Information Theory from the Tree of Continua: Probability, Inequalities, Theorems, and Applications from the Topological Theory of Distinguishability III

The Topological Theory of Distinguishability establishes the Tree of Con-tinua as the universal topological structure for all distinguishable information.This paper develops the entirety of classical information theory from that singlestructure. Beginning from probability as a ratio of counts in the tree, we derive:the foundational inequalities (KL divergence, Gibbs, subadditivity, concavity ofentropy), the Law of Large Numbers as a counting theorem about multinomialcoefficients, and Shannon’s three fundamental theorems (Source Coding, Chan-nel Capacity, Rate-Distortion). The Binary Symmetric Channel is treated as thecanonical illustration: its capacity C = 1 − H(p) emerges as the depth remain-ing after the noise tree’s depth is subtracted from the input tree’s depth. Theuniqueness theorem identifies entropy as the only function on branching distri-butions that the tree’s geometry forces into existence; Shannon’s three axiomsbecome consequences rather than postulates. We conclude with an application tothe human hemoglobin beta chain (HBB): the period-3 codon structure emergesfrom the coordination surface without biological assumptions, and the sicklecell mutation E6V is shown to be informationally minimal—one branch change,∆LDI ≈ 0.003—while being physiologically catastrophic. No metric space andno measure theory are required. The only analytic content beyond combinatoricsis the univariate continuity of t 7→ −t log t, which reduces to an elementary con-sequence of the combinatorial inequality Log x ≤ x − 1. The measure-theoreticapparatus of classical probability theory is a faithful representation of these com-binatorial facts in a more powerful but less transparent language. Shannon’sentropy was always a depth. The Tree of Continua makes this visible.