The constant-Lagrangian (CL) condition DΓ/Dt = 0 is stated at the rapidity level, with Γrotor = cosh χr cosh χφ; in the Galilean-coframe metric its slow-flow counterpart is γrs, the river’s Lorentz factor relative to the static Killing observers (the BQ-Pauli page CL in the metric). At the Dirac level both are the same Clifford operation: the product of two unit time vectors. That product is the square of the rotor between the two observers; its scalar part is the γ factor, its bivector part the direction of the relative velocity.
CL then says, at both levels, that the rotor between the static and the river observer keeps its rapidity along the flow: an equi-redshift flow. The two levels differ only in how the radial and the azimuthal motion combine: the rotor composes the rapidities relativistically, the metric adds the river velocities in the Galilean way. Choose the parameters.
Units: c = 1, rs = 2GM/c² = 1. Radial river vr = √(2GM/r) − Hr (paper chart, Hubble term only when switched on); Keplerian azimuthal river vφ = √(GM/r); CL river vφ = √(K − vr²) for r ≥ R, rigid inside R.
γrs − 1 is the fractional clock slowing of the river relative to the static observers. With the Keplerian azimuthal river it falls as r grows; the CL river keeps it constant beyond R: the river flows along an equi-redshift level.
Galilean coframe of the gravity-rotor page, paper chart: θt = cβ0 − vrβr − vφβφ, θr = βr, θθ = rβθ, θφ = r sθβφ, with gμν𝟙 = ½{θμ, θν} and gtt = −(c² − vr² − vφ²). Here βr = βI, βφ = βK, αr = β0βr, αφ = β0βφ.
CL condition, isorapidity Γ = cosh χr cosh χφ and the 3GM/R profile from de Haas (2026), The Gravitational Rapidity Potential and Relativistic Gravitational Fluid Dynamics; the BQ-Pauli version is CL in the metric, the rotor and coframe are those of the Dirac gravity rotor. Every identity is checked exactly in bq_dirac_cl_redshift.py. Part of the BQ algebra at the Dirac level.