Astrophysics Coronal Lyapunov Time Reference Time

The reference tick

The referenceTime declaration in the Recognition Science framework's machine-checked library of formal theorems sets a baseline unit: it defines the reference timescale as exactly 1 in the framework's native units, and identifies that unit with one Alfvén crossing time in the solar corona. The Alfvén crossing time is the time a magnetic disturbance takes to travel across a region, here roughly one second at the solar surface where the magnetic field is about 100 gauss. This is a definitional choice, a model, not a measured value; the declaration simply fixes the scale from which other timescales are counted.

From that single reference point, the framework derives a ladder of coronal timescales. The definition coronalTime multiplies the reference by powers of the golden ratio phi, about 1.618, so that each rung of the ladder is phi times longer than the one below it. The formal theorems prove that each successive timescale is strictly larger than the previous one, and that the ratio of adjacent timescales is exactly phi. The ladder runs from the Alfvén crossing at about one second, through granulation at about a minute, chromospheric evaporation at about a thousand seconds, coronal loop lifetimes at about an hour, and active region emergence at about a day. These are predictions, not established measurements.

The framework's library proves the internal consistency of this ladder: the positivity, the strict increase, and the exact phi ratio are all theorem-checked. What the library does not prove is that the solar corona actually behaves this way. The physical claim is a hypothesis with a named falsifier: if two adjacent coronal Lyapunov timescales are measured to differ by a ratio systematically outside the range 1.5 to 1.8 across at least three active regions, the ladder fails. The referenceTime itself claims nothing about the real Sun; it only sets the counting unit. The leap from a formal definition to a physical prediction is the hypothesis, and it is the part that observations can still break.

/-- Reference Alfvén-crossing timescale (RS-native 1). -/
def referenceTime : ℝ := 1

/-- Coronal timescale at φ-ladder rung `k`. -/
def coronalTime (k : ℕ) : ℝ := referenceTime * phi ^ k

theorem coronalTime_strictly_increasing (k : ℕ) :
    coronalTime k < coronalTime (k + 1) := by
  rw [coronalTime_succ_ratio]
  have hk : 0 < coronalTime k := coronalTime_pos k
  have hphi_gt_one : (1 : ℝ) < phi := by
    have := Constants.phi_gt_onePointFive; linarith
  have : coronalTime k * 1 < coronalTime k * phi :=
    mul_lt_mul_of_pos_left hphi_gt_one hk
  simpa using this

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

structure CoronalLyapunovCert where
  time_pos : ∀ k, 0 < coronalTime k
  one_step_ratio : ∀ k, coronalTime (k + 1) = coronalTime k * phi
  strictly_increasing : ∀ k, coronalTime k < coronalTime (k + 1)
  adjacent_ratio_eq_phi : ∀ k, coronalTime (k + 1) / coronalTime k = phi

What this page does not claim

The referenceTime declaration does not assert that the solar corona actually exhibits these timescales. The framework does not derive the golden ratio from solar physics; it applies a pre-existing constant to a new domain. No measurement of coronal Lyapunov timescales is presented or claimed in the pack.

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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