Slow Passage and Prepared Hysteresis in Adaptive Spectral Carriers

How does an adaptive system switch between stable states when its external input changes slowly? This paper derives preparation-dependent switching delays and a prepared quasistatic hysteresis limit for a finite adaptive spectral carrier, retaining its complete complex-amplitude and dynamic-memory equations. The analysis establishes a two-thirds-power switching-delay law with coefficients calculated directly from the full system. Incoming stable-branch preparation and initialization exactly at a fold select different Airy solutions and therefore different leading delays. Instantaneous stationary-memory elimination preserves the static folds but changes the predicted delay coefficients, demonstrating why stationary geometry alone does not determine switching dynamics. A complete-system connection and capture argument establishes the forward-and-reverse branch selection underlying prepared hysteresis. The paper also distinguishes geometric loop area from physical work: the loop integral approaches a positive finite limit, while accumulated maintenance work grows inversely with ramp speed. Four uninterrupted numerical cycles, preparation and uniqueness controls, an independent symmetry-restricted dynamic calculation, and energy-balance checks support the theoretical results. Explicit finite-rate error constants and attraction to a periodic orbit at a fixed nonzero driving rate remain future work. Developed within the Canonical Spectral-Curvature Field (CSCF) programme and Shadow Theory’s source-to-readout architecture, the paper provides a concrete mathematical account of evolving carriers, reciprocal memory and delayed switching. The accompanying public verification package includes executable code, numerical inputs and outputs, correction records, and reproduction instructions.