Transported Structural Compression: From Static Rank Bounds to Nonlinear Frequency Classification

This paper presents a unified theory of transported structural compression for smoothly deforming covariance systems. Companion papers established three structural components: Paper G: Static rank bounds arising from ODE order (rank(C) ≤ n). Paper H: Spectral-gap-controlled transport law for the leading principal direction under smooth deformation. Paper I: Frequency-dependent nonlinear classification of transport error (absorption, quadratic regime, and rank-1 breakdown). The present work integrates these results into a single theorem — the Unified Transported Structural Compression Theorem — showing that low-dimensional manifolds are simultaneously: Preserved (static structural bound), Transported (spectral-gap-controlled drift), Conditionally disrupted (frequency-dependent nonlinear perturbation). A two-parameter structural phase diagram (spectral gap Δλ vs perturbation frequency k) delineates the absorption, quadratic, and breakdown regimes. This framework applies beyond toy models and cosmological growth observables to general smoothly evolving dynamical systems exhibiting spectral separation.