{ "counts": { "adjacent": 13, "core": 4, "errors": 0, "negative": 73, "total": 90 }, "date": "2026-08-25", "errors": [], "fresh_content_days": 21, "generated_at": "2026-08-25T16:09:36Z", "items": [ { "age_days": 1, "arxiv_id": "2608.23281", "authors": [ "Ludovic Tagnon" ], "content_date": "2026-08-24", "freshness": "fresh", "id": "arxiv:2608.23281", "kind": "paper", "label": "core", "matched_signals": [ "lean_formal_proving_agents", "verifier_guided_reasoning" ], "published": "2026-08-24", "score": 6.5, "source": "arxiv-ai4math-core", "summary": "Hermite asked in 1848 for a representation of real numbers whose eventual periodicity characterizes cubic irrationals. The totally real case was solved by Karpenkov's $\\sin^2$-algorithm; the complex case, signature (1,1), is his Problem 4. We study a deterministic algorithm implementing his suggested analytic extension: the score expression is strictly negative on (1,1) data (closed form proved), the most negative score is selected, and exact score ties are resolved by a declared ordering. On a sample of 205 complex cubic polynomials, every run closes projectively with an exact unit certificate, each transition certified by exact comparisons in $\\mathbb{Q}(α)$. An exhaustive campaign over the full box $[-3,3]^3$ closes 194/194. Across 457 deformed bases, the terminal cycle is an invariant of the marked lattice. Certified finite transition graphs are computed for four fields; the plastic case is machine-checked in Lean 4, kernel-only. All data ship in a public archive with a portable verifier.", "title": "A deterministic sin^2-type algorithm for complex cubic irrationalities with exact periodicity certificates", "updated": "2026-08-24", "url": "https://arxiv.org/abs/2608.23281" }, { "age_days": 4, "arxiv_id": "2608.20793", "authors": [ "Sihai Jin" ], "content_date": "2026-08-21", "freshness": "fresh", "id": "arxiv:2608.20793", "kind": "paper", "label": "core", "matched_signals": [ "lean_formal_proving_agents", "verifier_guided_reasoning" ], "published": "2026-08-21", "score": 6.5, "source": "arxiv-ai4math-core", "summary": "We report a Lean 4 formal verification accompanying the paper \"Subregular Affine Cells and the Level -1 Vertex Algebra of Type D\" (arXiv:2608.11997). The formalization kernel-checks substantial internal parts of the proof architecture, including the Section 4 membership/descent chain, the type-D norm-gap argument, zero-orbit energy and signed-permutation rigidity, node-weight and numerical rigidity calculations, the exhaustion logic, the passage to the candidate quotient classification, the simple-object count, the final additive Grothendieck-group comparison, and the coefficient-substitution layer of the uniform character formula. Higher representation-theoretic results whose foundational infrastructure is not presently constructed in the file are isolated as explicit semantic interfaces rather than introduced as Lean axioms. Thus the precise claim is a kernel-checked internal deduction from explicit representation-theoretic boundary inputs, not a from-scratch formalization of vertex algebras, BRST reduction, finite W-algebras, or affine Hecke theory inside Mathlib.", "title": "A Lean 4 Verification Report for Subregular Affine Cells and the Level $-1$ Vertex Algebra of Type $D$", "updated": "2026-08-21", "url": "https://arxiv.org/abs/2608.20793" }, { "age_days": 5, "arxiv_id": "2608.20432", "authors": [ "Shane Caldwell" ], "content_date": "2026-08-20", "freshness": "fresh", "id": "arxiv:2608.20432", "kind": "paper", "label": "core", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-08-20", "score": 6.5, "source": "arxiv-ai4math-core", "summary": "Formal proofs in Lean 4 that pass the kernel's type checker can nonetheless vary widely in quality. We introduce ProofJudge, an agentic LLM-as-judge system that scores formal proof quality along five dimensions beyond correctness: library leverage, automation fit, structural clarity, statement quality, and Mathlib conventions. We evaluate ProofJudge on a novel dataset of 218 declarations drawn from distinct Mathlib PRs. The judge agent is grounded by tool access to the commit the PR is applied to, enabling it to query the library state when scoring. A judge is considered aligned with human preferences when it rates the version of the PR Mathlib accepted above the initial version that was sent back for revision. All six judge models evaluated recover the reviewers' preference well above chance, from 80.8% to 63.5%, and two open-weight judges reach roughly 70% at a tenth of the best judge's cost. We release the judge harness, evaluation dataset, and evaluation traces as open-source artifacts to support further research.", "title": "ProofJudge: Tool-Grounded LLM Evaluation of Formal Proof Quality in Mathlib", "updated": "2026-08-20", "url": "https://arxiv.org/abs/2608.20432" }, { "age_days": 6, "arxiv_id": "2608.19009", "authors": [ "Yajie Yin" ], "content_date": "2026-08-19", "freshness": "fresh", "id": "arxiv:2608.19009", "kind": "paper", "label": "core", "matched_signals": [ "lean_formal_proving_agents", "verifier_guided_reasoning" ], "published": "2026-08-19", "score": 6.5, "source": "arxiv-ai4math-core", "summary": "Large language models (LLMs) are increasingly paired with verifiers (step checkers, self-consistency filters, tool-based fact checkers, formal proof assistants) that claim to detect the model's errors. Yet the verification literature uses the word \"level\" to mean at least five different things: verification granularity, concept abstraction, risk tier, system-stack layer, and the epistemic source of the ground truth. We propose Verification Autonomy Levels (VAL), a meta-standard that classifies any verification scheme along a single axis: where does the verification spec come from, and what does the verdict guarantee? VAL ranges from L0 (LLM self-declaration; no deterministic anchor) through L2 (objective ground truth; correctness only) to L3/L4 (decidable systems with single-property or domain-level completeness), with L5 impossible in the unrestricted case. Central to VAL is the completeness blind spot: substitution- and sampling-based verifiers can confirm that proposed candidates hold, but cannot prove that no candidate was missed. We further identify a dichotomy the literature has not stated: completeness is reachable only for formally specifiable properties, whereas empirical open-world verification (fact-checking, diagnosis) caps at anchored correctness (L2). We document this gap empirically across four domains (symbolic mathematics, behavior monitoring, medical diagnosis, and code generation, the last a reverse validation with predictions stated before evidence) and in the strongest formal-verification baseline in our survey, whose authors note the verifier focuses on the correctness of each step. We show the levels of granularity, concept hierarchy, risk, and system stack are orthogonal to VAL, resolving a systematic conflation across 17 surveyed papers. Code and full assessment are released as supplementary material.", "title": "Grading the Graders: Verification Autonomy Levels (L0-L5) for LLM Reasoning", "updated": "2026-08-20", "url": "https://arxiv.org/abs/2608.19009" }, { "age_days": 6, "arxiv_id": "2608.19093", "authors": [ "Mengwei Su", "Kaiwen Yang", "Hao Xu", "Chih-Lin I" ], "content_date": "2026-08-19", "freshness": "fresh", "id": "arxiv:2608.19093", "kind": "paper", "label": "adjacent", "matched_signals": [ "general_ai_math_reasoning", "lean_formal_proving_agents" ], "published": "2026-08-19", "score": 4.9, "source": "arxiv-ai4math-core", "summary": "Among $n+1$ equiprobable equal-energy signals in $\\R^n$ under additive white Gaussian noise with maximum-likelihood decoding, which arrangement maximizes the probability of correct decoding? The question is Shannon's, recorded by Rice in 1950. Mulgund proved in 2026 that the regular-simplex value bounds the correct-decoding probability of every signal set at every signal-to-noise ratio, leaving open whether the simplex is the only maximizer. This paper determines the equality cases in a form stronger than uniqueness. A signal set other than a regular simplex falls strictly below the bound at every positive signal-to-noise ratio. Hence a code meeting the bound at one positive operating point is already a regular simplex, up to vertex relabeling and an orthogonal map. In probabilistic form, among the correlation matrices that signal sets induce, any matrix other than the identity gives a lower-orthant probability strictly above its independent counterpart at every finite threshold, leaving no room for a nontrivial equality. No code of ambient dimension below $n$ attains the bound. Under an energy budget $E$ with unrestricted blocklength the optimal codebook is uniquely the regular simplex of circumradius $\\sqrt{E}$. Every optimal codeword therefore exhausts its allowance. Equality in the Simplex Mean Width Conjecture likewise occurs only at the regular simplex. The proof strengthens the first self-convolution step of Mulgund's argument with Royen's correlation theorem. The single-parameter rigidity is machine-checked in Lean 4.", "title": "The Equality Cases of the Weak Simplex Conjecture", "updated": "2026-08-19", "url": "https://arxiv.org/abs/2608.19093" }, { "age_days": 1, "arxiv_id": "2608.23247", "authors": [ "Ryuta Arisaka", "Hirotaka Ono" ], "content_date": "2026-08-24", "freshness": "fresh", "id": "arxiv:2608.23247", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-08-24", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "We introduce multi-winner voting with argumentative ballots (MVArg) and investigate theoretical properties. As our conceptual contribution, we generalise approval ballots to argumentative ballots, thereby allowing voters to express defeasible preferences over candidates. We accordingly generalise voter cohesion and justified representation axioms JR, PJR and EJR. As our theoretical contribution, we establish several key results. First, MVArg is strictly more expressive than multi-winner voting with approval ballots (MV). Second, our notions of cohesion and justified representation are conservative generalisations of their counterparts in MV. Third, the MVArg counterpart of JR can always be satisfied, whereas the counterparts of PJR and EJR cannot always be. Fourth, although verifying whether a winner set satisfies the MVArg counterpart of JR is already coNP-hard, such a winner set can be constructed in polynomial time. All definitions, propositions, auxiliary lemmas and theorems have been formalised and mechanically checked in Lean 4.", "title": "Multi-Winner Voting with Argumentative Ballots", "updated": "2026-08-24", "url": "https://arxiv.org/abs/2608.23247" }, { "age_days": 3, "arxiv_id": "2608.21712", "authors": [ "Deyi Li", "Qi Xu", "Lingyao Li", "Tiansheng Wang", "Muxuan Liang", "Mei Liu" ], "content_date": "2026-08-22", "freshness": "fresh", "id": "arxiv:2608.21712", "kind": "paper", "label": "adjacent", "matched_signals": [ "verifier_guided_reasoning" ], "published": "2026-08-22", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "Transformer-based models are widely used for clinical prediction from electronic health records (EHRs), yet their architectures still require substantial manual tuning, and the optimal configuration may vary across tasks and hospitals. Neural architecture search (NAS) automates architecture design, but conventional methods are computationally costly for Transformer-based EHR models. Recent large language model (LLM)-guided NAS methods reduce manual search design but typically conduct each search independently, without reusing architecture knowledge across hospitals. In this study, we propose ATHENA (Agentic Transfer across Hospitals for EHR Neural Architecture Search), a knowledge-guided agentic NAS framework for Transformer-based EHR modeling. ATHENA uses a weight-sharing supernet that is pretrained once per hospital, allowing candidate architectures to be instantiated as inherited subnetworks and evaluated through fine-tuning rather than independent pretraining. It also incorporates a two-layer cross-hospital architecture prior. The first layer retrieves high-performing architecture examples from source sites based on task descriptors, while the second estimates the effects of architectural components using SHapley Additive exPlanations (SHAP)-based meta-regression. These priors guide a multi-agent LLM search together with validation feedback from the target hospital. Across six clinical prediction tasks and two independent health systems, ATHENA matches or outperforms four NAS baselines in 9 of 12 hospital-task evaluations at a search budget of 30. It also shows more consistent architecture selection across repeated searches. ATHENA provides a practical approach for reducing manual architecture tuning in Transformer-based EHR modeling. Code is publicly available at https://github.com/GatorAIM/ATHENA.", "title": "ATHENA: Knowledge-guided agentic neural architecture search for AutoFormer-based electronic health record modeling", "updated": "2026-08-22", "url": "https://arxiv.org/abs/2608.21712" }, { "age_days": 4, "arxiv_id": "2608.21356", "authors": [ "Jason Hickey" ], "content_date": "2026-08-21", "freshness": "fresh", "id": "arxiv:2608.21356", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-08-21", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "For sixty years, machine verification has been a major cost overhead, affordable only for exceptional artifacts. Here we report that generative AI inverts this relationship: at AI speed, machine verification is not only economical but essential to productivity --- it is the incorruptible referee that lets one person safely direct autonomous machine work at scale. In five weeks, one researcher on consumer AI subscriptions directed a small fleet of AI agents from application code, through a verified compiler and executive, to a RISC-V processor taped out on a community silicon shuttle; no proof passed through human review, and no RTL was written by a human. The working discipline --- the Salt method --- rests on a proof kernel no hallucinated proof can pass: mathematical claims travel between agents as kernel-checked artifacts, and human attention is reserved for statements, designs, and rulings. Verification is stated link by link, from the Lean 4 kernel to SAT-checked equivalence at the silicon boundary. We publish the complete accounting: theorem provenance, a pre-registered token meter, floor-bounded human time, and an error ledger whose catch numbering runs to #256 --- a monotone counter over the mathematics campaign's append-only flags ledger, maintained 2026-07-07 to 2026-07-20 (one number, #79, was never assigned; later catches are recorded un-numbered) --- against zero incorrect proofs reaching the record.", "title": "AI with Authority, from Application to Silicon", "updated": "2026-08-21", "url": "https://arxiv.org/abs/2608.21356" }, { "age_days": 5, "arxiv_id": "2608.20153", "authors": [ "Dingzirui Wang", "Xuanliang Zhang", "Keyan Xu", "Qingfu Zhu", "Wanxiang Che" ], "content_date": "2026-08-20", "freshness": "fresh", "id": "arxiv:2608.20153", "kind": "paper", "label": "adjacent", "matched_signals": [ "autoformalization" ], "published": "2026-08-20", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "Large language models (LLMs) have shown growing potential for automated theoretical computer science (TCS) research, yet existing benchmarks remain far from realistic research settings. We introduce \\ourbenchmark, an expert-validated benchmark for evaluating LLMs on frontier, end-to-end TCS research. \\ourbenchmark contains $175$ instances drawn from papers accepted to STOC, FOCS, SODA, and COLT in 2025-2026, preserving paper-specific definitions, assumptions, and proof dependencies, with expert-verified Lean formalizations and proofs. Evaluations of leading LLMs reveal that current models remain far from reliably completing the full research pipeline. In particular, autoformalization is the sharpest bottleneck: the best model achieves only $11.5$ on translating natural-language claims into formal theorem statements, compared with $28.6$ Pass@8 when proving human-provided formal statements. Building on \\ourbenchmark, we further develop an automated TCS research framework that generates, formalizes, filters, and proves new claims. Of $64$ generated claims, only $6$ ultimately pass expert evaluation and proof verification, indicating that beyond formalization, limited research taste remains another major barrier to autonomous TCS research.", "title": "FormalTCS: Benchmarking End-to-End Frontier Formal Theoretical Computer Science Research of Large Language Models", "updated": "2026-08-20", "url": "https://arxiv.org/abs/2608.20153" }, { "age_days": 5, "arxiv_id": "2608.19847", "authors": [ "Benqi Liu", "Kangkang Deng", "Zichen Wang", "Zaiwen Wen" ], "content_date": "2026-08-20", "freshness": "fresh", "id": "arxiv:2608.19847", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-08-20", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "Augmented Lagrangian methods are effective for nonlinear equality-constrained optimization, but solving their nonlinear primal subproblems can be expensive. For smooth nonconvex problems with deterministic or stochastic objectives, we propose a nonlinear-residual linearized augmented Lagrangian method (NR-LALM) that replaces this subproblem by a regularized Gauss-Newton-type step while retaining the classical multiplier update based on the nonlinear constraint residual. The resulting step is computed from one symmetric positive-definite linear system, but the mismatch between the linearized primal model and the nonlinear-residual update produces a quadratic constraint-linearization error in the multiplier identity. We show that this error can be controlled under local regularity; multiplier boundedness and trajectory localization are derived rather than assumed. With fixed, accuracy-independent parameters, deterministic NR-LALM finds an $\\varepsilon$-approximate Karush-Kuhn-Tucker (KKT) pair in $O(\\varepsilon^{-2})$ iterations and first-order oracle evaluations. For stochastic objectives, a projected stochastic path-integrated differential estimator with safeguarded restarts requires, in expectation, $O(\\varepsilon^{-3})$ stochastic-gradient evaluations and $O(\\varepsilon^{-2})$ constraint and Jacobian evaluations. Compactness and a Kurdyka-Lojasiewicz condition further yield finite-length convergence of the deterministic primal-dual sequence. An optional minimum-norm second-order correction reduces the constraint-linearization error from second to fourth order without changing the complexity orders. All theoretical results are formalized in Lean 4. Numerical experiments confirm the predicted error orders and show favorable performance on high-dimensional deterministic and stochastic problems.", "title": "A Fixed-Penalty Linearized Augmented Lagrangian Method with Classical Multiplier Updates", "updated": "2026-08-20", "url": "https://arxiv.org/abs/2608.19847" }, { "age_days": 6, "arxiv_id": "2608.19058", "authors": [ "Mahadee Al Mobin", "Md. Shariful Islam" ], "content_date": "2026-08-19", "freshness": "fresh", "id": "arxiv:2608.19058", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-08-19", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "We state and prove the Theorem: for primes $p < 10^n$ with base-$10$ expansion $p = \\sum_{k=0}^{n(p)-1} d_k(p) 10^k$, the positional digit probabilities $P_n(d \\mid k)$ satisfy \\[ \\lim_{n \\to \\infty} P_n(d \\mid k) = \\begin{cases} 1/10, & k \\ge 1,\\ d \\in \\{0,\\dots,9\\}, \\\\[4pt] 1/9, & k = \\mathrm{lead},\\ d \\in \\{1,\\dots,9\\}. \\end{cases} \\] The limiting behavior splits cleanly into two distinct mechanisms: an arithmetic regime for interior digits and an Archimedean regime for the leading digit. For fixed interior positions ($k \\ge 1$), digit extraction modulo $10^{k+1}$ reduces the problem to prime counts in reduced residue classes, where uniform distribution follows from Siegel--Walfisz (with Bombieri--Vinogradov allowing $k$ to grow with $n$). For the leading digit, the $1/9$ limit is not a Benford-type scale invariance, but arises from the local near-constancy of the prime density $1/\\log t$ within single decades combined with a Toeplitz-type error averaging. Explicit classical and conditional error bounds are recorded for both regimes.", "title": "The Position-wise Prime Digit Distribution Theorem: A Formal Proof of Position-wise Digit Equidistribution in the Prime Numbers", "updated": "2026-08-19", "url": "https://arxiv.org/abs/2608.19058" }, { "age_days": 6, "arxiv_id": "2608.18723", "authors": [ "Ghislain Dorian Tchuente Mondjo" ], "content_date": "2026-08-19", "freshness": "fresh", "id": "arxiv:2608.18723", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-08-19", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "Telecom operators traditionally offer predefined tariff grids, forcing users to choose from a limited set of plans. This paper proposes BFTR (Budget-First Tariff Recommendation), a complete algorithmic framework integrating eight Budget-First strategies, including two original hybrid approaches: Recursive Hybrid (conditional interpolation) and Knapsack-First Hybrid (priority knapsack). Unlike existing approaches that adjust prices upward to guarantee a minimum margin, BFTR guarantees the absence of overcharging by systematically aligning the final price with the catalog reference price. We mathematically formalize each strategy, prove the existence of an offer for any positive budget, and prove that the price deviation (surcharge) is zero for all strategies that do not use interpolation with correction. A detailed comparative analysis confronts BFTR to ten main existing tariff models on ten dimensions. Experiments on a dataset of 974 customers inspired by the Nigerian MTN market show that: (i) Recursive Hybrid is optimal for the customer (100% budget used, 29.9 GB volume, utility 0.946, 0% overcharging), (ii) Piecewise offers the highest volume (39.7 GB) with 0% overcharging, (iii) Power Law provides an excellent compromise (99.9% budget, 38.1 GB, 0% overcharging). All strategies achieve a zero surcharge, confirming the theoretical guarantees. A sensitivity analysis on the weighting parameter alpha (0.2 - volume priority, 0.5 - balance, 0.8 - budget priority) shows that utility rankings evolve logically. Execution times (< 10 ms) and very low failure rates (0% for robust strategies) confirm the operational viability of the system. The formal proof of the absence of overcharging constitutes a major theoretical contribution.", "title": "Budget-First Tariff Recommendation (BFTR): A Complete Algorithmic Framework for Telecom Plan Recommendation without Overcharging", "updated": "2026-08-19", "url": "https://arxiv.org/abs/2608.18723" }, { "age_days": 7, "arxiv_id": "2608.17741", "authors": [ "Olga Mashkova", "Asaad Mohammedsaleh", "Fernando Zhapa-Camacho", "Robert Hoehndorf" ], "content_date": "2026-08-18", "freshness": "fresh", "id": "arxiv:2608.17741", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-08-18", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "OWL 2 DL ontologies, grounded in the description logic $\\mathcal{SROIQ}$, express large knowledge bases in biomedicine and the Semantic Web. Neuro-symbolic (NeSy) learners over description logics either embed the ontology in a continuous space, abandoning classical entailment, or restrict to the Horn fragment $\\mathcal{EL}^{++}$, which has a single canonical model. We present Baobab, which compiles a $\\mathcal{SROIQ}$ ontology with a finite ABox into a Sentential Decision Diagram (SDD): it saturates a propositional core under a consequence-based calculus and instantiates the remaining $\\mathcal{SROIQ}$ features (nominals, number restrictions, and the role axioms) over the active domain. The SDD's evidence-conditioned weighted model count then trains a perception network to recognize real images under partial ABox supervision: on an ontology that exercises every distinctive $\\mathcal{SROIQ}$ feature, a CNN learns to read MNIST digits coupled by a successor relation and recovers latent ontology concepts that an independent perception leaves at chance. When the supervision admits several ontology-consistent completions, an independent perception collapses onto one, a reasoning shortcut: we show that a mixture indexed by the query's justifications can represent the calibrated posterior no independent perception can, and that seeding it from the circuit's enumerated completions attains the Bayes-optimal posterior on a real-image MNIST task where single-WMC and learned mixtures (the BEARS-ensemble hypothesis class) do not: to our knowledge the first to characterize and mitigate reasoning shortcuts in a non-Horn description logic. Soundness of the compiler and the representation result are machine-checked in Lean 4. Code is available at https://github.com/bio-ontology-research-group/baobab.", "title": "Neuro-symbolic learning over OWL 2 DL via consequence-based compilation to differentiable circuits", "updated": "2026-08-18", "url": "https://arxiv.org/abs/2608.17741" }, { "age_days": 7, "arxiv_id": "2608.17634", "authors": [ "Satpreet Makhija" ], "content_date": "2026-08-18", "freshness": "fresh", "id": "arxiv:2608.17634", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-08-18", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "The $\\operatorname{do}$-operator is described graphically by deleting arrows into its targets and functionally by replacing their mechanisms with constants. To call these operations equivalent is not yet a mathematical statement: one returns a graph and remembers only the targets, whereas the other returns mechanisms and also remembers the imposed values. We make a dependency-level comparison precise for deterministic acyclic structural causal models with finitely many endogenous variables. If $\\operatorname{Graph}(F)$ extracts the dependencies of a mechanism family $F$, our main theorem is $\\operatorname{Graph}(F^ι)=\\operatorname{Surg}(\\operatorname{Graph}(F),T_ι)$. Thus replacing target mechanisms removes exactly the dependencies removed by graph surgery. For a model $M=(G,F)$ whose graph may contain unused arrows, we characterize when the same equality holds with $G$ in place of $\\operatorname{Graph}(F)$; it holds for every intervention exactly when $G$ records the dependencies of $F$ exactly. We then define the intervened model, characterize its run, show how sequential interventions combine, and prove that an outcome depends only on interventions at its actual dependency ancestors. All principal results are machine-checked in an accompanying Lean 4 development.", "title": "Graph Surgery and the Do-Operator: A Precise Correspondence for Acyclic Structural Causal Models", "updated": "2026-08-20", "url": "https://arxiv.org/abs/2608.17634" }, { "age_days": 7, "arxiv_id": "2608.17880", "authors": [ "Ibrahim Mian", "Shayaan Siddique" ], "content_date": "2026-08-18", "freshness": "fresh", "id": "arxiv:2608.17880", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-08-18", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "Erdős problem 647 asks whether any $n > 24$ satisfies $\\max_{m isTopologicalGroup in names (#43027)", "updated": "2026-08-25", "url": "https://github.com/leanprover-community/mathlib4/commit/2d94cd3bf16b7be873a47af22be13d543f8eb6fe" }, { "age_days": 0, "authors": [ "Stefan Kebekus" ], "content_date": "2026-08-25", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:661ab0bdd381", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-08-25", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: chore: remove LLM from author line (#43098)", "updated": "2026-08-25", "url": "https://github.com/leanprover-community/mathlib4/commit/661ab0bdd38137bee88d60f44a8dd9ff1978a9c9" }, { "age_days": 0, "authors": [ "Michael Rothgang" ], "content_date": "2026-08-25", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:49eff4d8f7bf", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-08-25", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: chore: convenience constructor for `IsImmersionAtOfComplement` (#41646)", "updated": "2026-08-25", "url": "https://github.com/leanprover-community/mathlib4/commit/49eff4d8f7bf56485e211314a90729122e9324f8" }, { "age_days": 0, "authors": [ "mathlib-splicebot[bot]" ], "content_date": "2026-08-25", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:d8ffbd6ba089", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-08-25", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: chore(RingTheory/Polynomial/IntegralNormalization): automated extraction from #28013 (#43093)", "updated": "2026-08-25", "url": "https://github.com/leanprover-community/mathlib4/commit/d8ffbd6ba089ec684ea1570ef5523305666bd26c" }, { "age_days": 0, "authors": [ "Francesco Chotuck" ], "content_date": "2026-08-25", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:37876f96d706", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-08-25", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: chore(Order/Interval): define `Unique (Iic 0)` directly, drop a defeq option in Traj (#42739)", "updated": "2026-08-25", "url": "https://github.com/leanprover-community/mathlib4/commit/37876f96d7069323d1e5d1258e0e9dec844decd2" }, { "age_days": 0, "authors": [ "Moritz Doll" ], "content_date": "2026-08-25", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:c404bb7edd38", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-08-25", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: chore(Analysis/Normed): move `normSeminorm` into new file (#42887)", "updated": "2026-08-25", "url": "https://github.com/leanprover-community/mathlib4/commit/c404bb7edd38d35b2d8b8fd3bbf44ebe6b11caff" }, { "age_days": 0, "authors": [ "Bingyu Xia" ], "content_date": "2026-08-25", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:bfc23088f200", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-08-25", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: chore(Algebra/Order/Antidiag): relax an assumption in the definition of `HasAntidiagonal` (#41521)", "updated": "2026-08-25", "url": "https://github.com/leanprover-community/mathlib4/commit/bfc23088f200983e6192d7979f01abee7475ac08" }, { "age_days": 1, "authors": [ "Henrik Böving" ], "content_date": "2026-08-24", "freshness": "fresh", "id": "github:leanprover/lean4:7169e0500dda", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-08-24", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: perf: reduce aggressiveness of BitVec e-matching (#14905)", "updated": "2026-08-24", "url": "https://github.com/leanprover/lean4/commit/7169e0500ddaf19c1f0d79b2b4c23c854cf50d43" }, { "age_days": 1, "authors": [ "Henrik Böving" ], "content_date": "2026-08-24", "freshness": "fresh", "id": "github:leanprover/lean4:b464d0396fa1", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-08-24", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: perf: less aggressive e-matching for `List.Nodup.getElem_inj` (#14903)", "updated": "2026-08-24", "url": "https://github.com/leanprover/lean4/commit/b464d0396fa1400bce86f3e68c5bda601201b746" }, { "age_days": 1, "authors": [ "Henrik Böving" ], "content_date": "2026-08-24", "freshness": "fresh", "id": "github:leanprover/lean4:b182d58dfa3a", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-08-24", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: fix: mdata in mvcgen (#14889)", "updated": "2026-08-24", "url": "https://github.com/leanprover/lean4/commit/b182d58dfa3a2249a69ae5c5164907fb12524de7" }, { "age_days": 1, "authors": [ "Mac Malone" ], "content_date": "2026-08-24", "freshness": "fresh", "id": "github:leanprover/lean4:dd9cb93def1f", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-08-24", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: fix: lake: sync `--fail-fast` test via job dependencies (#14907)", "updated": "2026-08-24", "url": "https://github.com/leanprover/lean4/commit/dd9cb93def1f4678434f84afb72ee8378c0800e4" }, { "age_days": 1, "authors": [ "Mac Malone" ], "content_date": "2026-08-24", "freshness": "fresh", "id": "github:leanprover/lean4:04b67f75fbeb", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-08-24", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: fix: lake: `meta import` after `import all` (#14902)", "updated": "2026-08-24", "url": "https://github.com/leanprover/lean4/commit/04b67f75fbeba062e7285caa9d7b6bf2c1c72360" }, { "age_days": 1, "authors": [ "yhx-12243" ], "content_date": "2026-08-24", "freshness": "fresh", "id": "github:leanprover/lean4:19260e81d1ca", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-08-24", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: fix: add lawful `BEq` instances for `Sum` and expose the derived `BEq` (#14895)", "updated": "2026-08-24", "url": "https://github.com/leanprover/lean4/commit/19260e81d1ca2ea41aea9cb9946fccdfc7474a0b" }, { "age_days": 1, "authors": [ "Kim Morrison" ], "content_date": "2026-08-24", "freshness": "fresh", "id": "github:leanprover/lean4:8b6143a341d6", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-08-24", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: feat: add the recall command (#14899)", "updated": "2026-08-24", "url": "https://github.com/leanprover/lean4/commit/8b6143a341d6d5111e6e98e8e3745e1eb220bff2" }, { "age_days": 1, "authors": [ "Henrik Böving" ], "content_date": "2026-08-24", "freshness": "fresh", "id": "github:leanprover/lean4:5c6bce120d03", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-08-24", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: feat: add ground evaluation Nat.log2 support (#14915)", "updated": "2026-08-24", "url": "https://github.com/leanprover/lean4/commit/5c6bce120d03401245d25ae79a4bb0d6654a5df8" }, { "age_days": 1, "authors": [ "Wojciech Różowski" ], "content_date": "2026-08-24", "freshness": "fresh", "id": "github:leanprover/lean4:15e960e6c8ac", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-08-24", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: feat: add `monotonicity_by` clause to `coinductive` predicates (#14861)", "updated": "2026-08-24", "url": "https://github.com/leanprover/lean4/commit/15e960e6c8ac6d1829992b7a4e947d909b59a066" }, { "age_days": 1, "authors": [ "Henrik Böving" ], "content_date": "2026-08-24", "freshness": "fresh", "id": "github:leanprover/lean4:1dbcfb17854f", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-08-24", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: feat: add BitVec.ofNatClamp (#14916)", "updated": "2026-08-24", "url": "https://github.com/leanprover/lean4/commit/1dbcfb17854ffb313419ef05578c95fbd4136859" }, { "age_days": 1, "authors": [ "Lean stage0 autoupdater" ], "content_date": "2026-08-24", "freshness": "fresh", "id": "github:leanprover/lean4:71a4d9a3e573", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-08-24", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: chore: update stage0", "updated": "2026-08-24", "url": "https://github.com/leanprover/lean4/commit/71a4d9a3e573916b2bbcb49ad9e1ca62851b77da" }, { "age_days": 1, "authors": [ "Sebastian Ullrich" ], "content_date": "2026-08-24", "freshness": "fresh", "id": "github:leanprover/lean4:923390e2f934", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-08-24", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: chore: CI: reuse the Lake artifact cache in non-release builds (#14665)", "updated": "2026-08-24", "url": "https://github.com/leanprover/lean4/commit/923390e2f93471c98ac318f986f6a07186255f99" }, { "age_days": 3, "authors": [ "Sebastian Graf" ], "content_date": "2026-08-22", "freshness": "fresh", "id": "github:leanprover/lean4:e3b3b04971c2", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-08-22", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: fix: apply specs to goals with a registered WP instance in vcgen (#14883)", "updated": "2026-08-22", "url": "https://github.com/leanprover/lean4/commit/e3b3b04971c27058eb917160d9f2e8bb23296e72" }, { "age_days": 3, "authors": [ "Kyle Miller" ], "content_date": "2026-08-22", "freshness": "fresh", "id": "github:leanprover/lean4:47680c2e6283", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-08-22", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: fix: add missing terminfo on structure fields (#14834)", "updated": "2026-08-22", "url": "https://github.com/leanprover/lean4/commit/47680c2e62831da4daa01abe567b3e4fc509fbae" }, { "age_days": 3, "authors": [ "Dennj" ], "content_date": "2026-08-22", "freshness": "fresh", "id": "github:leanprover/lean4:fd0efc4306a7", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-08-22", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: feat: lake: add --fail-fast build option (#14797)", "updated": "2026-08-22", "url": "https://github.com/leanprover/lean4/commit/fd0efc4306a7773c2cd4e079ddaa907426d0f5da" }, { "age_days": 4, "authors": [ "Sebastian Graf" ], "content_date": "2026-08-21", "freshness": "fresh", "id": "github:leanprover/lean4:f6c7d68c7fc2", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-08-21", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: refactor: move `PredTrans` combinators into the `PredTrans` namespace (#14882)", "updated": "2026-08-21", "url": "https://github.com/leanprover/lean4/commit/f6c7d68c7fc27e3b60585f30740c7d141d3c5b36" }, { "age_days": 4, "authors": [ "Sebastian Graf" ], "content_date": "2026-08-21", "freshness": "fresh", "id": "github:leanprover/lean4:5adc41ebee85", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-08-21", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: feat: deprecate the `mvcgen` tactic (#14874)", "updated": "2026-08-21", "url": "https://github.com/leanprover/lean4/commit/5adc41ebee85e17f75c528168058be83e525d989" }, { "age_days": 4, "authors": [ "Kim Morrison" ], "content_date": "2026-08-21", "freshness": "fresh", "id": "github:leanprover/lean4:5377fdd2a337", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-08-21", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: chore: credit the running user in generated release notes (#14879)", "updated": "2026-08-21", "url": "https://github.com/leanprover/lean4/commit/5377fdd2a337d8faee05500d126070c1f9aceddb" }, { "age_days": 1, "arxiv_id": "2608.22928", "authors": [ "Yusheng Zheng", "Xiaoyu Song", "Yanpeng Hu", "Lebin Cheng", "Yuxi Huang", "Wei Zhang" ], "content_date": "2026-08-24", "freshness": "fresh", "id": "arxiv:2608.22928", "kind": "paper", "label": "negative", "matched_signals": [], "published": "2026-08-24", "score": 0.5, "source": "arxiv-ai4math-core", "summary": "Agent runtimes can Checkpoint an execution, Fork it, Restore a checkpoint, or Merge branches without restarting a task. We call these operations execution edits, with Checkpoint recording the current execution for later use and Fork, Restore, and Merge changing what the Agent will do next. An execution edit cannot undo an earlier authorization or a tool request already sent. An unsafe edit can therefore authorize the same tool action twice, discard a result the task still requires, or conflict with a call that began before the edit. The Agent is untrusted, so the runtime uses its execution record to determine which past actions an edit must account for and which required results it must preserve to keep the subsequent execution safe. Yet existing Agent systems support such operations without deriving what each edit must preserve from the running execution, whereas prior methods for computing safe behavior take that requirement as input. We give an algorithm that decides exactly whether an edit is safe. It returns all safe ways to continue, or proves that none exists. To make this decision, the algorithm lists every way the task can finish without violating policy. It removes any way that could make a still-required result impossible to finish later. If none remain, it returns a checkable proof that no safe implementation exists. Otherwise, the remaining ways describe exactly what the runtime may allow. Our formal results cover Checkpoint and the six forms of Fork, Restore, and Merge, together with extensions, atomic enforcement, and the information every exact checker needs. Lean mechanizes the finite checker and runtime invariant, and tests validate all six edit forms. The source code, Lean proofs, and executable tests are available in the public GitHub repository at https://github.com/eunomia-bpf/agent-check-restore-safety.", "title": "When Can Agents Safely Checkpoint, Fork, Restore, and Merge? Exact Checking for Execution Edits", "updated": "2026-08-24", "url": "https://arxiv.org/abs/2608.22928" }, { "age_days": 1, "arxiv_id": "2608.23413", "authors": [ "Seiichiro Kusuoka", "Shuta Nakajima" ], "content_date": "2026-08-24", "freshness": "fresh", "id": "arxiv:2608.23413", "kind": "paper", "label": "negative", "matched_signals": [], "published": "2026-08-24", "score": 0.5, "source": "arxiv-ai4math-core", "summary": "We consider the Sherrington--Kirkpatrick model with inverse temperature $β>0$ and deterministic external field $h>0$. Let $q$ be the replica-symmetric fixed point: $q=\\mathbb E{\\rm tanh}^2 (h+β\\sqrt q\\,Z)$, where $Z$ is a standard normal. We prove that, uniformly on compact subsets of the strict de Almeida--Thouless region: $β^2\\mathbb E{\\rm sech}^4(h+β\\sqrt{q} Z) <1,$ the overlap satisfies the concentration: $$ \\mathbb E\\langle (R_{12}-q)^2\\rangle=O(N^{-1}). $$ The result of the present paper provides an alternative proof of the replica-symmetric free energy formula in de Almeida--Thouless region recently established by Lopatto [arXiv:2604.11921]. Moreover, our approach yields the stronger conclusion of an explicit quantitative bound. As a consequence, we obtain an $O(N^{-1})$ replica-symmetric free-energy correction and identify the finite-volume replicon susceptibility. Our proof is self-contained and does not use the identification of the limiting free energy with the Parisi variational formula. The main result supersedes the corresponding result in our recent preprint arXiv:2607.23427, extending the replica-symmetric bounds to the strict de Almeida-Thouless region. However, we keep the previous preprint, since its argument is different and substantially simpler than the one given here.", "title": "A quantitative replica-symmetric bound of Sherrington--Kirkpatrick model in the entire de Almeida--Thouless region", "updated": "2026-08-24", "url": "https://arxiv.org/abs/2608.23413" }, { "age_days": 2, "arxiv_id": "2608.22550", "authors": [ "Jan Snellman" ], "content_date": "2026-08-23", "freshness": "fresh", "id": "arxiv:2608.22550", "kind": "paper", "label": "negative", "matched_signals": [], "published": "2026-08-23", "score": 0.5, "source": "arxiv-ai4math-core", "summary": "We consider families of lines that cover every point of the integer lattice $\\mathbf{Z}^2$ in the plane, subject to the constraint that no two lines of different direction in the family meet at a lattice point. Restricting to \\emph{lattice lines} (lines containing at least two, hence infinitely many, lattice points, equivalently of rational direction), we show that the set of directions occurring in such a covering can be made dense in the space of line directions. The construction is a recursive splitting of $\\mathbf{Z}^2$ into nested rank-2 sublattice cosets, each handed off to a freshly chosen direction; the key technical point is a steering lemma showing that at every stage of the recursion a new direction arbitrarily close to any prescribed target can still be realized, via an elementary sieve bound.", "title": "On the directions occurring in lattice-line coverings of the integer plane", "updated": "2026-08-23", "url": "https://arxiv.org/abs/2608.22550" }, { "age_days": 3, "arxiv_id": "2608.22117", "authors": [ "Joshua Nunley" ], "content_date": "2026-08-22", "freshness": "fresh", "id": "arxiv:2608.22117", "kind": "paper", "label": "negative", "matched_signals": [], "published": "2026-08-22", "score": 0.5, "source": "arxiv-ai4math-core", "summary": "A standard Transformer block separates cross-token interaction in self-attention from a nonlinear feed-forward network applied independently at each position. We introduce the TANGO model (Token-Aggregated Nonlinear Gating Operators), which replaces these two sublayers with one cross-token gated residual update. Each source token produces a SwiGLU gate vector. Query-key similarities determine a weighted average of source gates for each destination, and the resulting gate rescales projected destination features. TANGO assigns a separate weight to every causally visible source and is quadratic in sequence length. The WANGO model (Windowed Aggregation of Nonlinear Gating Operators) retains the same unnormalized scores within a recent window and uses positive feature-map prefix statistics for older sources, giving linear sequence-length complexity for fixed window and feature dimensions. We compare TANGO and WANGO with Recurrent and Untied Transformer++, full-attention GAU, and FLASH. All models have approximately 44.3M nonembedding parameters and are trained in three matched runs. TANGO, WANGO, and Recurrent Transformer++ apply one shared block four times; the other architectures use four independent blocks. TANGO obtains the lowest mean validation negative log-likelihood on FineWeb-Edu, Lean, and DeepMind Mathematics, although it has the largest analytical forward-pass operation count. WANGO obtains the lowest mean FineWeb-Edu NLL among the architectures with computation linear in sequence length and outperforms Recurrent Transformer++ at nearly the same analytical forward-pass multiply-accumulate count.", "title": "TANGO: Token-Aggregated Nonlinear Gating Operators for Natural and Formal Language Modeling", "updated": "2026-08-22", "url": "https://arxiv.org/abs/2608.22117" }, { "age_days": 4, "arxiv_id": "2608.21502", "authors": [ "Bennett Chow", "Yuan Liao", "Ziyang Qin" ], "content_date": "2026-08-21", "freshness": "fresh", "id": "arxiv:2608.21502", "kind": "paper", "label": "negative", "matched_signals": [], "published": "2026-08-21", "score": 0.5, "source": "arxiv-ai4math-core", "summary": "We describe a Lean formalization of Hamilton's 1982 theorem on closed, connected three-manifolds with positive Ricci curvature. The development contains a short-time existence theorem for Ricci flow and substantial geometric-analysis infrastructure: Riemannian tensor calculus, the Levi--Civita connection, Ricci-flow evolution equations, scalar and tensor maximum principles, three-dimensional curvature algebra, preservation of Ricci pinching, and Hamilton's improved pinching estimate. The formalization follows an alternative blow-up route, rather than Hamilton's original normalized-flow proof. Its time-uniform short-time existence, maximal continuation, no-local-collapsing, and Cheeger--Gromov--Hamilton compactness pipelines have been formalized and are included in the artifact, while we give only a brief account of these companion developments and record the interfaces and consequences used by the Hamilton argument; a detailed exposition of their full constructions is deferred to the second author's forthcoming thesis. We interweave representative Lean declarations with their mathematical meaning and record the status and provenance of every major component. All source-level status claims are tied to the source release identified below.", "title": "A Lean Formalization of Hamilton's Three-Manifold Theorem", "updated": "2026-08-21", "url": "https://arxiv.org/abs/2608.21502" }, { "age_days": 5, "arxiv_id": "2608.21458", "authors": [ "Hiroyuki Nakahata" ], "content_date": "2026-08-20", "freshness": "fresh", "id": "arxiv:2608.21458", "kind": "paper", "label": "negative", "matched_signals": [], "published": "2026-08-20", "score": 0.5, "source": "arxiv-ai4math-core", "summary": "In a software architecture, each service can obey its own conventions and each handoff between adjacent services can hold, and yet a semantic inconsistency may remain that appears only on a full traversal of the system. This paper independently constructs two cohomologies measuring this gap between local and global correctness, and proves that they agree. The first construction speaks the language of repair: from the semantic repair options admitted in each local context and their equivalence relation, it generates the coefficient $M_{\\mathrm{sem}}$. The second speaks the language of equations: it organizes the constraints of the architecture as a simultaneous equation system and generates the quotient coefficient $Q_E$ by its obstruction ideal. Over a selected finite cover $U$ in Algebraic Architecture Theory (AAT), which constructs software architecture as algebraic geometry, and under finitely many selection conditions matching the local data, the comparison map induces the isomorphism $H^1_{\\mathrm{sem}}(U) \\cong \\check{H}^1(U, Q_E)$ together with a correspondence of residual classes. We call this the SAGA comparison theorem. The obstructions measured in the two languages are the same cohomology class, so semantic diagnosis and geometric computation translate into each other. Moreover, when the family of repair states satisfies the sheaf condition, a global repair exists if and only if the obstruction class vanishes on both sides. The paper presents this result in three layers: the mathematical proof; the Lean formalization status at release time; and a diagnosis in which the measurement tool ArchSig, on a real open-source microservice system, reproducibly walks the full circle from a measured nonzero obstruction to its disappearance after repair. The three layers refer to the same release identity, and each claim is connected to primary evidence.", "title": "SAGA: A Comparison Theorem for Local-to-Global Software Architecture - From Semantic Repair Cohomology to Algebraic-Geometric Descent", "updated": "2026-08-20", "url": "https://arxiv.org/abs/2608.21458" }, { "age_days": 5, "arxiv_id": "2608.23237", "authors": [ "Simon Daniel", "Timon Böhler", "David Richter", "Pascal Weisenburger", "Mira Mezini" ], "content_date": "2026-08-20", "freshness": "fresh", "id": "arxiv:2608.23237", "kind": "paper", "label": "negative", "matched_signals": [], "published": "2026-08-20", "score": 0.5, "source": "arxiv-ai4math-core", "summary": "With growing complexity, distributed software systems become increasingly challenging to maintain and reason about. When implementing a distributed protocol, developers must ensure manually that the different components fit together. Choreographic programming addresses this challenge by specifying global protocols in a single program and projecting them into communicating processes, so-called endpoints. Recent choreographic approaches are designed as programming libraries that embed this paradigm into a host language like Haskell or Rust. In these designs, we observe common cases of partiality: unreachable branches in endpoint projection (EPP) and located-value access can trigger runtime errors or undefined behavior, relying on manual discipline of library maintainers rather than being statically type-checked. Also, some programs require users to write down dummy branches that should not be reachable, for example when branching on sum types. To close this gap, we use the dependently typed Lean programming language to implement a similar choreographic library. We show how we are able to move from a partial EPP to a total EPP function, and also eliminate cases of partiality in user-written code with pattern matching on sum types. ChorLean ensures total EPP and safe value access via proof-carrying located values, passing Lean's totality checker without undefined cases, while supporting the same feature set as libraries like MultiChor.", "title": "On Eliminating the Impossible with Dependent Types: Choreographic Libraries with Proof-Carrying Located Values", "updated": "2026-08-20", "url": "https://arxiv.org/abs/2608.23237" }, { "age_days": 5, "arxiv_id": "2608.19844", "authors": [ "Liyan Wang" ], "content_date": "2026-08-20", "freshness": "fresh", "id": "arxiv:2608.19844", "kind": "paper", "label": "negative", "matched_signals": [], "published": "2026-08-20", "score": 0.5, "source": "arxiv-ai4math-core", "summary": "For $n\\geq 4$, let $c(n)$ be the minimum number of distinct circles containing at least three points of an $n$-point set in the Euclidean plane, where the set is neither collinear nor concyclic. Put[F(n)=1+\\binom{n-1}{2}-\\left\\lfloor\\frac{n-1}{2}\\right\\rfloor.]We determine $c(n)$ for every $n\\geq 4$: it equals $F(n)$ apart from three exceptional orders. We also solve the variant in which no three points are collinear; that variant has a single exceptional order. The proofs and exact finite verifications were developed through a collaboration between human researchers and artificial-intelligence systems.", "title": "Circles determined by planar point sets", "updated": "2026-08-20", "url": "https://arxiv.org/abs/2608.19844" }, { "age_days": 5, "arxiv_id": "2608.20589", "authors": [ "Andrey Itkin" ], "content_date": "2026-08-20", "freshness": "fresh", "id": "arxiv:2608.20589", "kind": "paper", "label": "negative", "matched_signals": [], "published": "2026-08-20", "score": 0.5, "source": "arxiv-ai4math-core", "summary": "The Marketron model of \\cite{HalperinItkin2025Mark} and its option pricing extension in \\cite{HalperinItkinMarketron2} suffer from structural non-identifiability: an eighteen-parameter space traps solvers in suboptimal local minima and renders economic quantities unmeasurable. By removing exact scaling gauges and sign symmetries, freezing non-financial parameters by explicit criteria, and adiabatically eliminating the fast hidden signal, we derive a robust nine-parameter reduced model. A Gauss-Newton Hessian with empty null space and a manifold-boundary analysis confirm that the reduced core carries no exact symmetry and admits no further reduction. A diffusive correlation between flow and return innovations captures the short-maturity skew. A staged calibration from the physical measure to the risk-neutral measure, illustrated on SPX options, fits the whole surface with a single parameter set. The same reduction turns the wedge between the physical and pricing values of the flow block into a well-defined market price of flow risk rather than a ridge artifact, identifiable here for the first time, though a single surface constrains its level only weakly. Finally, our analysis reveals that in the Marketron model the log-price obeys a generalized Langevin equation with a closed-form, state-modulated memory kernel, and that the memory variable itself is the exact Markovian lift of this kernel. This mapping also yields a testable condition, the equality of the signal and memory relaxation rates, which on the SPX surface come out well separated, though both weakly identified, placing the fitted market tentatively in the driven, non-equilibrium regime and turning the active-matter reading from an analogy into a falsifiable constraint.", "title": "Calibrating Inelastic Markets to Options: The Lean Marketron and the Generalized Langevin Equation", "updated": "2026-08-20", "url": "https://arxiv.org/abs/2608.20589" }, { "age_days": 6, "arxiv_id": "2608.18999", "authors": [ "Danila Litvinov" ], "content_date": "2026-08-19", "freshness": "fresh", "id": "arxiv:2608.18999", "kind": "paper", "label": "negative", "matched_signals": [], "published": "2026-08-19", "score": 0.5, "source": "arxiv-ai4math-core", "summary": "Radial clipping does two things at once: it removes the part of a vector outside a ball and retains a bounded vector inside the ball. The removed part produces bias, while the squared norm of the retained part produces energy. We determine the exact joint price of these effects when only a $p$-moment, $1