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

arXiv: https://arxiv.org/abs/2507.16620 · HTML: https://mlsanity.com/paper/2507.16620/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Jul 2025.

## Built on by (most cited first, 1 of 1)

- [Transfinite Fixed Points in Alpay Algebra as Ordinal Game Equilibria in Dependent Type Theory](https://mlsanity.com/paper/2507.19245/) (2025)

## Abstract (arXiv)

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.