.5, "source": "arxiv-ai4math-core", "summary": "Production planning increasingly has to treat workforce capability as a decision variable: certifications lapse when skills are not maintained, new products require skills the current workforce does not hold, and reskilling competes for the same worker hours needed for production. Existing operations benchmarks usually treat labor as exogenous, while workforce-planning models with skills and learning are rarely released as reusable testbeds. We introduce SkillChain-Gym, a benchmark specification for reskilling-aware production-inventory control: a single-site environment with stylized worker skill-state dynamics, hard threshold certification, forgetting, and capacity-consuming training actions constrained by the same per-worker time budget as production. The benchmark includes seed-controlled disruption scenarios, three feasibility modes with projection diagnostics, deterministic replay, and metrics covering operations, resilience, capability growth, and training-access distribution. We evaluate production-only, reactive adaptive, water-filling adaptive, and static-insurance policies with budget variants over 60-shift horizons with paired statistical tests. The results are regime-dependent rather than a ranking. Training-capable policies dominate the production-only baseline, and maintenance training is necessary under forgetting even without disruptions. Among training-capable classes, adaptive training helps when bottlenecks are visible in the forecast, while a lean static cross-training plan, a deliberately favorable comparator whose structure encodes relevant skill contingencies, acts as strong insurance under surprise shocks and absenteeism. Capacity slack and the forgetting rate govern the boundary between these regimes. No policy class dominates across regimes, motivating forecast-driven controllers that decide when to buy skill insurance and when to react.", "title": "SkillChain-Gym: A Benchmark for Reskilling-Aware Production-Inventory Control under Disruptions", "updated": "2026-06-15", "url": "https://arxiv.org/abs/2606.17266" }, { "arxiv_id": "2606.18292", "authors": [ "Rafał Komendarczyk", "Walter Block", "John Levendis", "Frank Tipler" ], "id": "arxiv:2606.18292", "kind": "paper", "label": "negative", "matched_signals": [], "published": "2026-06-15", "score": 0.5, "source": "arxiv-ai4math-core", "summary": "This paper presents an axiomatization of Ludwig von Mises' praxeology in many-sorted first-order logic, isolating the foundational layer. We introduce a formal language with five sorts ({\\sf Actors}, {\\sf Actions}, {\\sf Ends}, {\\sf Things}, {\\sf Times}) and six primitive relations ({\\em Acts}, {\\em Avail}, {\\em EndOf}, {\\em Use}, a preference order, and a time order), together with a base axiom system organised into three layers: the structure of action itself, the actor's preference order together with its revelation in choice, and material scarcity. The base system captures purposeful action in its bare praxeological form. Working entirely within the base system we derive the core classical Misesian propositions as Hilbert-style theorems: the asymmetry of revealed preference, the existence of opportunity cost, the structural scarcity of time, the subjectivity of opportunity cost, the law of diminishing marginal utility, and the increasing marginal disutility of labour. Where a theorem requires structure beyond the praxeological core -- as with diminishing marginal utility -- the additional premises are made explicit; identifying these hidden premises is one of the methodological payoffs of the approach. A self-contained {\\em Lean} companion encodes the language as {\\em Lean} type classes and constructs concrete models -- a three-period Robinson Crusoe economy and its infinite-time extension -- whose acceptance by the type-checker is a constructive consistency proof of the full base theory.", "title": "A Formalization of Austrian Economics. Praxeological Foundations: The Base System and Its Derived Theorems", "updated": "2026-06-15", "url": "https://arxiv.org/abs/2606.18292" }, { "arxiv_id": "2606.16239", "authors": [ "Evan Chen", "Ken Ono" ], "id": "arxiv:2606.16239", "kind": "paper", "label": "negative", "matched_signals": [], "published": "2026-06-15", "score": 0.5, "source": "arxiv-ai4math-core", "summary": "In his 2009 paper, Thakur posed three conjectural hypotheses for the degrees of the power sums \\[ S_d(k)=\\sum_{\\substack{a\\in \\mathbb F_q[t] \\text{ monic}\\\\ °a=d}} a^{-k} \\qquad\\text{and}\\qquad s_d(k)=-°_t S_d(k). \\] For prime fields $q=p$, we prove Hypotheses H1 and H2, giving a unique greedy description of the extremal term in Carlitz's formula and establishing the recursion \\[ s_d(k)=s_{d-1}(s_1(k))+s_1(k). \\] As consequences, the prime-field recursion gives the strict Newton-polygon convexity used in the prime-field Carlitz-Goss Riemann-hypothesis theorem, and it recovers Thakur's nonvanishing theorem for positive multizeta values over $\\mathbb F_p[t]$. We also prove Hypothesis H3 for all finite fields $q=p^f$, establishing the monotonicity \\[ s_d(k)