prove well-posedness

This work addresses initial/boundary-value problems for differential equations lacking rigorous existence-uniqueness theory. We propose a data-driven framework for well-posedness assessment, diverging from classical theoretical analysis that relies on exact boundary conditions. Instead, our method leverages sparse, non-boundary, and multi-solution observational data—common in real-world measurements—and introduces the first integration of data assimilation with implicit operator learning. By synergistically combining manifold learning, supervised and self-supervised learning, and uncertainty-aware regression, the framework enables learnable inference of existence, uniqueness, and stability of solutions. Evaluated across multiple canonical PDE settings, it successfully identifies well-posed regimes and reduces estimation error by 37% compared to conventional heuristic criteria. Results demonstrate the feasibility, robustness, and generalizability of data-driven well-posedness evaluation.

This study addresses the stability of the solution operator with respect to perturbations in the input parameter distribution within the framework of nonparametric Bayesian computer model calibration. By integrating nonparametric Bayesian inference, weak convergence theory of probability measures, and total variation metric analysis, the work establishes—for the first time—a systematic continuity theory for the solution operator in this calibration setting. The primary contributions include proving the uniform continuity of the solution operator under the total variation metric and demonstrating its continuity under the weak topology for a broad class of prior distributions. These results provide a rigorous theoretical foundation for the robustness of nonparametric Bayesian calibration methods in complex scientific applications.

This work addresses the construction of most general solutions for parameterized constraint formulas of the form ∃x₁…∃xₙ φ(x₁,…,xₙ,y₁,…,yₘ) within theories 𝒯 that admit elimination of specific existential quantifiers, where φ is a quantifier-free conjunction of literals and the yᵢ are parameters. By introducing conditional function symbols that capture “if-then-else” constructs, the authors generalize existing results on the existence of most general unifiers in discriminator clusters. Integrating parameterized constraint solving with algebraic semantic characterizations, they establish a unified framework for constructing most general solutions. This approach substantially broadens the scope of applicability compared to prior methods, and its effectiveness and generality are demonstrated through illustrative examples.

Existing mathematical AI benchmarks suffer from three key limitations: narrow coverage of mathematical complexity, neglect of proof motivation and reasoning processes, and evaluation distortion due to Goodhart’s Law. To address these, we propose a “process-oriented” paradigm for modeling mathematical competence, organizing data around *mathematical workflows*—structured sequences of reasoning steps—and integrating Polya’s principle of *motivated proofs*. Our contributions are threefold: (1) the first computable, formalized mathematical workflow data structure; (2) a proof representation schema supporting explicit motivation annotation and stepwise chain-of-thought tracing; and (3) a domain-specific Datasheet protocol for rigorous, transparent benchmarking. This framework significantly enhances models’ understanding of proof discovery trajectories and underlying cognitive logic. Empirically, it enables standardized, reproducible training and evaluation of mathematical AI systems, establishing a foundational infrastructure for next-generation mathematical reasoning research.

Traditional control theory neglects computational uncertainty—such as mathematical object distortion induced by finite-precision arithmetic—leading to reliability gaps between Lyapunov stability analysis and digital controller implementation. Methodologically, this paper introduces the first constructive control framework that explicitly treats computational uncertainty as an independent modeling dimension in controller synthesis and system analysis. Leveraging tools from computability theory, constructive analysis, and measurable selection, we establish a constructive Danskin theorem and provide computable reconstructions of fundamental objects—including control Lyapunov functions (CLFs), Carathéodory trajectories, and eigenvalue problems. Our primary contribution is a computationally feasible paradigm for stability and stabilization proofs: all mathematical constructs are uniformly approximable by finite-precision algorithms while rigorously preserving required properties. This ensures robustness and implementability of digital controllers under realistic computational constraints.

This work proposes a fully automated, interpretable novelty metric for Lean proof paths that requires no human annotations or technical ontologies, objectively quantifying their deviation from historical mathematical knowledge. The method analyzes the dependency footprint of proof terms, constructs a temporally anchored hierarchical smoothing prior using early Mathlib snapshots, and computes a weighted surprisal score to assess novelty. Experimental evaluation demonstrates strong alignment with expert judgment, achieving 69.7% agreement on 76 blind-rated pairs and 84.2% accuracy in the subset with maximal score separation, thereby validating both the reliability and endpoint calibration of the proposed measure.

This work addresses the efficient computation of fixed points for specific variables in systems of equations over Noetherian partially ordered sets with a bottom element—a problem commonly arising in program verification. The paper proposes a local fixed-point algorithm based on a dependency oracle that dynamically identifies variable dependencies and explores only the subsystem influencing the target variable. By leveraging the Noetherian structure, the method ensures sound termination guarantees. The designed dependency oracle supports customization, composition, and approximation, enabling flexible trade-offs between precision and performance while preserving correctness. Experimental evaluation demonstrates that a prototype implementation outperforms existing approaches across multiple scenarios, offering both superior efficiency and a clean, adaptable architecture suitable for diverse application domains.

This work addresses the absence of a formally verified foundational library in mathematical finance, which has hindered rigorous and reusable theoretical development. Building upon Mathlib and the BrownianMotion package in Lean 4, the authors construct a comprehensive formal library spanning eleven core areas, including continuous-time stochastic calculus, derivative pricing, and risk and portfolio theory. The library comprises over 200 theorems proved without gaps in assumptions. Notably, it presents the first formal construction of the L² Itô integral and derives the risk-neutral measure within a proof assistant. Additionally, a fidelity auditing mechanism is introduced to explicitly track the axioms and assumptions underlying each theorem. This effort establishes the most extensive machine-verified infrastructure for mathematical finance to date, enabling unified certification and reliable reuse of classical results.

This work addresses the challenge of verifying mathematical proofs generated by large language models by formally encoding, for the first time, an entire advanced undergraduate probability textbook—including its measure-theoretic foundations—into Lean. To bridge the semantic gap between the textbook’s exposition and the abstract formalism of the Mathlib library, the authors introduce an “interface lemma” strategy. Combined with structured proof engineering and formalization techniques specific to measure theory, this approach yields a reusable, machine-verifiable infrastructure spanning fourteen textbook chapters. The resulting formalization not only provides rigorous verification of all stated theorems and explicit articulation of their assumptions but also establishes a robust foundation for reliable AI-assisted mathematics, educational applications, and future formalization efforts in probability theory.