A Computable Phase-Gradient Observable from Quantum Fourier Transform with Numerical Evidence Across Entangled Systems and Cosmological Data

We define and numerically evaluate a relational observable extracted from the phasegradient of the momentum-space wavefunction obtained after a forward Space-to-MomentumQuantum Fourier Transform (QFT). The metric is τ = ⟨|∇pΘ|⟩, where Θ(p) = arg( ˜Ψ(p)).We test this observable on five distinct classes of data:1. Synthetic Gaussian entangled states (unwrapped phase ratio 6.24× between maximallyentangled and separable states).2. Real two-qubit polarization density matrices from SPDC (James et al., Altepeter,Motazedifard), where entangled states yield τ ≈ 0.87–6.7 while separable states yieldτ ≈ 0.3. High-dimensional Choi matrices from process tomography (Goel et al.), showing scalingwith dimension (r = 0.959 for total gradient vs. dimension).4. Momentum-entangled 4He* atoms (Athreya et al.), where the inverse (Momentum →Space) QFT shows suppression, confirming directional asymmetry.5. Planck 2018 CMB polarization maps, yielding phase-structure scores consistent withGaussianity (null result, z ≈ 0.20).While the observable exhibits consistent and reproducible behavior across these datasets,the interpretation as a physical time parameter remains open. The dynamical equivalence(i.e., whether τ satisfies the Schr¨odinger equation as a generator of time translations) is notproven here. We present this work as a well-defined, experimentally grounded numericalobservable and a candidate for further theoretical development.