Beyond the Ring: How Flat, and How High
Real velocity points normalized by a single visible ring — and the residual that remains
When every real rotation point is divided by its galaxy’s reference ring (one visible disk+gas ring at its mass-weighted radius), the curves flatten beyond R/R_ring ≈ 1 — they stop rising and settle into a band. But the band sits above one: the plateau ratio V_obs/V_ring has median 1.37, with 105/132 galaxies above unity. The visible ring sets the right shape and scale, but a residual of order +37% remains — the missing velocity, still there.
1. The construction (from frozen tables)
Each galaxy’s measured rotation points (SPARC) are divided, axis by axis, by the reference values computed for that galaxy: radius by R_ring, velocity by V_ring, where the ring carries the disk+gas mass at its mass-weighted radius. Both quantities come from the fixed per-galaxy table — not recomputed on the fly — so the plot and the numbers are guaranteed consistent. The ring reference is anchored, by construction, at the point (1, 1).
2. Flat — but above one
The flattening is real: past the ring radius the curves stop climbing and run nearly horizontal. The eye is drawn to how close to 1 they look. But measured properly — the median of V_obs/V_ring beyond R/R_ring > 1.5 for each galaxy — the plateau is not at 1. It is at 1.37, and most galaxies lie above unity.
3. What it means
Two things are true at once. First, the single visible ring is a good reference: it sets the radius where the curve flattens and the order of magnitude of the plateau, across four decades of mass. The curves do collapse onto a shared, flat shape beyond it. Second, that flat shape sits above the ring’s own velocity: typically the observed plateau exceeds what the visible ring alone would produce by about 37%. That excess is the missing velocity — the same deficit, now expressed as a clean ratio.
The split by type is telling: bulged galaxies sit closer to 1 (median 1.18), because their concentrated mass is better captured by a compact ring; disk+gas galaxies sit higher (median 1.43), their extended, diffuse mass leaving a larger residual.
A single visible ring already explains the shape of every rotation curve — flat beyond one ring-radius — and most of its scale. What it does not explain is the last 37%: the plateau sits consistently above the ring. This residual, tight and repeatable rather than random, is exactly what a wave-mass term must supply. The ring gets the curve flat; the missing piece is how high.
All velocity points are real SPARC data (Lelli, McGaugh & Schombert 2016). Reference values (R_ring, V_ring) are read from the fixed per-galaxy table, where masses use the velocity decomposition and the wave mass is held at 15% of the visible (Solar-System ratio), with total mass conserved. The “flat near 1” impression of the calibration subset becomes, on the full 132 galaxies, a plateau at 1.37 — flat in shape, but above unity in level.
BeeTheory.com — Beyond the ring: how flat, how high · Data: Lelli, McGaugh & Schombert 2016 · Initial generation: 21 May 2026 with Claude.ai · © Technoplane S.A.S. 2026