Kraft's Inequality: Bridging Prefix Codes, Measure, and Computation — E8 Intelligence Research

Abstract

FINDING: Prefix-free codes obey Kraft's inequality, which partitions the unit interval into dyadic subintervals, linking algorithmic information theory to measure-preserving partitions and universal computation. | MATH: Kraft inequality: ∑_{i} D^{-l_i} ≤ 1 (D-ary alphabet, codeword lengths l_i); equality holds for complete/compact prefix-free sets. For binary (D=2), this is ∑ 2^{-l_i} ≤ 1. The measure of the set of infinite sequences with a given prefix w is 2^{-|w|}. Kraft–McMillan theorem extends to uniquely decodable codes. Optimal prefix-free machine U: H(s) = min{ |p| : U(p)=s } over prefix-free domain. | CONNECTION: The dyadic partition of [0,1] by prefix-free codes is a binary tree — its leaves correspond to intervals of length 2^{-l_i}. The golden ratio φ = 1.618 appears in optimal binary trees (Fibonacci coding, where codeword lengths follow Fibonacci numbers, giving asymptotic efficiency ~φ). The constant 0.618 = φ−1 emerges in the redundancy of Fibonacci codes. The Kraft sum Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com