Astrophysics Picsimulation Lyapunov Reference Exponent

Reference exponent

The reference exponent is the base value from which a predicted family of chaos measures is built. In the framework, the declaration sets this base to exactly 1. From that anchor, the value at any resolution level k is defined as the reference exponent multiplied by phi raised to the power negative k, where phi is the golden ratio. The first level, k equal to zero, therefore has the reference value 1 itself.

This definition produces a strict geometric ladder. Each step to the next resolution level multiplies the previous value by phi to the power negative one, so the ratio of any two adjacent levels is always that same factor. The framework's machine-checked library of formal theorems proves both that every level is positive and that this adjacent ratio holds for every level. A certificate structure packages these two properties together, and the library constructs that certificate from the proofs.

The setting of the reference exponent to 1 is a definitional choice, not a derived result. The ladder structure that follows is a theorem within the framework, but the initial value itself is chosen. The framework's library does not prove that any real plasma simulation has a Lyapunov exponent of exactly 1 at any resolution. The empirical bench cited in the framework documentation, from Dawson 1983 and Birdsall-Langdon 2004, reports that adjacent doubling of macro-particles per Debye cell reduces numerical heating by phi squared, which the framework treats as matching its canonical scaling. That comparison is an empirical check, not a theorem.

In plain terms, the declaration establishes the shape of a prediction: chaos measures in particle-in-cell simulations should fall on a golden-ratio ladder, with the reference level set to 1. What it does not claim is that any measured exponent equals 1, or that the ladder has been experimentally confirmed. The framework offers the ladder as a structural prediction, with the reference value as its chosen starting point.

/-- Reference Lyapunov exponent at `N_ppc` rung 0. -/
def referenceExponent : ℝ := 1

theorem lyapunovAt_adjacent_ratio (k : ℕ) :
    lyapunovAt (k + 1) / lyapunovAt k = phi⁻¹ := by
  rw [lyapunovAt_succ_ratio]
  field_simp [(lyapunovAt_pos k).ne']

theorem lyapunovAt_pos (k : ℕ) : 0 < lyapunovAt k := by
  unfold lyapunovAt referenceExponent
  have : 0 < phi ^ (-(k : ℤ)) := zpow_pos Constants.phi_pos _
  linarith [this]

theorem lyapunovAt_succ_ratio (k : ℕ) :
    lyapunovAt (k + 1) = lyapunovAt k * phi⁻¹ := by
  unfold lyapunovAt
  have hphi_ne : phi ≠ 0 := Constants.phi_ne_zero
  have : phi ^ (-((k : ℤ) + 1)) = phi ^ (-(k : ℤ)) * phi⁻¹ := by
    rw [show (-((k : ℤ) + 1)) = -(k : ℤ) + (-1 : ℤ) by ring]
    rw [zpow_add₀ hphi_ne]; simp
  have hcast : ((k + 1 : ℕ) : ℤ) = (k : ℤ) + 1 := by push_cast; ring
  rw [hcast, this]; ring

/-- Reference Lyapunov exponent at `N_ppc` rung 0. -/
def referenceExponent : ℝ := 1

What this page does not claim

No real plasma simulation has been measured to have a Lyapunov exponent of exactly 1 at any resolution. The empirical bench from Dawson 1983 and Birdsall-Langdon 2004 is not a theorem in the framework's library. The ladder structure does not claim that all chaos measures in plasma physics follow this pattern.

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/PICSimulationLyapunov.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: