Astrophysics Coronal Timescale From Phi Ladder Coronal Timescale Count

The five-rung count

The solar corona spans a vast range of timescales, from the roughly ten seconds an Alfvén wave takes to cross a magnetic loop to the roughly six days an active region persists. Between those extremes sit granulation convection, chromospheric evaporation, and coronal loop lifetimes. A recognition ledger, a discrete record of events the framework uses to track physical structure, organizes these five durations as consecutive rungs on a ladder.

The framework's machine-checked library of formal theorems proves that the count of these five named timescales is exactly five. That is the declaration coronalTimescaleCount: a theorem, checked by the kernel, that the inductive type CoronalTimescale has exactly five members. The five members are Alfvén crossing, granulation, chromospheric evaporation, coronal loop, and active region.

The same library proves that adjacent rungs on this ladder scale by the golden ratio φ. Formally, for any natural number k, the ratio of timescale at rung k+1 to timescale at rung k equals φ. This is a theorem about the definition timescaleAtRung, which maps a rung index k to φ^k. The proof is a short algebraic computation using the definition and the positivity of φ.

These two results combine into a certificate structure, CoronalTimescaleCert, that bundles the count of five and the φ-ratio property. The certificate is a single object asserting both facts, and the library constructs it directly from the two theorems. The docstring notes that the adjacent-step ratios in observed solar timescales are approximately 10, which is close to φ^5 ≈ 11.09, and that the five rungs span five decades.

What the declaration does not claim is empirical confirmation. The theorem establishes a formal count and a formal ratio for the named ladder. It does not assert that the Sun's corona actually exhibits these exact ratios; that comparison against observed solar timescales is an empirical check, not a theorem. The docstring's numbers, such as 10 seconds and 600 seconds, are illustrative observations, not formal claims.

theorem coronalTimescaleCount : Fintype.card CoronalTimescale = 5 := by decide

theorem timescaleRatioPhiRung (k : ℕ) :
    timescaleAtRung (k + 1) / timescaleAtRung k = phi := by
  unfold timescaleAtRung
  have hpos := pow_pos phi_pos k
  rw [pow_succ, div_eq_iff hpos.ne']
  ring

What this page does not claim

The theorem does not assert that observed solar timescales actually follow the phi-ratio; that is an empirical check. The declaration does not derive the physical bridge from recognition events to coronal dynamics. The five named timescales are a definitional choice, not a forcing result.

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/CoronalTimescaleFromPhiLadder.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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