Astrophysics Galactic Rotation Curve From Rs Rotation Regime

The five rotation regimes

Astronomers who map how fast stars and gas orbit a galaxy's center have long sorted the resulting rotation curve into recognizable phases. Moving outward from the center, a typical curve rises steeply, then flattens out to a roughly constant speed, and eventually declines or is truncated at the galaxy's edge. The standard description also includes an inner region that rotates like a rigid body and a transitional rising segment. In the Recognition Science framework, these five phases are collected into a single formal object called RotationRegime, a discrete classification with exactly five cases: rigidBodyInner, rising, flat, declining, and truncation.

The framework's machine-checked library of formal theorems proves that this classification has exactly five members, and it defines a transition radius for each regime boundary. The key structural claim is that consecutive transition radii are separated by a constant ratio equal to the golden ratio φ, approximately 1.618. This is not an empirical fit but a proved consequence within the framework: the theorem transitionRadius_ratio derives that the ratio of successive radii is exactly φ. The framework also proves that every transition radius is positive, so the sequence of regime boundaries is well-ordered and never collapses to zero.

In Recognition Science, this structure is presented as a model of how a galaxy's rotation curve is organized, not as a measurement of any particular galaxy. The declaration bundles the five-regime count, the φ-ratio property, and the positivity condition into a single certificate object. That certificate is a definitional choice: it packages the framework's claims about rotation curves into one verifiable unit. The framework's contribution is to show that if you accept its starting principles, the five-regime structure with golden-ratio boundaries follows by formal proof, with no gaps and no extra assumptions.

What the declaration does not claim is just as important. It does not assert that any observed galaxy actually exhibits all five regimes, nor does it predict the specific radius in kiloparsecs where a given galaxy's curve flattens. The framework provides a formal template for the shape of a rotation curve, not a catalog of real galaxies. Whether real rotation curves match this template is an empirical question that the declaration itself does not address. The framework's claim is about the internal consistency of its own model, not about the astronomical data.

theorem rotationRegime_count : Fintype.card RotationRegime = 5 := by decide

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

No observed galaxy is asserted to exhibit all five regimes. No prediction is made for the radius in kiloparsecs where a specific galaxy's rotation curve flattens. The declaration does not address whether real rotation curves match the framework's template.

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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