A Geometric Law for Disk+Gas Galaxies
Two frozen coefficients, calibrated on half the sample, tested blind on the other half
For bulgeless (disk+gas) galaxies, the plateau velocity can be predicted from two purely geometric reference quantities — the ring velocity V_ring and the extent R_max/R_ring — through a law with two fixed coefficients. Calibrated on half the galaxies and applied blind to the other half, it predicts the plateau to a median error of 17% (44/71 within 20%). The visible ring carries most of the signal; the geometric term refines it.
1. The inputs — all geometric, from the reference model
Each galaxy is reduced to one visible ring (disk+gas at its mass-weighted radius). Three reference quantities follow, none of them fitted to the rotation curve:
| Symbol | Meaning |
|---|---|
| M_ring | total disk+gas mass |
| R_ring | mass-weighted radius of the ring |
| V_ring | √(G·M_ring/R_ring), the ring’s own circular velocity |
| R_max/R_ring | extent: how far the galaxy is seen relative to the ring radius |
2. The law and its coefficients
The deficit between the observed plateau and the ring velocity, expressed as a relative excess, follows a simple two-coefficient form fitted on the calibration half:
and the predicted plateau is V_plateau = V_ring · (1 + ΔV/V_ring). The two coefficients on the geometric terms are -0.101 (extent) and -0.222 (ring velocity); the constant is 1.428. The negative extent coefficient means more extended galaxies show a smaller relative excess — their diffuse mass is already well captured by the ring.
3. Blind test
The coefficients are then frozen and applied to the other half of the galaxies — never seen during calibration. No per-galaxy tuning.
The blind prediction lands within 20% for 44 of 71 galaxies and within 30% for 56. The agreement is carried mainly by V_ring: the visible ring alone fixes the order of magnitude of the plateau, and the geometric term adjusts the rest.
The strong claim is the blind test: two frozen coefficients, no per-galaxy freedom, predict the plateau of unseen galaxies to ~17%. The honest qualifier: most of that success comes from V_ring (the visible ring), with the geometric extent term adding a modest correction. The residual scatter — the part neither V_ring nor R_max/R_ring captures — is real and reminds us the reduction to a single ring loses some information. This is a calibrated empirical law for disk+gas galaxies, not yet a first-principles derivation.
All velocities are real SPARC data (Lelli, McGaugh & Schombert 2016). Reference quantities (M_ring, R_ring, V_ring, R_max) come from the fixed per-galaxy table. The 143 disk+gas galaxies were split in two by name: 72 for calibration, 71 held out for the blind test. The coefficients shown were fitted only on the calibration set.
BeeTheory.com — A geometric law for disk+gas galaxies · Data: Lelli, McGaugh & Schombert 2016 · Initial generation: 21 May 2026 with Claude.ai · © Technoplane S.A.S. 2026