Astrophysics Galactic Rotation Curve From Rs Transition Radius

The transition radius

A galactic rotation curve plots how fast stars and gas orbit a galaxy's center against their distance from it. Observed curves show distinct stretches: a rigid-body rise near the core, a rising segment, a flat portion where orbital speeds stop increasing, a decline, and finally a truncation where the disk ends. The Recognition Science framework models each of these five regimes as a discrete state, and it defines a transition radius, a distance where the curve moves from one regime to the next.

The definition is starkly simple: for any whole-number step k, the transition radius equals phi raised to the k-th power, where phi is the golden ratio, about 1.618. The framework's machine-checked library of formal theorems proves two facts about this definition. First, each transition radius is positive. Second, the ratio of one transition radius to the previous one is exactly phi; each step up the ladder multiplies the radius by the golden ratio. This is not a fitted parameter. It is a consequence of the definition itself, and the library checks the proof with no gaps.

The framework's library also bundles these facts into a certificate structure, a formal object that packages the five-regime count, the phi ratio, and the positivity proof together. The certificate exists to let other formal developments rely on the rotation-curve setup without rechecking its pieces. It is a bookkeeping device inside the library, not a new physical law.

What the declaration does not claim is just as important. It does not assert that any real galaxy's rotation curve actually follows this ladder. It does not predict the measured flatness of observed curves or explain dark matter. The definition is a modeling choice inside the framework, and the theorems only describe the arithmetic of that choice. Whether the ladder matches astrophysical data is an empirical question the declaration leaves open.

inductive RotationRegime where
  | rigidBodyInner
  | rising
  | flat
  | declining
  | truncation
  deriving DecidableEq, Repr, BEq, Fintype

noncomputable def transitionRadius (k : ℕ) : ℝ := phi ^ k

theorem transitionRadius_ratio (k : ℕ) :
    transitionRadius (k + 1) / transitionRadius k = phi := by
  unfold transitionRadius
  have hpos : (0 : ℝ) < phi ^ k := pow_pos phi_pos k
  rw [div_eq_iff hpos.ne', pow_succ]
  ring

theorem transitionRadius_pos (k : ℕ) : 0 < transitionRadius k :=
  pow_pos phi_pos k

noncomputable def galacticRotationCert : GalacticRotationCert where
  five_regimes := rotationRegime_count
  phi_ratio := transitionRadius_ratio
  radius_always_pos := transitionRadius_pos

What this page does not claim

The declaration does not assert that any real galaxy's rotation curve follows the phi-ladder. It does not explain the flatness of observed rotation curves or the dark matter problem. The transition radius definition is a modeling choice, not a derived physical prediction.

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/Astrophysics/GalacticRotationCurveFromRS.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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