Astrophysics Galactic Rotation Curve From Rs Transition Radius Pos

The positive transition radius

Galactic rotation curves plot how fast stars and gas orbit a galaxy's center against their distance from it. Observations show a common sequence of shapes: a rigid-body rise near the center, a rising segment, a flat stretch (the part that motivates dark matter or modified gravity), a decline, and a truncation. In the Recognition Science framework, these five regimes are modeled as five discrete states, and the radius where the curve moves from one regime to the next is called a transition radius, a boundary between two adjacent rotation behaviors.

The framework defines each transition radius as phi^k, where phi is the golden ratio (about 1.618) and k is a whole number. The declaration transitionRadius_pos proves that for every whole number k, this radius is strictly greater than zero. That is a small but necessary fact: a radius must be a positive distance, and the proof guarantees the definition never produces a zero or negative boundary. The companion theorem transitionRadius_ratio shows that the ratio of consecutive transition radii is exactly phi, so the boundaries are spaced by a constant multiplicative step.

In Recognition Science, the golden ratio arises from the framework's central cost function, and the five regimes here match the framework's five-dimensional configuration space. The machine-checked library of formal theorems records these results with no unproved assumptions. But the declaration does not claim that real galaxies actually show this exact spacing, nor that the five regimes are the only possible ones, nor that the golden ratio is measured in any rotation curve. It only establishes a formal property of a defined quantity.

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

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

What this page does not claim

No claim that observed galaxies exhibit golden-ratio transition radii. No claim that the five regimes are the only possible rotation-curve shapes. No claim about the physical mechanism that sets the transition radii.

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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