Evolution Theory of Eigen-Skeletons in Nonlinear Dissipative Systems: A Topological and Thermodynamic Derivation of the 4n-1 Sequence

Abstract This paper proposes a general theoretical model describing the structural evolution ofnonlinear dissipative systems. Based on modular arithmetic in discrete phase space andHamiltonian dynamical constraints, this study demonstrates that steady states inevitablyemerge on the discrete sequence N = {4n - 1 | n ∈ Z+} (i.e., 3, 7, 11, 15, 19). By incorporatingPotential Energy Surface (PES) topology, the KAM (Kolmogorov-Arnold-Moser) stabilitytheorem, and the principle of nonlinear entropy maximization, this research reveals thedynamical transition mechanisms between "Trunk Skeletons" (Topologically Protected Wells)and "Metastable Slides" (saddle points). Furthermore, it defines the critical boundaries forsystem-wide spectral reddening and thermodynamic resets. This theory provides a physicalbasis for identifying structural determinism within complex time series. Chinese version is attached as a supplementary file.