field extension methods

This paper addresses the structured lifting problem by proposing the first systematic abstract framework that uniformly characterizes its mathematical structure and computational semantics. Methodologically, it integrates lifting property analysis from category theory, structured abstract modeling, and type-theoretic semantic techniques. The work establishes, for the first time, a rigorous proof that lifting solutions within this framework are universally existent, closed under relevant operations, and unique. These results not only uncover fundamental regularities underlying structured lifting but—crucially—bridge categorical lifting with type-theoretic computation. In particular, they provide a rigorous semantic foundation for axiomatizing computational rules in cubical type theory, thereby advancing formalization and computability research in higher-order type theory.

This work proposes a novel method for constructing linear codes over finite fields by leveraging the geometric and topological structure of simplicial complexes. By establishing an explicit connection between codeword weights and the combinatorial structure of the complex, the study systematically analyzes how topological operations influence key code parameters—particularly the minimum distance. For the first time, tools from algebraic topology and combinatorial geometry are employed to enable controlled design of linear codes. The approach successfully yields multiple families of optimal linear codes over $\mathbb{F}_2$, with precise characterizations of their dimensions and minimum distances, thereby offering a new geometric perspective and a constructive framework for coding theory.

This work addresses the long-standing challenge of explicitly constructing near-optimal lossless rank extractors, weak subspace designs, and strong $s$-blocking sets over small finite fields whose size depends only on the rank or codimension. By integrating tools from function field theory, polynomial identity testing, and Fourier analysis based on $\varepsilon$-biased sets, the authors achieve the first explicit near-optimal constructions of these objects over non-prime fields with $q \geq \mathrm{poly}(s)$. Notably, the resulting strong $s$-blocking set has size $O(s(k - s)q^s)$, improving upon the previous exponential bound $2^{O(s^2 \log s)} q^s k$ and matching the non-explicit optimal asymptotics. The paper also presents the first explicit near-optimal constructions for both lossless rank extractors and weak subspace designs in this setting.

Large language models (LLMs) generate scientific computing code—such as Fast Fourier Transform (FFT) implementations—lacking formal guarantees on numerical stability, floating-point precision, and algorithmic correctness. Method: This paper proposes a stepwise semantic lifting approach grounded in an extended SPIRAL framework, integrating symbolic execution with interactive theorem proving (Coq/Isabelle) to systematically lift LLM-generated floating-point code to high-level, mathematically precise specifications. The method explicitly encodes FFT domain knowledge and constraints for faithful floating-point arithmetic semantics. Contribution/Results: It introduces the first semantic lifting pathway tailored to scientific computing kernels. Experiments demonstrate end-to-end lifting of GPT-generated FFT code into verifiable mathematical specifications, with formal verification of both algorithmic equivalence and numerical stability. This establishes a novel paradigm for formally certifying AI-generated scientific software.

This paper establishes an algebraic semantics foundation for programming languages with computational effects (e.g., state, I/O, exceptions). To address this, it introduces the *effect multicategory* as the central semantic model and—crucially—first defines and studies multicategories enriched over a *duoidal category*, rigorously proving their equivalence. This equivalence naturally yields definitions of effect algebras and 2-morphisms. Methodologically, the work integrates duoidal category theory, pre-multicategories, tensor and Cartesian product structures on functor categories valued in sets, and enriched category theory. The main contributions are: (i) a unified algebraic semantic framework subsuming both pure and effectful computation; (ii) the first general algebraic model for effectful programming languages based on higher-order categorical structures—specifically, 2-morphisms; and (iii) a rigorous theoretical foundation enabling axiomatization, logical reasoning, and principled language design for effect systems.

This work addresses the absence of constructive semantic models for extensions of dependent type theory—such as the univalence axiom and higher inductive types—in synthetic mathematics. It presents the first construction of a higher topos model within a fully constructive metatheory, thereby providing a unified semantic foundation for emerging formal systems like simplicial homotopy type theory and synthetic algebraic geometry. By establishing this model, the paper not only fills a critical gap in constructive semantics but also demonstrates the broad applicability and expressive power of higher topos models across diverse frameworks of synthetic mathematics, confirming their viability as a general semantic infrastructure.

This work addresses the underdeveloped state of algebraic geometry codes under the sum-rank metric by introducing, for the first time, linearized algebraic geometry codes tailored to this metric. The construction leverages the quotient structure of algebraic function fields and Ore polynomial rings, yielding an explicit encoding scheme that achieves theoretically optimal parameters and exhibits asymptotic goodness. By extending the classical framework of algebraic geometry codes, this study fills a critical gap in the literature concerning efficient code constructions for the sum-rank metric.

Traditional text-based code editing struggles to satisfy syntactic and semantic constraints, leading to inefficient performance by large language model (LLM) agents when performing cross-file or non-local modifications. This work proposes Source Code Algebra (SCA), a novel framework that introduces compositional, idempotent, and commutative algebraic operations into code editing, replacing character-level edits with semantically complete atomic operations. Built upon abstract syntax trees, SCA defines semantics-aware edit primitives and formally models their algebraic properties, integrating them into an LLM interface. Experimental evaluation of the prototype system SCAS demonstrates that this paradigm significantly improves modification success rates while reducing communication token overhead by one to two orders of magnitude, highlighting its potential as an efficient and accurate new approach to code editing.

This work unifies various extensional constructs—such as path types and Riehl–Shulman extension types—within two-level type theory (2LTT) without introducing new axioms, providing them with a solid semantic foundation. By reusing existing models of homotopy type theory (HoTT), it automatically derives new models that support extension types, thereby offering the first rigorous definition of extension types together with a verification of their full inference rules. Consequently, several key principles previously treated as axioms become provable theorems. The development is fully formalized in Agda, where the equivalence between cubical glue types and the univalence axiom is established. This result opens a new avenue toward proving the long-standing conjecture that cubical type theory is conservative over HoTT.

This work aims to formalize self-referential structures and streamline the proof of Löb’s theorem within a categorical framework. To this end, the author introduces a Gödel numbering structure over fibrations, reconstructs geminal category theory, and establishes a categorical semantics for self-referential logic. This approach not only yields a significantly simplified proof of Löb’s theorem but also derives a novel categorical counterpart of the Gödel–Löb axiom. By integrating category theory, fibrations, modal logic, and theories of self-reference, the study constructs a concise yet rigorous semantic framework and uncovers its potential connections to modal type theory.