Spectral Gap Topology, Pseudospectral Control, and the Thermodynamic Cost of Classical Information

This work develops a unified operator-theoretic framework linking spectral gaps, pseudospectral stability, and thermodynamic bounds on classical information in open quantum systems. We define a classical record as a state whose distinguishability remains stable under bounded non-selfadjoint perturbations. Within this framework, we prove that such stability is equivalent to resolvent-controlled exclusion of the ε-pseudospectrum from a spectral gap. This establishes a precise mathematical criterion for the persistence of macroscopic states. Building on this equivalence, we derive a gap-modified Landauer bound in which the minimal work required to stabilize information depends explicitly on both thermal population imbalance and dynamical broadening. In the limit of gap collapse or strong environmental coupling, the effective protection efficiency vanishes and the required stabilization cost diverges, implying that gapless records are thermodynamically inaccessible. We further prove a superpolynomial suppression of ultraviolet–infrared transitions for states with bounded momentum support. Using Paley–Wiener-type arguments, we show that highly oscillatory ultraviolet operators weak-* converge to zero against smooth infrared states, yielding effective decoupling across extreme energy scales. Taken together, these results provide a model-independent explanation for the stability of macroscopic structure based solely on spectral topology and resolvent control, without requiring symmetry protection or fine-tuning. This manuscript is accompanied by explicit analytical proofs and visualization of pseudospectral geometry and thermodynamic scaling behavior.