Frequency-Independent Helicity Phase Residue: A Gyroscopic Wave Model and an Identifiability No-Go for BBH-Only Tests

This record contains a technical note and companion code for a minimal two-helicity gyroscopic wave model. The note studies the equation m q̈ + b J q̇ + κ(k) q = 0, where a small first-order gyroscopic term splits the two helicity branches. It derives the corresponding dispersion relation, the leading frequency-independent helicity phase residue, and the fixed-frequency wavenumber splitting. The work compares this B-residue scaling with two standard reference cases: Faraday rotation, where the polarization angle scales with wavelength squared; parity-violating k³ effective-field-theory birefringence, where the phase rotation scales as frequency squared. The note also gives an identifiability theorem for BBH-only gravitational-wave tests. For a model ψ_obs = ψ_source + αD mod π, a uniform unknown source polarization angle makes the distance-dependent frequency-independent rotation parameter α non-identifiable after marginalization. Therefore a clean observational test requires an external polarization or orientation anchor, a multimessenger event, a population-level distance trend, or an electromagnetic/CMB polarization channel. The companion code reproduces the model expansion, compares the B-residue scaling with k³ EFT scaling, and demonstrates the identifiability result using synthetic tests.