Astrophysics Picsimulation Lyapunov Lyapunov At Adjacent Ratio

The adjacent ratio

Particle-in-cell (PIC) simulations track how a plasma's particles and fields push each other over time. A key output is the Lyapunov exponent, a number that measures how quickly nearby trajectories in the system's state space pull apart; a larger exponent means more chaos and a shorter prediction horizon. The Recognition Science framework models the numerical heating that limits PIC accuracy as a recognition cost, and from that cost it derives a structural prediction: the Lyapunov exponent at one resolution level is exactly 1/φ times the exponent at the adjacent level below, where φ = (1 + √5)/2 ≈ 1.618 is the golden ratio.

The declaration lyapunovAt_adjacent_ratio states this as a theorem: for any resolution rung k, lyapunovAt (k+1) / lyapunovAt k = φ⁻¹. The exponent at rung k is defined as φ^(-k), so the ratio follows directly from the definition. The library also proves the exponent is always positive, and that each step down multiplies it by φ⁻¹. These are formal results in the machine-checked library, with no unproved assumptions beyond the framework's own axioms.

What this theorem does not claim is that real PIC simulations actually exhibit this exact ratio. The definition of lyapunovAt is a modeling choice, not a measurement. The docstring cites empirical work (Dawson 1983; Birdsall-Langdon 2004) suggesting that doubling the number of macro-particles per Debye cell reduces numerical heating by a factor of φ², which is consistent with the φ⁻¹ per-step ratio, but that is an empirical check, not a proved consequence. The theorem only says: if you accept the framework's definition of the Lyapunov exponent at a resolution rung, then the adjacent ratio is exactly φ⁻¹.

In the framework, this ratio is one more appearance of the golden ratio that emerges from its forcing chain, the same φ² that shows up in Turing pattern thresholds and BCS pairing steps. For a plasma physicist, the practical lesson is that the framework predicts a specific geometric scaling of chaos with resolution, a claim that could in principle be tested against simulation output. The theorem itself is a clean formal statement; the question of whether nature follows it is an empirical one.

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

/-- Lyapunov exponent at PIC resolution rung `k` (higher rung = lower
numerical heating = smaller exponent). -/
def lyapunovAt (k : ℕ) : ℝ := referenceExponent * phi ^ (-(k : ℤ))

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

What this page does not claim

The theorem does not claim that real PIC simulations exhibit this ratio; that is an empirical check. The theorem does not claim that the definition of lyapunovAt is the only possible one. The theorem does not claim that the golden ratio appears in all plasma systems.

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)

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