Astrophysics Galactic Rotation Curve From Rs Galactic Rotation Cert

The rotation certificate

A galactic rotation curve plots how fast stars and gas orbit a galaxy's center against their distance from it. Astronomers have long recognized five standard shapes along such a curve: a rigid-body inner rise, a rising segment, a flat segment, a declining segment, and a truncation at the edge. The Recognition Science framework models these five regimes as a single finite type, and its GalacticRotationCert declaration packages the claim that there are exactly five of them.

The certificate also records a structural rule about where one regime hands off to the next. Each transition radius, the distance at which the curve changes shape, is set to phi raised to an integer power, where phi is the golden ratio, about 1.618. The certificate proves that the ratio of consecutive transition radii is always phi, and that every transition radius is positive. These are formal theorems in the framework's machine-checked library of formal theorems, with zero unproved assumptions.

What the certificate does not do is assert that real galaxies actually follow this pattern. It defines the five regimes and proves the ratio property for the framework's own model of transition radii. The connection between this formal object and observed galactic rotation curves is an empirical question, not a theorem. The certificate is a clean package of definitions and proofs about the framework's internal model, not a measurement of the night sky.

The practical payoff is a compact, auditable object. Anyone working inside the framework can import GalacticRotationCert and rely on the five-regime count and the phi-ratio property without rechecking the arithmetic. It is a building block for further astrophysics work in the framework, not a claim about the universe's actual rotation curves.

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

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

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

What this page does not claim

The certificate does not claim real galaxies follow the phi-ladder pattern. No claim that the five regimes are the only possible rotation-curve shapes in nature. The certificate does not derive the flat rotation curve from dark matter or MOND.

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)

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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