AI 中文总结
该研究旨在超越阿基米德智能，提出从超度量空间结构内在推导非阿基米德学习理论，制定相关原理并引入公理框架，证明了关于\(\Zp\)和有限商的基本结构结果，揭示层次已编码在信息空间内在几何中。
AI 中文摘要
当代机器学习几乎完全建立在阿基米德数学之上。近期工作表明p进数和超度量方法可丰富神经网络等。本文倡导更基础观点，提出真正的非阿基米德学习理论应从超度量空间的拓扑、几何、代数和测度结构内在推导。为此制定内在非阿基米德学习原理并引入公理框架，其中信息状态由嵌套超度量球组织，学习由信息的分层细化和重新分布表示，尺度由赋值深度决定。还证明了基本结构结果，表明在\(\Zp\)上细化深度与公共前缀深度一致，有限商\(\mathbb Z/p^N \mathbb Z \)通过N级保留根球层次结构。
英文摘要
Contemporary machine learning is founded almost entirely on Archimedean mathematics: data are embedded in real vector spaces, similarity is measured by Euclidean-type metrics, learning is formulated through real-valued loss functions, and optimization is driven by differential or gradient-based methods. Recent work has shown that p-adic and ultrametric methods can enrich neural architectures, hierarchical representation, classification, and information processing. Rather than constructing a p-adic analogue of existing neural networks, we advocate a more fundamental viewpoint. We propose that a genuinely non-Archimedean theory of learning should be derived intrinsically from the topology, geometry, algebra, and measure structure of ultrametric spaces. To this end, we formulate the Principle of Intrinsic Non-Archimedean Learning and introduce an axiomatic framework in which information states are organized by nested ultrametric balls, learning is represented by hierarchical refinement and redistribution of information, and scale is determined by valuation depth. In this framework, neurons, layers, activation functions, gradients, and backpropagation are not assumed to be primitive concepts but possible computational realizations of a deeper mathematical theory. We prove a basic structural result showing that, on \(\Zp\) refinement depth coincides with common-prefix depth and that the finite quotients \(\mathbb Z/p^N \mathbb Z \) preserve the rooted ball hierarchy through level N. Consequently, hierarchy is not an auxiliary structure to be learned but is already encoded in the intrinsic geometry of the underlying information space
