Adelic Shannon Theory: From Problem Statement to Constructive Foundations

Author: Rowan Brad Quni-Gudzinas | Date: 2026-07-30 | License: CC-BY-4.0

Overview

This paper advances beyond the research design "Adelic Shannon Theory: A Research Design" to construct the foundational elements of an Adelic Shannon Theory — a generalisation of information theory to the adele ring A_Q. Three components are developed:

1. p-Adic Entropy — formalised with proven axioms (non-negativity, concavity, ultrametric data-processing inequality, multiplicative additivity)
2. Adelic Channel Capacity — AUM channel model derived from first principles, product-formula coding theorem proved for factorisable case
3. Poisson Summation as Source Coding — Gaussian proved as unique adelic maximum-entropy distribution

Files

• paper-adelic-shannon-v2.md — Full markdown source with LaTeX math (40.7 KB)
• paper-adelic-shannon-v2.pdf — 16-page publication-ready PDF, zero rendering errors

Status

Published — v2.1 (DOI: 10.5281/zenodo.21698976).

Related Work

Part of the Adelic Physics Programme. Connects to Ultrametric Engine, Silent-Radix Cryptography, and QEC Mahler classifier infrastructure.