Cadence Balance Tour — Regime 1 ⭐
Rotation Curves & the BTFR: Where Gravity First Cracks
If you want to see where gravity first slips, look at a spiral galaxy.
Newton gives a simple prediction:
Beyond the first few kiloparsecs, the farther out you go, the slower the stars should orbit.
A neat falloff:
LaTeX: v(r) \propto \frac{1}{\sqrt{r}}
But galaxies don’t fall off.
Instead, the expected rotation curve flattens into a straight line.
Galaxies spin like they’re on a string
Across thousands of observations, the outer stars settle into a nearly constant rotation speed — as if the whole disk were guided by a taut invisible line rather than pulled by ordinary gravity.
Different shapes.
Different sizes.
Same behavior.
Already, something is wrong with the classical picture.
But the deeper clue is the pattern hiding in those speeds.
Astronomers found a line too precise to ignore
When you take any galaxy and compare:
its total visible (baryonic) mass, M
the flat rotation speed the galaxy settles into, v∞
you get an astonishingly clean relation:
LaTeX: v_{\infty}^4 \propto M_{\mathrm{baryonic}}
This is the Baryonic Tully–Fisher Relation.
You don’t need the name; the meaning is simple:
Once you know a galaxy’s visible mass, you can predict its flat rotation speed — with almost no scatter.
And the exponent behind this relation — roughly 1/4 — shows up everywhere rotation curves bend.
Gravity doesn’t predict this.
But it is what we see.
The standard fix: invisible matter, tuned per galaxy
The ΛCDM model handles this by placing every galaxy inside an undetected “dark matter” halo.
These halos are shaped — one galaxy at a time — so the curves flatten and the mass–speed line comes out right.
It works numerically.
But it works because the halo is tuned to make it work.
Each galaxy must be tuned on its own, and there is no underlying pattern to the tuning that anyone has been able to find.
Light Frame Cadence doesn’t tune — it balances
Light Frame Cadence starts with one principle:
galaxies balance two kinds of curvature at a universal acceleration floor, a₀ — the point where their behavior shifts.
inward curvature (TD)
outward curvature (TS)
That balance alone forces the flat outer speed and the fourth-power mass relation.
Light Frame Cadence’s mass-scaling rules are simple:
Because TS grows with mass as M¹ᐟ² while TD grows more slowly as M¹ᐟ⁴, the two naturally meet at the shared acceleration floor a₀.
Balance them at a₀, and the 0.25 BTFR value appears automatically:
LaTeX: v_{\infty}^4 \propto M
No dark matter.
No tuning.
No per-system adjustments.
Just the geometry settling where it must.
The reason this slipped past every model is simple: no one was looking for two different kinds of curvature. TS grows like M¹ᐟ², TD only like M¹ᐟ⁴, and the moment you force them to meet at the same floor a₀, the observed quarter-power law falls out. GR has no such balance principle, so the BTFR looked “mysterious” instead of inevitable.
Three galaxies that show the rule at work
DDO 154 (dwarf)
Its baryonic mass is tiny.
Plug that mass value into the cadence formula — you get the correct flat speed.
ΛCDM needs a creative halo; cadence needs only the observed mass and the balance condition.
NGC 2403 (mid-sized spiral)
Bright center, extended disk.
Inner region: TD-dominant, Newton-like.
Outer region: TS-dominant, flat.
The transition radius predicted from its baryonic mass matches the observed curve.
UGC 128 (LSB galaxy)
Very little light.
Very extended.
Rotation curve flattens almost immediately.
Cadence predicts this: weak TD means TS takes over early; the flat speed follows from its baryonic mass.
And it’s not just these examples — it’s essentially the entire catalog
From the 175 SPARC galaxies (171 with complete data), every one follows the same cadence scaling, tightening around 0.256.
Flat speeds fall out of baryonic mass.
The BTFR slope appears with Δ = 1/4.
The TS–TD balance shows up at the expected 0.25 transition.
No halo tuning anywhere.
The galaxies behave as if the geometry is doing the work.
(Data: SPARC rotation curves, Lelli et al. 2016)
Why this is first in the tour
Rotation curves are the cleanest demonstration of cadence balance.
The universe shows the same rhythm —
the same exponent, the same flattening, the same mass–speed law.
Gravity alone struggles.
The data behave as if Light Frame Cadence — not invisible matter — is setting the rules.
It lands on the pattern without effort.
This is where the story begins.
Next: Regime 2 — The Deep Radial Acceleration Relation
A square-root law appears in the data — one no gravitational theory predicted,
and one Light Frame Cadence explains in a single line.