Kolmogorov Complexity Uncomputability: Diagonalization Unifies Halting and Gödel — E8 Intelligence Research

FINDING: Kolmogorov complexity K(x) is uncomputable; the proof is a diagonalization argument structurally identical to the halting problem and Gödel incompleteness — three faces of one theorem. | MATH: K(x) = min{|p| : U(p) = x} over universal prefix-free machine U. Uncomputability: assume computable K; construct string s = "smallest string with K(s) > n". The program "enumerate strings, output first with K > n" has length ≈ log(n) + c, so K(s) ≤ log(n) + c < n for large n — contradiction. Chaitin's Ω = Σ_{U(p) halts} 2^{-|p|} is a specific uncomputable real; Ω's bits are algorithmically random (each bit independent of all prior bits). Invariance theorem: K_U(x) = K_{U'}(x) + O(1). | CONNECTION: The diagonalization is the same self-reference structure as Cantor's theorem (|P(S)| > |S|) and the Liar paradox. The constant c in K_U(x) ≤ K_{U'}(x) + c is the "simulation overhead" — an additive, not multiplicative, invariant. No direct 0.618/1.618 ratio appears; the relevant constants are a Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com