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从模糊到精确：Halo 架构如何通过有理数运算实现无限深度推理

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

arXiv ID: 2601.18702 [cs.LG]
Subjects: Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Hardware Architecture (cs.AR)
Authors: Hansheng Ren
Submitted: 26 Jan 2026 (Last revised: 20 Oct 2026, version v5)

文章背景与核心概要

当前主流的大语言模型都建立在“模糊”的浮点数运算底座之上，为了防止长序列推理过程中的数值崩溃，研究人员不得不堆叠复杂的残差连接、归一化和梯度截断等“脚手架”，极大地浪费了宝贵的算力。本文提出了颠覆性的 Halo 架构，首次将严谨的有理数域 (\(\mathbb{Q}\)) 运算引入模型，并辅以专用的精确推理单元 (EIU)。通过独特的“微环-宏环”双环拓扑结构，Halo 不仅巧妙解决了有理数位宽无限膨胀的内存难题，更实现了对传统 Transformer 繁重数值结构的“大拆解”，还原出极为纯粹的代数本质。这一突破性设计成功消除了梯度噪声，使采用超大宏观学习率成为可能，为实现兼具严谨逻辑与无限深度的通用人工智能推理开辟了全新的硬软件协同范式。

内容提要

Summary

现代大语言模型 (Large Language Models, LLMs) 的计算完全依赖于近似的浮点数（即“模糊”）算术。为了抵消这种数值基础天生带来的不稳定性，现代神经网络架构不得不引入沉重的结构启发式设计——例如复杂的残差连接 (Complex Residuals)、Pre-RMSNorm、注意力缩放 (Attention Scaling) 以及梯度截断 (Gradient Clipping) 等机制——这些结构仅仅是为了防止数值崩溃，就消耗了大量宝贵的计算资源。

Modern Large Language Models (LLMs) rely on approximate floating-point ("fuzzy") arithmetic. To counteract the inherent instability of this numerical foundation, modern neural architectures require heavy structural heuristics—such as Complex Residuals, Pre-RMSNorm, Attention Scaling, and Gradient Clipping—which consume valuable computational resources strictly to prevent numerical collapse.

为了解决这一根本问题，本论文提出了从近似计算转向精确算术的范式变革，隆重推出了 Halo 架构。该架构扎根于数学上的有理数域 (\(\mathbb{Q}\)) ，并由定制的精确推理单元 (Exact Inference Unit, EIU) 提供底层算力支撑。针对有理数运算过程中位宽呈指数级膨胀的难题，Halo 通过创新的双环拓扑 (Dual-Ring Topology) 进行了巧妙破解：

This paper proposes a paradigm shift from approximate computation to exact arithmetic by introducing the Halo Architecture. Grounded in the Rational Field (\(\mathbb{Q}\)) and powered by a custom Exact Inference Unit (EIU), Halo addresses the exponential bit-width growth of rational arithmetic via a Dual-Ring Topology:

  1. 微环 (连续统维护，Continuum Maintenance)：利用丢番图逼近 (Diophantine Approximation) 严格限制内存复杂度的增长上限；
  2. 宏环 (符号对齐，Symbolic Alignment)：通过周期的状态坍缩机制，强制确保推理过程中的逻辑一致性。
  1. The Micro-Ring (Continuum Maintenance): Strictly bounds memory complexity using Diophantine Approximation.
  2. The Macro-Ring (Symbolic Alignment): Enforces logical consistency via periodic state collapse.

这一坚实稳定的双环底层基座，推动了对传统数值“脚手架”的“大拆解” (Great Dismantling)，成功将 Transformer 模块剥离精简至其最纯粹的代数形式（犹如一张未受污染的白板，Tabula Rasa）。此外，论文还证实了“效率悖论” (Efficiency Paradox)：彻底消除梯度噪声 (\(\sigma \to 0\)) 后，模型可以采用前所未有的宏观超大学习率 (Macro-Learning Rates)，从而有望将模型的总收敛时间 (Total Time-to-Convergence) 降低数个数量级。

This stable dual-ring substrate enables the "Great Dismantling" of numerical scaffolding, stripping the Transformer block down to its clean algebraic form (Tabula Rasa). Furthermore, the paper verifies the "Efficiency Paradox": eliminating gradient noise (\(\sigma \to 0\)) allows for Macro-Learning Rates, potentially reducing total time-to-convergence by orders of magnitude.

摘要

Abstract

当今大语言模型 (LLMs) 主流的参数扩展范式，完全建立在“模糊”的浮点算术基石之上。为了缓解这一近似计算基础固有的不稳定性，现代网络架构不得不搭建起一套极其复杂的结构与数值启发式“脚手架”——包括复杂的残差连接、Pre-RMSNorm、注意力缩放以及梯度截断等——仅仅是为了防止数值崩溃，就耗费了相当可观的算力开销。

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.

对此，我们提出向“精确”计算演进的范式转变，隆重介绍了基于有理数域 (\(\mathbb{Q}\)) 构建并由专用精确推理单元 (EIU) 驱动的 Halo 架构。为了解决有理数运算中位宽呈指数级急剧增长的顽疾，Halo 采用了一种双环拓扑结构，巧妙融合了两种互补的控制机制：(1) 微环（连续统维护），通过丢番图逼近严格限定内存复杂度；(2) 宏环（符号对齐），通过周期的状态坍缩强力约束逻辑一致性。

We propose a paradigm shift to the "Exact". We introduce the Halo Architecture, grounded in the Rational Field (\(\mathbb{Q}\)) 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.

这种高度稳定的双环底座为数值“脚手架”的“大拆解”创造了可能，使得 Transformer 模块能够彻底洗净铅华，回归至干净纯粹的代数形式 (Tabula Rasa)。不仅如此，我们还验证了“效率悖论”：完全消除梯度噪声 (\(\sigma \to 0\)) 允许模型采用极大的宏观学习率，从而有望使总收敛时间缩短数个数量级。Halo 架构雄辩地证明，通用人工智能 (AGI) 的实现，离不开在严谨数学框架下对连续场与离散推理链的高效混合与统一。

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 (\(\sigma \to 0\)) 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.

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