p-adic analysis

This study investigates the computational complexity of $p$-adic optimization problems beyond the binary ($p=2$) case. By generalizing Baker’s forcing method to arbitrary primes $p$, the authors establish a reduction framework linking integer optimization to $p$-adic optimization. Leveraging this framework, they provide the first unified proof that several prominent $p$-adic models—including $p$-adic linear regression, 2-adic dynamic neural networks, and various van der Put–based $p$-adic neural network architectures—are all NP-hard. This work bridges $p$-adic analysis, computational complexity theory, and neural network modeling, significantly extending the applicability of forcing techniques and offering a foundational complexity-theoretic characterization for $p$-adic machine learning.

Prior to this work, no formalization of the Mason–Stothers theorem—a polynomial analogue of the ABC conjecture—or its key diophantine consequences existed in Lean 4 or mathlib4. Method: Following Snyder’s elementary proof, we developed a verified foundation of polynomial number theory in Lean 4, systematically comparing and back-porting results from Isabelle and Lean 3 formalizations. Contribution/Results: We present the first complete formalization of the Mason–Stothers theorem and three core corollaries in Lean 4/mathlib4: unsolvability of the polynomial Fermat–Cartan equation, non-parametrizability of a specific elliptic curve, and Davenport’s theorem. All proofs are machine-checked, fully integrated into the mathlib4 main branch, and publicly available under an open-source license. This work fills a foundational gap in polynomial ABC-type inequality formalization within Lean 4 and substantially enhances the trustworthiness and reusability of mechanized reasoning for polynomial Diophantine problems.

This study investigates the arithmetic properties of hypergeometric functions over the p-adic numbers, with a focus on their p-adic valuations and reduction behavior modulo primes. Building upon Christol’s theorem and integrating p-adic analysis with algebraic algorithms, the work achieves the first exact computation of p-adic valuations within arbitrary disks of convergence and establishes a systematic, effective criterion for determining the mod-p reducibility of hypergeometric functions. Furthermore, it introduces an algorithm to construct annihilating polynomials for the reductions modulo p. These contributions provide practical computational tools for the theory of arithmetic D-modules and significantly advance the algorithmic understanding of the arithmetic properties of hypergeometric functions.

This paper addresses fundamental open problems in computational complexity—such as P vs NP and the nonexistence of polynomial-size circuits for SAT—within weak formal systems like bounded arithmetic $S^1_2$. Using a synthesis of proof complexity, model theory, recursion theory, and propositional logic simulation techniques, it establishes, for the first time, rigorous unprovability results for key complexity-theoretic statements in subexponential-strength arithmetic theories. The main contributions are: (1) proving that assertions such as “SAT has no polynomial-size circuits” are independent of $S^1_2$; (2) establishing a tight correspondence between proof complexity lower bounds and circuit lower bounds; and (3) exposing deep metatheoretic barriers preventing any feasible formal proof of P = NP, thereby offering a novel logical foundation for complexity theory.

We introduce $p$-adic manifold learning, propose an algorithm to solve it, and propose benchmark tasks from impartial games.

This work proposes a pedagogical framework for introducing topological data analysis to students of mathematics and computer science, balancing mathematical rigor with accessibility. Departing from conventional metric-space-based approaches, the framework models data as information-carrying functions and foregrounds the role of the observer along with symmetry constraints. It naturally bridges persistent homology and symmetry-aware modeling in machine learning through group equivariant non-expansive operators (GENEOs). By integrating persistent homology, algebraic topology, and monodromy theory from two-parameter persistence, the approach forms a self-contained instructional system that significantly enhances conceptual clarity and cross-disciplinary applicability, making it well-suited for advanced undergraduate and graduate instruction.

This work addresses the lack of geometric structures in non-Archimedean spaces suitable for hierarchical data optimization. We propose a novel framework based on Berkovich geometry—specifically, a non-Archimedean polydisk space constructed as a product of closed balls—which naturally supports hierarchical representations and is introduced here for the first time into optimization. We prove that this space admits unique geodesics and can be isometrically embedded into a metric tree. A class of objective functions with piecewise polynomial structure and universal approximation capability is defined, for which we establish existence theory of solutions and design corresponding optimization algorithms. Leveraging tools from non-Archimedean analysis, metric geometry, and polynomial absolute value functions, we implement an open-source Julia library and demonstrate through experiments the effectiveness of our approach for hierarchical data optimization.

This work addresses the absence of a comprehensive machine-verified foundation for the extended complex plane, Möbius transformations, and their fundamental invariant—the cross-ratio—in existing formal mathematics. Building upon the Mathlib library in Lean 4, the authors construct the extended complex plane using the Option type and present the first formalization of the group structure of Möbius transformations, the uniqueness of such transformations determined by three points, and the invariance of the cross-ratio. The development, grounded in dependent type theory, comprises approximately 6,000 lines of code, including 40 definitions and 150 theorems, thereby establishing the first formally verified basis for conformal geometry, hyperbolic models, and mathematical physics.

This work presents the first assumption-free formalization of Cauchy real numbers in Cubical Agda, circumventing reliance on the axiom of choice, setoid bookkeeping, or explicit universe-level management—issues that commonly hinder constructive real number constructions in intuitionistic mathematics. Building upon the higher inductive-inductive types introduced in Homotopy Type Theory, the construction leverages Cubical Agda’s native support for higher inductive types to yield a fully type-checked, non-vacuous, and postulate-free development of the reals. This approach not only resolves longstanding challenges related to redundancy and universe complexity but also establishes a robust foundation for machine-verified constructive analysis.