From Fuzzy to Exact: The Halo Architecture for Infinite-Depth Reasoning via Rational Arithmetic

TL;DR

This paper introduces the Halo Architecture, a novel approach for infinite-depth reasoning in large language models using rational arithmetic, aiming to replace fuzzy floating-point computations with exact, stable, and efficient algebraic methods.

Contribution

The paper presents the Halo Architecture with a custom Exact Inference Unit and dual-ring topology, enabling exact rational computations and reducing reliance on heuristic numerical scaffolding in LLMs.

Findings

Halo achieves stable exact inference in LLMs.

Eliminates the need for complex numerical heuristics.

Potential for faster convergence and improved stability.

Abstract

The prevailing scaling paradigm of Large Language Models (LLMs) rests on a substrate of "Fuzzy" floating-point arithmetic. To mitigate the inherent instability of this approximate foundation, modern architectures have erected a complex scaffolding of structural and numerical heuristics--Complex Residuals, Pre-RMSNorm, Attention Scaling, and Gradient Clipping--consuming significant compute solely to prevent numerical collapse. We propose a paradigm shift to the "Exact". We introduce the Halo Architecture, grounded in the Rational Field (Q) and powered by a custom Exact Inference Unit (EIU). To resolve the exponential bit-width growth of rational arithmetic, Halo employs a Dual-Ring Topology that unifies two complementary control mechanisms: (1) The Micro-Ring (Continuum Maintenance), which strictly bounds memory complexity via Diophantine Approximation; and (2) The Macro-Ring (Symbolic Alignment), which enforces logical consistency via periodic state collapse. This stable dual-ring substrate allows for the "Great Dismantling" of numerical scaffolding, reducing the Transformer block to its "Clean" algebraic form (Tabula Rasa). Furthermore, we verify the "Efficiency Paradox": the elimination of gradient noise (sigma -> 0) allows for Macro-Learning Rates, potentially reducing the Total Time-to-Convergence by orders of magnitude. Halo demonstrates that General Intelligence requires the hybridization of continuous fields and discrete chains under a rigorous mathematical framework.