Astrophysics Solar Wind From Phi Ladder Solar Wind Type

The five wind bands

The solar wind, the stream of charged particles the Sun releases, is usually described by its speed. Astronomers recognize three broad bands: slow wind around 300 to 400 km/s, fast wind around 600 to 800 km/s, and extreme events such as coronal mass ejections above 1000 km/s. A finer, five-way split adds intermediate and quiet states, giving the five canonical types: quiet, slow, intermediate, fast, and extreme.

In Recognition Science, a machine-checked library of formal theorems defines a type called SolarWindType with exactly those five constructors. A proved theorem in the library, solarWindTypeCount, states that the count of these types equals 5. The library also defines a speed function on natural numbers, solarWindSpeed, which assigns to each index k the value phi^k, where phi is the golden ratio, approximately 1.618. A second proved theorem, solarWindSpeedRatio, shows that the ratio of speeds at adjacent indices is exactly phi: solarWindSpeed(k+1) / solarWindSpeed(k) = phi. The framework models the five observed bands as five adjacent rungs on this phi-ladder, so the ratio of a fast band to its slower neighbor is the golden ratio.

What the declaration does not claim is equally important. The type and the two theorems establish a count and a ratio within a formal model; they do not assert that the actual measured solar wind speeds match this ladder. The docstring offers a rough comparison, slow/fast wind ratio approximately phi, but that is an empirical check, not a proved theorem. The framework also does not claim that the five-way classification is the only correct one; it is a definitional choice that the model adopts. The theorems are about the formal objects, not about the physical plasma.

The practical upshot is a compact, internally consistent vocabulary for the solar wind's speed structure. A reader of the framework's library can rely on the count of five types and the golden-ratio spacing between adjacent speeds as proved facts within that system. The physical question, whether the real solar wind obeys this spacing, remains a matter for observation and is not settled by the formal library.

theorem solarWindTypeCount : Fintype.card SolarWindType = 5 := by decide

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

inductive SolarWindType where
  | quiet | slow | intermediate | fast | extreme
  deriving DecidableEq, Repr, BEq, Fintype

What this page does not claim

The measured solar wind speeds match the phi-ladder. The five-way classification is physically necessary rather than a modeling choice. The theorems apply to the real solar wind without further empirical assumptions.

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/SolarWindFromPhiLadder.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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: