lyapunovAt
plain-language theorem explainer
The definition sets the Lyapunov exponent at PIC resolution rung k to the reference value scaled by phi to the power minus k. Plasma kineticists running particle-in-cell codes would cite this when mapping numerical heating to the phi-ladder. It is a direct one-line scaling from the rung-zero reference exponent.
Claim. The Lyapunov exponent at resolution rung $k$ is defined by $L(k) = L_0 phi^{-k}$, where $L_0$ is the reference exponent at rung zero and $phi$ is the golden ratio.
background
Particle-in-cell simulations track the Lyapunov exponent of the particle-field system on the phi-ladder, with rung index k corresponding to the number of macro-particles per Debye cell. The module states that adjacent doubling of this number reduces numerical heating by phi squared, reproducing the same scaling that appears in Turing patterns and BCS pairing. The upstream referenceExponent supplies the base value 1 at rung zero.
proof idea
One-line definition that multiplies the reference exponent by phi raised to the negative integer power of rung k.
why it matters
This supplies the rung-dependent exponent used by the PICLyapunovCert structure and the adjacent-ratio and successor-ratio theorems. It encodes the structural claim that PIC Lyapunov exponents decrease geometrically with resolution according to the phi-ladder, consistent with the self-similar fixed point and eight-tick octave of the Recognition framework. The module reports zero sorrys and zero axioms.
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