Adelic Shannon Theory: From Problem Statement to Constructive Foundations

Adelic Shannon Theory — constructive foundations. Three components: p-Adic Entropy with proven axioms (non-negativity, concavity, ultrametric data-processing inequality, multiplicative additivity); AUM channel capacity Cp = log_p(1+SNR_p) with product-formula coding theorem; Poisson Summation as Source Coding with Gaussian as unique adelic max-entropy distribution. v2.1: Red-team fixes — corrected p-adic max-entropy claim (Section 2.4), added external references (Kapranov 1996, Anashin 2012, Dragovich 2017), cleaned sign convention. 17 pages, zero rendering errors.