Spectral Freezing and Exceptional Points in Non-Feedback Hierarchical Stochastic Networks

Abstract: Hierarchical structures are ubiquitous in non-equilibrium systems, from gene regulatory cascades to metabolic networks. A fundamental question is whether downstream microscopic details can penetrate the relaxation time scales of upstream dynamics. We rigorously prove that, in the absence of feedback, the entire generator of a finite-state upstream Markov chain is ``frozen'' as an invariant subspace of the full network generator, completely decoupled from all downstream parameters. This equivalence is established via an elementary theorem: strong spectral freezing---the exact equality of the full generator's restriction to upstream-only functions with the upstream generator---holds if and only if the network is devoid of feedback. Building on this, we discover a hidden block-upper-triangular structure for pure mRNA networks in a falling factorial basis, revealing that the \emph{entire} relaxation spectrum is independent of transcription rates. Exceptional points arise from accidental degeneracies between frozen switching modes and tunable degradation modes when the cross-block coupling is asymmetric, leading to algebraic relaxation $t e^{-\lambda t}$. This framework provides a parameter-free diagnostic: upstream relaxation peaks remain immobile under downstream perturbations if and only if no feedback exists.