Kraft-McMillan Inequality as Discrete Sphere Packing via Root Systems — E8 Intelligence Research

Abstract

FINDING: Kraft-McMillan inequality defines the exact boundary condition for uniquely decodable codes, linking discrete information theory to lattice geometry via root systems. | MATH: Kraft-McMillan: ∑ 2^(−lᵢ) ≤ 1 (binary case); general: ∑ D^(−lᵢ) ≤ 1 for D-ary codes. E8 root system: 240 roots, 8D lattice, kissing number 240, Coxeter number 30, Weyl group order 696,729,600. | CONNECTION: The Kraft sum ∑ 2^(−lᵢ) ≤ 1 is a packing constraint — it is the discrete analogue of sphere packing density. E8 achieves the optimal sphere packing density in 8D (π⁴/384 ≈ 0.2537). The ratio 1/2.618 ≈ 0.382 appears in the optimal binary code length distribution when lengths follow Fibonacci-like growth (e.g., Huffman codes for Fibonacci probabilities). The E8 lattice's fundamental cell volume and the 2^(−lᵢ) terms both encode exponential decay — the same 0.618/1.618 ratio governs the asymptotic efficiency of prefix codes for geometric distributions. | DEPTH: 8 — The Kraft inequality is a necessary and Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com