Ultrametric Structures and Norm-Based p-Adic Operations for Hierarchical and Robust Information Processing

In this article, we propose a non-archimedean framework for information processing based on the arithmetic and geometry of p-adic numbers. Exploiting the ultrametric structure of Qp, we introduce norm-based p-adic operations inspired by classical logical connectives that exhibit strong stability under perturbations. The non-archimedean nature of the p-adic norm induces natural scale separation and clustering properties, leading to robust information encoding and spectral localization. Using p-adic Fourier analysis, we show that hierarchical signals decompose into disjoint frequency bands, enabling multiscale information processing with reduced interference. These results provide a mathematical foundation for robust and hierarchical information architectures, with potential applications in signal processing, error-tolerant computation, and non-classical information systems.