Weak Mixing Proof of Pisot Substitution Conjecture via Prefix-Suffix Automaton — E8 Intelligence Research

FINDING: A proof of one version of the Pisot Substitution Conjecture via weak mixing in the prefix-suffix automaton, establishing pure discrete spectrum for substitution dynamical systems. | MATH: Pure discrete spectrum ⇔ measurable conjugacy to a translation on a compact abelian group (Rauzy fractal tiling); key tool: weak mixing of the derived subshift of finite type; Pisot number λ > 1 (algebraic integer, all conjugates |λ'| < 1) governs substitution matrix eigenvalues; diffraction measure μ̂ = Σ_{k∈L} |â(k)|² δ_k — pure point iff Bragg peaks indexed by a finitely generated Z-module (lattice L). | CONNECTION: Pisot numbers include the golden ratio φ = 1.618 (root of x²−x−1), whose conjugate 1−φ = −0.618; also plastic number ρ ≈ 1.3247 (x³−x−1). Pure point spectra yield crystallographic-like diffraction — 2D/3D quasicrystals with 5-fold, 8-fold, 10-fold, 12-fold rotational symmetries (forbidden in periodic crystals) emerge from such substitutions. Rauzy fractals are self-similar tili Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com