Cosmology Galaxy Rotation Tully Fisher

The Tully-Fisher relation

The Tully-Fisher relation is an observed correlation in astronomy: a spiral galaxy's total luminosity L is proportional to the fourth power of its rotation velocity v, so L ∝ v⁴. Brighter galaxies spin faster, and the relation holds tightly enough that astronomers use it to estimate distances to galaxies too far for parallax. The scatter around the relation is small, under 0.1 dex, meaning the prediction is accurate to within about 25 percent.

The standard explanation for the relation runs through dark matter. A galaxy's visible stars and gas account for only a fraction of its mass; the rest is thought to be a halo of dark matter. The observed flat rotation curves, where orbital speed stays roughly constant far from the center instead of falling off, require this extra mass. The Tully-Fisher relation then reflects a deeper link between the total mass of a galaxy and the speed of its outer rotation.

In Recognition Science, the framework models dark matter as ledger shadows, discrete records of recognition events in odd phases of an eight-tick cycle. The framework's library, a machine-checked collection of formal theorems, contains a declaration named tully_fisher. That declaration records the intent to explain the observed L ∝ v⁴ relation from the framework's J-cost equilibrium distribution. The declaration itself, however, concludes in a tautology, the trivial statement True. It establishes the target, not the result.

The framework also records a falsifier for the broader galaxy rotation account. The structure GalaxyRotationFalsifier names the conditions that would refute the framework's picture: rotation curves that are not flat, a competing theory that works everywhere, or a mismatch between ledger predictions and observation. This is the honest shape of a scientific claim: a prediction with a named way to fail.

What the declaration does not claim is the establishment of the Tully-Fisher relation itself. The observed relation is a measurement, not a framework theorem. The framework's contribution is a target for future proof, a statement of what a complete derivation would need to show. Until that proof exists, the relation remains an observed fact awaiting a framework explanation.

/-- The baryonic Tully-Fisher relation is VERY tight:
    M_baryon ∝ v⁴

    Scatter is remarkably small (< 0.1 dex).
    This suggests a fundamental relationship.

    CDM has trouble explaining the tightness.
    MOND explains it naturally. -/
def btfrData : String :=
  "M_baryon ∝ v⁴ with scatter < 0.1 dex"

/-- The Tully-Fisher relation:
    L ∝ v⁴ (luminosity scales as 4th power of rotation velocity)

    MOND predicts this naturally.
    CDM requires it to arise from galaxy formation. -/
theorem tully_fisher :
    -- L ∝ v⁴ observed and predicted by MOND
    True := trivial

/-- The Tully-Fisher relation:
    L ∝ v⁴ (luminosity scales as 4th power of rotation velocity)

    MOND predicts this naturally.
    CDM requires it to arise from galaxy formation. -/
theorem tully_fisher :
    -- L ∝ v⁴ observed and predicted by MOND
    True := trivial

/-- The derivation would be falsified if:
    1. Rotation curves not flat (already confirmed)
    2. No dark matter (MOND works everywhere)
    3. Ledger distribution doesn't match observations -/
structure GalaxyRotationFalsifier where
  curves_not_flat : Prop
  mond_works_everywhere : Prop
  ledger_mismatch : Prop
  falsified : curves_not_flat ∨ mond_works_everywhere → False

What this page does not claim

Not claimed: the framework proves the Tully-Fisher relation; the declaration is a stated target, not a proved theorem. Not claimed: dark matter as ledger shadows is established; it is a modeling choice within the framework. Not claimed: the framework provides a complete alternative to MOND or dark matter; it records a falsifier, not a finished theory.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Cosmology/GalaxyRotation.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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