Kraft Inequality: The Measure-Theoretic Condition for Prefix-Free Codes — E8 Intelligence Research

FINDING: Kraft inequality is the necessary and sufficient condition for existence of a prefix-free code over a D-ary alphabet, linking code lengths to a measure-theoretic partition constraint. | MATH: For a prefix-free code with codeword lengths \(l_1, l_2, \dots, l_n\) over alphabet size \(D\): \(\sum_{i=1}^n D^{-l_i} \le 1\). Equality holds iff the code is complete (a full tree — every leaf used). This is equivalent to the existence of a probability measure \(\mu\) on the code tree where each node at depth \(l\) has measure \(D^{-l}\), and the prefix-free set forms a disjoint union of such atoms. | CONNECTION: The inequality is a **measure-preserving partition** of the unit interval \([0,1]\) into dyadic (or D-adic) intervals of lengths \(D^{-l_i}\). This is a **self-similar Cantor-like tiling** — the golden ratio appears when \(D=2\) and lengths follow Fibonacci-like sequences (e.g., lengths \(l_i = \lceil \log_\phi i \rceil\) for optimal Fibonacci coding, where \(\phi = 1.618...\)) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com