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

TL;DR

This work challenges the prevailing emphasis on throughput over precision in deep learning by introducing the Exactness Hypothesis: AGI requires Arbitrary Precision Arithmetic, moving from $\mathbb{R}$ to $\mathbb{Q}$. The Halo Architecture pairs an Infinite-Precision Light stream with a Ring-based state via the Exact Inference Unit to achieve zero numerical divergence in infinite-depth reasoning, addressing issues like Semantic Drift and non-associativity of floats. The paper provides theoretical guarantees (Halo Boundedness Theorem) and empirical validation on the Huginn-0125 prototype, showing that rational arithmetic maintains trajectory fidelity and stable gradients where BF16/FP32 fail, even at 600B parameters. It also discusses broader implications, arguing that exact computation is essential for reliable long-horizon reasoning and proposes a future where exactness enables universal causal simulation and potentially transformative applications in biology and medicine.

Abstract

Current paradigms in Deep Learning prioritize computational throughput over numerical precision, relying on the assumption that intelligence emerges from statistical correlation at scale. In this paper, we challenge this orthodoxy. We propose the Exactness Hypothesis: that General Intelligence (AGI), specifically high-order causal inference, requires a computational substrate capable of Arbitrary Precision Arithmetic. We argue that the "hallucinations" and logical incoherence seen in current Large Language Models (LLMs) are artifacts of IEEE 754 floating-point approximation errors accumulating over deep compositional functions. To mitigate this, we introduce the Halo Architecture, a paradigm shift to Rational Arithmetic ($\mathbb{Q}$) supported by a novel Exact Inference Unit (EIU). Empirical validation on the Huginn-0125 prototype demonstrates that while 600B-parameter scale BF16 baselines collapse in chaotic systems, Halo maintains zero numerical divergence indefinitely. This work establishes exact arithmetic as a prerequisite for reducing logical uncertainty in System 2 AGI.

Figures (6)

• Figure 1: Semantic Drift Analysis. Comparison of cumulative error in recursive inference. Standard BF16 (Red) exhibits immediate error accumulation ($>10^{-4}$), rendering deep causal chains unreliable. FP32 (Orange) delays but does not prevent drift. The Halo EIU (Blue Dashed), utilizing Rational Arithmetic ($\mathbb{Q}$), maintains zero error (plotted at $10^{-30}$) across 2000 steps.
• Figure 2: Logical Survival Limit. Divergence from the analytical truth in a chaotic system. BF16 models (Red) collapse into random noise within 10 steps. FP32 (Orange) survives until $\sim$20 steps. The Halo Architecture (Blue), anchored by exact rational arithmetic, maintains perfect trajectory fidelity indefinitely.
• Figure 3: The Derivative of Truth. Gradient magnitude vs. Network Depth. Standard BF16 (Red) gradients explode/vanish beyond 500 layers. Halo (Blue) maintains precise gradient flow indefinitely, enabling the training of arbitrarily deep reasoning chains.
• Figure 4: Needle in a Haystack Test. Retrieval error vs. Sequence Length. Due to cumulative "Semantic Drift," BF16 models (Red) lose the ability to recall specific tokens after $\sim$2,000 steps. Halo (Blue) exhibits perfect recall regardless of context length.
• Figure 5: The Curse of Dimensionality: 600B Model Instability. Semantic drift accumulation across model scales. Contrary to the belief that scale solves all, the 600B-scale model (Dark Red, $d=24576$) degrades faster than the 7B-scale model (Light Red, $d=4096$). Wider layers involve larger summation operations, amplifying floating-point rounding errors.

Theorems & Definitions (1)

• proof