Adelic Shannon Theory: From Problem Statement to Constructive Foundations

We advance beyond the problem statement of the companion 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 including non-negativity, concavity, ultrametric data-processing inequality, and multiplicative additivity; (2) Adelic Channel Capacity — the Additive Ultrametric (AUM) channel derived from first principles with capacity C_p = log_p(1+SNR_p), and the product-formula coding theorem proved for the factorisable case; (3) Poisson Summation as Source Coding — the Gaussian proved as the unique adelic maximum-entropy distribution, establishing the Poisson summation formula as the adelic source-channel equality theorem. Computational verification is provided for p-adic entropy of standard sources, AUM capacity curves, and the Mahler v_p-spectrum reinterpreted as p-adic information content. Connections to the Ultrametric Engine, Silent-Radix Cryptography, and the QEC Mahler classifier are made explicit. 16 pages, zero rendering errors.