group quotient 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.

Existing formal verification methods struggle to effectively validate correctness properties of programs characterized by symmetry. This paper introduces the first Hoare logic framework specifically designed for symmetry verification: it replaces conventional pre- and postconditions with group actions, defines a formal syntax for group-action specifications, and establishes a natural entailment relation—enabling rigorous, compositional reasoning about symmetry properties of imperative programs. The approach integrates group action theory, Hoare logic, and static analysis, and is implemented in the prototype tool SymVerif. Evaluation on multiple hand-crafted benchmarks confirms the framework’s effectiveness; notably, it uncovered a logical inconsistency in the symmetry formulation of an existing dynamical systems model. The core contribution is the first sound and complete Hoare logic for symmetry based on group actions—establishing a provably correct, implementable paradigm for symmetry-driven program verification.

This paper addresses the problem of determining whether a given algebraic variety (V) arises as the Zariski closure of an orbit of a point under the action of an (s)-generated commutative matrix group. To resolve this, the authors first formulate and solve the “decidability” problem for such orbit closures, establishing a unified framework integrating commutative algebra, structural theory of matrix groups, lattice theory, and algebraic-geometric analysis of orbit closures. They devise a decision algorithm that, given (V) and (s), determines in PSPACE whether (V) equals the orbit closure of some point under an (s)-generated commutative linear algebraic group. Moreover, they prove that this problem is PSPACE-complete—establishing a tight complexity characterization. The main contribution is the first proof of computability for this geometric decision problem, together with an optimal complexity bound, thereby filling a fundamental theoretical gap in the structural decidability of orbit closures under algebraic group actions.

This paper addresses the fragmentation and lack of interoperability among structural complexity measures across graph theory, geometric group theory, and dynamical systems. Methodologically, it introduces a unified “structured decomposition” framework grounded in category theory: (i) it is the first to formalize diverse domain-specific decomposition paradigms categorically; (ii) it establishes a general duality theory linking decompositions to object completions; and (iii) it defines composable width functors enabling cross-model quantification, comparison, and translation of structural complexity. Key contributions include: a unified categorical characterization of over ten complexity parameters—including treewidth, layered treewidth, and hypergraph treewidth—revealing their intrinsic structural relationships; and a novel parameterized tractability paradigm for NP-hard problems, grounded in decomposition width. The framework achieves both theoretical unification and algorithmic realizability.

This work introduces Razborov’s flag algebra framework into computer science from the perspective of logic and formal methods to address asymptotic subgraph density inequalities in extremal graph theory. We reformulate the framework as a formal system comprising syntax, semantics, and proof strategies, enabling inequality derivation through labeled variants and downward operators. A key innovation lies in interpreting the flag algebra’s shifting mechanism as an adjoint pair, thereby uncovering deep connections to Galois connections and category theory. The framework successfully reproduces Mantel’s theorem and Goodman’s lower bound on Ramsey multiplicities, demonstrating its efficacy for symbolic proofs and establishing a novel bridge between formal verification and programming language theory.

This work addresses the reliable computation of Galois and monodromy groups for parametrized polynomial systems. To this end, it introduces a novel framework that integrates certified homotopy path tracking with homotopy graphs, enabling—for the first time—the rigorous numerical verification of monodromy group actions. By combining certified numerical algorithms with techniques from numerical algebraic geometry, the proposed method guarantees the mathematical correctness of its computational results. The approach has been successfully validated on a range of examples drawn from both pure and applied mathematics, demonstrating its effectiveness, reliability, and practical utility in analyzing the group-theoretic structures of complex polynomial systems.

This work proposes the Graphical Algebraic Geometry (GAG) framework, which for the first time rigorously formalizes polynomials, ideals, and affine varieties from commutative algebra using a diagrammatic language. By integrating tools from category theory, (co)span semantics, and algebraic geometry, GAG establishes a universal and complete compositional reasoning system for polynomial constraint satisfaction problems (#CSP). The core contributions include establishing a formal correspondence between #CSP and graph rewriting, uncovering a deep connection between GAG and the qudit ZH quantum graphical calculus, and proving that constraint rewriting in GAG is #P-hard. Furthermore, it is shown that computing amplitudes in the qudit ZH calculus requires only a constant number of oracle queries to GAG, thereby opening a novel pathway for efficient modeling of quantum computations.

This work addresses the challenge of ensuring numerical stability, computational correctness, and physical consistency in high-stakes or scientific AI applications, where traditional post-training validation falls short. The authors propose embedding algebraic structural constraints directly into the model design phase to enable decidable correctness guarantees. Their key innovation lies in the first integration of Hindley-Milner type inference over finitely generated Abelian groups with a computable restriction of Solomonoff’s universal prior, yielding information-theoretically optimal hypotheses. This framework further combines dimensional type systems, program hypergraphs, graded Clifford algebraic inference, forward coeffect analysis, and exact posit accumulation to preserve model invariants while eliminating the cumulative computational overhead inherent in existing reliability approaches across deployment, inter-layer propagation, and inference stages.

This study addresses several long-standing conjectures by J. H. Davenport, A. Locatelli, G. K. Sankaran, and others concerning the topological properties of cells in cylindrical algebraic decomposition (CAD). For the first time, the paper systematically constructs explicit counterexamples that refute these conjectures. By integrating CAD theory from real algebraic geometry, topological construction techniques, and symbolic computation, the work reveals fundamental limitations in the existing assumptions. These findings not only clarify the theoretical boundaries of CAD but also provide a refined foundation for applications involving topological analysis of semi-algebraic sets, such as motion planning in robotics.