We develop a scale–independent framework for the classification of astrophysical dynamical systems based on intrinsic spectral–dynamical invariants reconstructed from observational time series. To each system P we associate a spectral invariant \ (P) = (r, , _, hₓ₎, D₂, , G), \ encoding frequency rank, Lyapunov spectrum, topological entropy, fractal dimension, invariant measure, and symmetry group of the reconstructed attractor. The invariant induces a partition of the space of admissible systems into spectral phases. We prove separating properties for hyperbolic limit cycles and quasi–periodic tori, establish conditional stability of under small observational noise, and characterize expansive phases via positive Lyapunov exponent or entropy. A structural trichotomy emerges: linear time evolution on compact attractors either factors through toroidal geometry (cyclic phases), or exhibits intrinsic exponential divergence (expansive phases). The hypersurface _=0 acts as a phase boundary between these regimes. This yields a precise mathematical formulation of a linear–cyclic closure principle: although time itself is linear and unbounded, its dynamical realization may close into compact cyclic geometry in non–expansive regimes, while remaining open to chaotic evolution in expansive regimes. Because is dimensionless and scale–free, it enables structural comparison of dynamical systems across cosmological distances, suggesting a universal spectral phase structure of astrophysical emitters.
Vasil Tsanov (Thu,) studied this question.