The structure PICLyapunovCert in module IndisputableMonolith.Astrophysics.PICSimulationLyapunov bundles three properties of the function lyapunovAt.
(1) In plain English it states: the Lyapunov exponent is positive at every natural-number rung k; the value at rung k+1 equals the value at k multiplied by the reciprocal of phi; and the ratio of adjacent values is exactly the reciprocal of phi.
(2) In Recognition Science this certifies that PIC-simulation Lyapunov times lie on the phi-ladder, reproducing the same phi^2 scaling that appears in the recognition lattice, Turing patterns, and EW mass ratios (see module docstring on PIC convergence with N_ppc).
(3) The formal statement is read as a Lean structure whose three fields are universally quantified predicates: lyapunov_pos : ∀ k, 0 < lyapunovAt k, one_step_ratio : ∀ k, lyapunovAt (k + 1) = lyapunovAt k * phi⁻¹, and adjacent_ratio : ∀ k, lyapunovAt (k + 1) / lyapunovAt k = phi⁻¹. The witness picLyapunovCert supplies the concrete proofs lyapunovAt_pos, lyapunovAt_succ_ratio, and lyapunovAt_adjacent_ratio.
(4) Visible dependencies are the auxiliary definitions referenceExponent and lyapunovAt together with the three supporting theorems listed above; the structure is instantiated directly by picLyapunovCert with zero sorry.
(5) The declaration proves only the internal algebraic properties of the model function lyapunovAt; it does not prove any statement about real plasma simulations, numerical heating rates, or empirical falsifiers.