Transordinal Fixed-Point Operators and Self-Referential Games: A Categorical Framework for Reflective Semantic Convergence

TL;DR

This paper introduces a categorical framework combining fixed-point theory, transfinite recursion, and game semantics to model how language interpretations stabilize through infinite self-reference, providing a rigorous account of semantic convergence.

Contribution

It develops a novel transordinal fixed-point construction and a hierarchy of reflective games to explain semantic stabilization in a formal, symbolic setting.

Findings

Existence and uniqueness of transordinal fixed points established

Hierarchical reflective games characterize semantic equilibrium

Framework offers guarantees for formal linguistics and language-aware systems

Abstract

We present a new theoretical framework that unifies category-theoretic fixed-point constructions, transfinite recursion, and game-based semantics to model how interpretations of language can stabilize through unlimited self-reference. By iterating a meaning-refinement operator across all ordinal stages, we isolate a unique "transordinal" fixed point and show, via a hierarchy of reflective games, that this same object is the sole equilibrium of an infinite dialogue between a text and its interpreter. The result delivers a mathematically rigorous account of semantic convergence without resorting to statistical training or empirical benchmarks, yet remains simple to explain: start with a rough meaning, let speaker and listener correct each other forever, and the process provably settles on a single, self-consistent interpretation. Because the construction is entirely symbolic, it offers both precise guarantees for formal linguistics and a blueprint for designing language-aware systems that can reason about their own outputs. The paper details the requisite transordinal machinery, proves existence and uniqueness theorems, and connects them to long-standing questions about reflection, truth, and equilibrium in formal systems and semantics.