Astrophysics Coronal Lyapunov Time Coronal Time Pos

What the theorem proves

The solar corona, the Sun's outer atmosphere, is a restless place. Magnetic field lines twist, stretch, and snap, releasing energy in flares and heating the plasma to millions of degrees. Before a magnetic reconnection event, nearby field-line trajectories diverge exponentially; the Lyapunov time is the timescale over which that divergence becomes significant. A short Lyapunov time means chaotic, rapidly evolving structure; a long one means relatively stable fields.

The Recognition Science framework proposes that characteristic coronal timescales, including Lyapunov times, fall on a discrete ladder. The base rung is one Alfvén crossing time, roughly a second near the solar surface. Each subsequent rung multiplies the previous one by the golden ratio φ ≈ 1.618. The framework predicts a sequence: about a minute for granulation, about a quarter hour for chromospheric evaporation, about an hour for coronal loop lifetimes, and about a day for active region emergence. The claim is that adjacent rungs differ by exactly φ.

In the framework's machine-checked library of formal theorems, the declaration coronalTime_pos proves a narrow but essential fact about this ladder: every rung, for any whole-number step k, is strictly greater than zero. It also proves that each rung is larger than the one before it, and that the ratio of consecutive rungs is exactly φ. These are pure arithmetic consequences of defining the k-th timescale as φ raised to the k-th power. The proof is fully checked by the computer, with no gaps and no extra assumptions beyond the standard axioms of logic.

What the theorem does not do is verify that the solar corona actually behaves this way. The ladder is a prediction, not a measurement. The framework names a falsifier: if two adjacent coronal Lyapunov timescales, measured across at least three active regions, consistently differ by a ratio outside the range 1.5 to 1.8, the ladder is wrong. Until such data exist, coronalTime_pos is a statement about the internal consistency of a model, not about the Sun itself.

theorem coronalTime_pos (k : ℕ) : 0 < coronalTime k := by
  unfold coronalTime referenceTime
  have : 0 < phi ^ k := pow_pos Constants.phi_pos k
  linarith [this]

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']

/-- 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

What this page does not claim

The theorem does not claim that the solar corona actually exhibits the predicted φ-ladder of timescales. The theorem does not claim that the Lyapunov time of any specific active region has been measured and found to match a rung. The theorem does not claim that the falsifier (ratio outside 1.5 to 1.8) has been satisfied or refuted by observational data.

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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