Rotation Curves as the Shape of a Label Space: the Label Framework Applied to Galaxies

Collisionless dark matter described by three comoving labels has a density fixed by the label measure. Around a centre the label blade splits into a radial factor, the enclosed mass, and an angular blade dual to the Newtonian field. We read rotation curves geometrically in these terms: a measure uniform in volume gives a core with rigid rotation, a measure uniform in radius gives a flat curve, and maximal mixing under the phase-space volume-preserving symmetry contains both in one isothermal state. The split does not predict a profile. If the dark-to-baryon ratio and the level of the k-label density at the onset of the plateau are common to a population, then M_bar ∝ v^(2k/(k−1)), with slope 4 for the angular density, consistent with low-acceleration baryonic systems. The deep-MOND law appears as a formal scaling relation with a0 as an orbit label.