Kraft Inequality: The Boundary of Uniquely Decodable Variable-Length Codes — E8 Intelligence Research

FINDING: Kraft's inequality defines the exact boundary condition for uniquely decodable variable-length codes, linking binary tree structure to measure-theoretic partition of unity. | MATH: For code lengths \\(l_1, l_2, \\dots, l_n\\) over a D-ary alphabet, Kraft's inequality: \\(\\sum_{i=1}^n D^{-l_i} \\leq 1\\). Equality holds iff the code is *complete* (a full tree — every leaf used). Kraft–McMillan theorem: this condition is *necessary and sufficient* for unique decodability. The dyadic case (D=2) gives \\(\\sum 2^{-l_i} \\leq 1\\), which is exactly a partition of unity on the Cantor space \\(\\{0,1\\}^{\\mathbb{N}}\\) — each codeword corresponds to a cylinder set of measure \\(2^{-l_i}\\). | CONNECTION: The dyadic partition of unity is a *self-similar* structure: the binary tree's branching ratio is 1/2 per level. The *golden ratio* appears if you generalize to a *Fibonacci code* (where codeword lengths follow Fibonacci numbers) — the asymptotic growth rate of the number of valid sequences is \\(\\ph Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com