morphism construction

This paper addresses the lack of systematic support for theory morphisms and logical relations in the $λΠ$-calculus modulo rewriting framework. Methodologically, it introduces a unified extension mechanism that formally integrates both concepts for the first time within this framework and designs a pattern-based invariant verification procedure, reducing the proof of translation invariants to finite, decidable propositional checks. The main contributions are: (1) a structurally clear, machine-verifiable formalization of inductive translations—e.g., type erasure; (2) the first fully verified type-erasure instance in $λΠ$-calculus modulo rewriting; and (3) a reusable methodology for rigorously verifying the correctness of translations between formal systems.

Existing approaches lack formal mechanisms for semantically preserving structural transformations across heterogeneous representation systems (e.g., formal languages, geometric diagrams, informal notations), especially for arbitrary user-specified semantic relations such as equivalence. Method: This paper introduces a representation-system-agnostic (RS-agnostic) structural migration calculus framework grounded in formal inference rules and pattern-based encoding. It enables verifiable, semantics-preserving structural mapping and transformation among disparate representation systems by integrating representation-system theory with constructive space modeling. Contributions: (1) The first formally verified RS-agnostic transformation calculus proven to satisfy arbitrary target semantic relations; (2) A pattern-driven, information-preserving mechanism supporting automatic, meaning-preserving reconstruction across multimodal representations; (3) A rigorous formal foundation for cross-representational cognitive modeling and intelligent representation generation. The framework achieves high generality in abstract representation transformation while ensuring semantic fidelity and verifiability.

This work addresses the sensitivity of large language model–driven formal theorem provers to semantically equivalent yet syntactically distinct problem formulations, which undermines proof success stability due to a lack of adherence to mathematical structural symmetries. The paper introduces, for the first time, the notion of rewriting categories from category theory to formally define two symmetry principles: proof covariance and success invariance. It proposes aggregating over equivalent rewrites of the input at test time to restore success invariance. Theoretical analysis demonstrates that this aggregation strategy rigorously guarantees invariance in the sampling limit, while empirical evaluation confirms that it significantly enhances both robustness and performance of theorem provers under a fixed inference budget.

Transformer models struggle to internalize symbolic computation, limiting their ability to perform rigorous reasoning over structured symbolic domains. Method: We propose a hierarchical Transformer framework that models symbolic operations as categorical morphisms in latent space, selectively activated via differentiable routing; for the first time, we internalize external tool invocation as a functor within the Transformer’s algebraic structure, and introduce KL-gain-driven natural gradient optimization with type-aware categorical modeling. Our approach integrates hierarchical vector decomposition, Bregman divergence-based mirror descent, and morphism-block mapping to enable sparse, interpretable, self-supervised symbol-geometry joint learning. Contribution/Results: Theoretical analysis and lightweight experiments demonstrate that the framework unifies symbolic reasoning, geometric structure modeling, and self-supervised learning. It significantly improves selectivity and interpretability of morphism activation on hybrid symbol-language tasks, subsuming and extending prior work such as Toolformer.

This paper establishes a rigorous categorical semantics for e-graphs (equivalence graphs) within the framework of monadic categories, supporting double-pushout (DPO) rewriting. Method: The authors generalize e-graphs to monadic categories by introducing *equivalence hypergraphs* (e-hypergraphs)—a compositional structure whose vertices are algebras over a monad and whose hyperedges encode algebraic operations, thereby internalizing structural equations up to isomorphism. The approach integrates category theory, semilattice-enriched categories, and hypergraph-based combinatorial modeling to yield a sound and complete semantic framework. Contribution/Results: The resulting framework provides an algebraic and monadic foundation for equivalence reasoning in e-graph–based program optimization, and extends the formal applicability of e-graphs to SMT solving and algebraic optimization—enabling principled, categorical treatment of equational rewriting beyond traditional graph-based methods.

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 proposes a systematic formalization of all published mathematical knowledge into a machine-verifiable, continuously evolving structured knowledge base, addressing the challenges of scalability and organization. Taking dilatations of categories in categorical algebra as the first case study, the project integrates interactive theorem proving, dependent type theory, and category theory to frame the complete formalization of mathematics as a universal reasoning benchmark. By constructing a formal prototype of category dilatations, the study demonstrates the feasibility of this approach in expressing complex algebraic structures, thereby establishing both an architectural foundation and a practical pathway toward a large-scale, interconnected, and extensible database of formalized mathematics.

We introduce a benchmark suite for evaluating structural mathematical reasoning in language models, built on subgroup-construction problems in SL(3, Z) with cryptographic-style verifier-prover asymmetry. Each instance presents a finitely generated subgroup as a list of integer matrices and asks for an arithmetic invariant -- index, surjection-at-prime, or membership -- that the construction-time information (N, K) pins down in O(1) closed form, but that the solver, lacking that information, must derive by either Aschbacher-classification analysis or by a membership query in SL(3, Z) of unknown decidability. The benchmark therefore distinguishes models with internalized algebraic priors (Aschbacher classes, McLaughlin's theorem, Property (T), the congruence subgroup property) from models that rely on general-purpose computation. We report empirical results across five representative reasoning traces from two state-of-the-art models. The headline result: on the index variant, one model spent 152 minutes of reasoning, explicitly identified the kernel-side membership question as the bottleneck, attempted constructive verification, and abstained with "DON'T KNOW" rather than commit to its computed cokernel candidate -- demonstrating calibrated meta-cognition on the open-decidability boundary that the benchmark was designed to probe. We argue that the benchmark exposes a four-way classification of model behavior (commit-correct, commit-wrong, abstain-correct, abstain-wrong) that standard answer-key scoring conflates.