Sharp Uniqueness and Bistability in Driven Adaptive Spectral Carriers

How can identical external input sustain two stable states with different amplitudes, memories and responses? This paper establishes sharp uniqueness and bistability results for driven adaptive spectral carriers within the Canonical Spectral-Curvature Field (CSCF) programme and Shadow Theory’s source-to-readout architecture. The model couples complex amplitudes to real edge memory through a positive spectral carrier, with an explicit external power contract. The main results sharpen the sufficient uniqueness condition to include equality and prove that its universal graph-norm coefficient is optimal. On the original three-node path, the paper determines the exact threshold for global attraction to a unique equilibrium for every constant input. Below that threshold, suitable inputs sustain two fully stable states. In a specified parameter regime, a complete analytical count establishes exactly three equilibria: two stable minima and one saddle. A general mobility–inertia theorem connects stationary curvature to full dynamical stability, including complex, transverse and memory perturbations. Both folds are proved to be nondegenerate saddle-node bifurcations. Independent modal cubic formulas account for all eight dynamical eigenvalues. A full-state tilted-energy criterion identifies sufficient attraction-basin regions. Numerical pulse experiments demonstrate switching between stable states after returning to the same external input, with physical work and dissipation tracked separately. The two stable states have identical mean amplitude, stationary supplied power and clamped static-response norm, yet different full-feedback responses. This supplies a precise example of information lost through a coarse observable readout. The results extend CSCF from uniquely attracting driven states to maintained state-dependent memory and bistability. Continuous external power sustains both alternatives. Physical implementation and a rigorous dynamic quasistatic hysteresis limit remain future work. The accompanying public verification package provides executable code, fixed specifications, retained numerical outputs, exact scalar checks, independent numerical formulations, generated tables and figures, integrity checks and reproduction instructions. Author: Jeremy RodgersDOI: 10.5281/zenodo.23932350Website: https://everythingequation.com