Kolmogorov Complexity, Integrated Information, and Algebraic Symmetry via Lattice Isomorphisms — E8 Intelligence Research

FINDING: Kolmogorov complexity measures the intrinsic randomness of a string via the length of its shortest description; integrated information theory (IIT) attempts to quantify consciousness via system integration; lattice partition isomorphisms and Boolean algebra root system A_n link these to algebraic structures of symmetry. MATH: - Kolmogorov complexity \( K_U(x) = minₚ \{ |p| : U(p) = x \} \) (universal Turing machine \(U\)). - IIT's integrated information \(Φ\) measures irreducibility of a system's causal structure; no single closed-form equation, but relies on partition analysis. - Boolean algebra: \( B^n \) with operations \(, , \); root system \(A_n\) has simple roots \( α_i = e_i - eᵢ₊₁ \) in \(Rⁿ⁺¹\), with Weyl group \(Sₙ₊₁\). - Lattice partition isomorphism: mapping between partitions of a set and subspaces of a vector space over GF(2) (e.g., Birkhoff's representation theorem for finite Boolean algebras). CONNEC Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com