Astrophysics Coronal Lyapunov Time Coronal Time Succ Ratio

The rung ratio

The solar corona, the Sun's outer atmosphere, is threaded by magnetic fields that twist and reconnect chaotically. One way to measure that chaos is the Lyapunov time, the timescale on which two nearly identical magnetic field lines diverge exponentially. The framework's library of formal theorems defines a ladder of such coronal timescales, starting from one Alfvén crossing time, the fastest coherent magnetic mode, roughly one second at the solar surface. Each rung of the ladder is a power of the golden ratio, approximately 1.618, times that reference time.

The declaration coronalTime_succ_ratio is a theorem stating that the timescale at the next rung equals the current timescale multiplied by the golden ratio. In symbols, for any rung index k, coronalTime(k + 1) = coronalTime(k) × φ. The proof is a direct consequence of the definition of coronal time as a power of φ; it is not an empirical measurement. A companion theorem states that the ratio of adjacent timescales is exactly φ, and another that the sequence is strictly increasing. Together they form a certificate: a packaged collection of these properties, all machine-checked with no unproven assumptions.

In Recognition Science, this ladder is not an isolated curiosity. The framework derives the golden ratio as the unique self-similar scaling from its cost function, and then finds the same φ-ladder structure across solar, stellar, and astrophysical timescales. The corona is one instance of a pattern the framework claims is universal. The theorem itself, however, establishes only the arithmetic of the ladder: it says that if you define coronal times as powers of φ, then adjacent rungs relate by φ. It does not establish that real coronal timescales actually follow this ladder.

That empirical question is a prediction with a named falsifier: if two adjacent coronal Lyapunov timescales, measured on a corpus of at least three active regions, differ by a ratio systematically outside the interval (1.5, 1.8), the prediction fails. The formal theorem is silent on whether the Sun obeys it. The framework's claim is that the structure is forced by its axioms; the check against the real corona remains an open experimental test.

theorem coronalTime_succ_ratio (k : ℕ) :
    coronalTime (k + 1) = coronalTime k * phi := by
  unfold coronalTime; rw [pow_succ]; ring

theorem coronal_adjacent_ratio (k : ℕ) :
    coronalTime (k + 1) / coronalTime k = phi := by
  rw [coronalTime_succ_ratio]
  field_simp [(coronalTime_pos k).ne']

/-- Coronal Lyapunov timescale certificate. -/
def coronalLyapunovCert : CoronalLyapunovCert where
  time_pos := coronalTime_pos
  one_step_ratio := coronalTime_succ_ratio
  strictly_increasing := coronalTime_strictly_increasing
  adjacent_ratio_eq_phi := coronal_adjacent_ratio

theorem coronalTime_succ_ratio (k : ℕ) :
    coronalTime (k + 1) = coronalTime k * phi := by
  unfold coronalTime; rw [pow_succ]; ring

What this page does not claim

The theorem does not establish that measured coronal Lyapunov times actually follow the φ-ladder. The theorem does not derive the golden ratio; it assumes the ladder's definition as powers of φ. No empirical measurement of coronal timescales is presented in this declaration.

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/CoronalLyapunovTime.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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