solvability analysis

This work addresses the absence of rigorous formalizations of abstract simplicial complexes and their stellar subdivisions in existing proof systems. It presents the first purely combinatorial formal framework for abstract simplicial complexes grounded in combinatorial topology, implemented in the Lean theorem prover. The framework encompasses fundamental operations such as morphisms, links, and joins, and systematically investigates their interaction with stellar subdivision. Key contributions include the first formalization of stellar subdivision in any proof assistant, the verification of several crucial identities—some previously undocumented in the literature—for the study of triangulated manifolds, and the proof of significant theorems such as the invariance of links under subdivision. This development establishes a reliable formal foundation for computational topology.

This work addresses the high formalization complexity of verifying map equivalences in homotopy type theory. We propose two lightweight techniques: (1) decomposing maps into composites of elementary equivalences, and (2) systematically leveraging the 3-for-2 property of equivalences to refine logical reasoning. To our knowledge, this is the first formal framework that jointly exploits both techniques, significantly reducing proof complexity and redundancy. Implemented within the Coq/Agda toolchain, we formally verify a foundational theorem in synthetic homotopy theory. The resulting proof is concise, highly modular, and strongly reusable—demonstrating the practical efficacy of our approach. Our methodology provides a transferable, principled foundation for equivalence reasoning in homotopical settings, advancing the automation and scalability of formal proofs in univalent mathematics.

本文通过引入正拓扑和可行精细化概念，解决信息细化与正实现问题，提出基于信息论和博弈论的解释框架，并探讨资源约束下的应用。

This study addresses the equivalence verification problem between two fundamental representations of finite closure systems—implicational and intersectional canonical bases—specifically, whether an intersectional basis fully captures all closed sets generated by a given set of implications. By integrating techniques from computational complexity theory, formal concept analysis, and functional dependency theory, the work establishes for the first time that this problem is coNP-complete, even when restricted to acyclic implication sets with premises of size at most three. This result precisely characterizes the computational complexity of verifying completeness in closure system representations, rules out the existence of output-polynomial algorithms even in restricted settings such as acyclic convex geometries, and provides new lower bounds for related problems including characteristic model identification.

This work addresses the critical problem of unexplainable unrealizability verification in program synthesis. We propose the first Hoare-style unrealizability logic—a formal deductive system that models the semantic behavior of program search spaces and systematically approximates the collective execution behavior of infinite program sets via sound, machine-checkable inference rules. This transforms opaque, black-box unrealizability proofs into human-understandable, machine-verifiable structured derivations. Our key contributions are threefold: (1) unifying and formalizing the implicit reasoning principles underlying existing unrealizability tools; (2) enabling inductive assertion synthesis and rigorous formal verification of unrealizability; and (3) yielding a compositional, extensible proof infrastructure. The logic enhances transparency, theoretical rigor, and tool-supported verifiability in program synthesis analysis—establishing a foundational framework for principled, explainable synthesis correctness reasoning.

This study addresses the lack of formalization infrastructure and low collaborative efficiency in the geometric analysis formalization of the Poincaré conjecture by proposing a milestone-decomposed parallel agent workflow. Leveraging Lean 4 as the verification framework, this approach integrates mathematical blueprint planning with multi-agent parallel reasoning to enable efficient human-AI collaborative formalization of complex mathematical proofs. The project successfully completes the AI-assisted formal verification of the Poincaré conjecture and establishes a reusable formalization infrastructure that significantly reduces the verification costs for subsequent related theorems. Ultimately, this work provides a scalable new paradigm for large-scale mathematical formalization.

This study investigates whether P = NP implies #P = FP by analyzing the topological structure of the solution space of 3SAT to uncover deep connections among complexity classes. The approach uniquely links higher-order Betti numbers—such as b₂—to complexity class collapses, integrating homological theory, Toda’s theorem, and non-relativizing techniques, supported by large-scale empirical analysis for N ≤ 500. Theoretically, the work establishes that P = NP ⇒ #P = FP ⇒ PH = P. Empirically, it identifies solution-space fragmentation as a topological barrier affecting five distinct algorithmic paradigms. These findings provide novel topological evidence supporting P ≠ NP and circumvent relativization barriers that have historically limited progress in this domain.

This study investigates a transfinite generalization of the Phoa principle in synthetic domain theory and its unification with Segal completeness and chain completeness. By introducing dual simplices, spine structures, and a novel notion of sobriomorphism, the authors reinterpret and extend the Phoa principle to transfinite settings, thereby proposing a unified framework of completeness that subsumes both Segal and chain completeness. The work axiomatizes the interval type in Cubical Agda and formally verifies the main theorems, achieving machine-checked proofs of the Phoa principle and its generalizations. This formalization not only confirms the theoretical developments but also offers new insights into topological structures within synthetic domain theory.

This work addresses the challenge of verifying safety properties for infinite-state parameterized programs under complex topologies by introducing a novel proof system called the “parameterized proof space.” Leveraging local symmetries inherent in program topologies, the approach enables efficient verification of entire families of parameterized programs through the reuse of proof arguments across isomorphic neighborhoods. The key contributions include the development of a relatively complete proof system that operates without requiring explicit axiomatization of the underlying topology, integration of the model-theoretic notion of limit programs to support automatic construction and verification of universally quantified invariants, and the establishment of decidability guarantees for the verification process under certain conditions.

本文通过构建两个同调理论来解决软件架构中局部与全局语义一致性的问题，并证明了它们的一致性，从而实现了从语义修复到代数几何的转换。