Covariance-Enforced Compatibility for Qubit Phase Observables and the Sharp Qutrit Onset of Specker Incompatibility

This work determines the sharp finite-dimensional onset of incompatibility for finite families of U(1)-covariant phase observables. In dimension two, every covariant phase observable is shown to be a convolutional postprocessing of a single canonical qubit phase observable. Consequently, every finite family of qubit covariant phase observables, including arbitrary finite coarse-grainings, is jointly measurable. In dimension three this universal compatibility breaks down. Four rank-two qutrit covariant phase observables are constructed whose binary interval coarse-grainings form a rigorously certified four-Specker family. Triple compatibility is established using explicit parent POVMs with exact-rational positivity tests and interval-bounded affine corrections, while four-way incompatibility is certified by independent dual semidefinite witnesses with exact rational normalization and interval evaluation. The work therefore establishes the sharp threshold d_min = 3 for finite-family incompatibility within this phase-covariant class. It also characterizes the qutrit canonical-smearing sector, proves robustness on a full-dimensional 28-parameter neighborhood, and provides a common-commutant certificate excluding a hidden qubit reduction. Supplementary material contains detailed analytic derivations and certificates. The accompanying reproducibility package provides verification code, exact/interval certificates, numerical summaries, and machine-readable outputs.