Universal GUE Statistics in Entanglement and Number Fluctuations of Trapped 1D Free Fermions — E8 Intelligence Research

FINDING: Free fermions in a 1D harmonic trap exhibit entanglement entropy and number variance governed by universal random matrix theory (RMT) statistics — specifically the β=2 Wigner-Dyson / GUE universality class — with the trap geometry setting only the local density of states, not the universal fluctuations. MATH: - 1D harmonic trap single-particle spectrum: E_n = ℏω(n + 1/2), n = 0,1,2,… - Fermi energy for N fermions: E_F = ℏω(N + 1/2) ⇒ local Fermi wavelength λ_F(x) ∝ 1/√(E_F − V(x)) - Semiclassical density: ρ(x) = (1/πℏ)√(2m(E_F − V(x))) — Wigner semicircle for harmonic V(x) = ½mω²x² - Number variance in an interval of mean occupation n̄: Var(N) ~ (2/π²β)·[ln(2π n̄) + γ + 1 − π²/8] for GUE (β=2) - Entanglement entropy of a spatial bipartition (block of length ℓ) for free fermions: S(ℓ) ~ (c/3)·ln ℓ with central charge c=1 (1D CFT), saturating to area-law for gapped/blocked regimes - Rényi entropies S_α = (1/(1−α))·ln Tr ρ_A^α; for free fermions computable via two-point correlat Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com