Sustained Adaptive Spectral States: Global Attraction, Nonlinear Gain, and Linked Coherent–Dissipative Response

What sustains an adaptive spectral system after its unforced amplitudes and memory would otherwise relax, and what does feedback contribute beyond the pattern imposed by an external input? This paper develops a finite driven realization of the Canonical Spectral-Curvature Field (CSCF) within Shadow Theory’s source-to-readout architecture. Complex amplitudes and real edge memory interact reciprocally through a positive spectral carrier. An explicitly defined external input supplies energy, while adaptive feedback changes interaction strengths, stationary amplitudes and observable response. The main theorem establishes a sufficient graph-curvature condition under which every constant complex input produces a unique globally asymptotically stable equilibrium. Stationary memory elimination yields a Lambert-W potential with a sharp scalar curvature bound. Returning to the complete amplitude–memory Hessian establishes local exponential stability with an explicit conservative rate, despite the failure of global joint convexity. For a three-node network, an exact nonlinear gain law predicts a 5.30% contrast-amplitude increase over the identically driven fixed carrier at the specified parameters. Uniform excitation supplies a control with no adaptive contrasts, while a one-port input produces unequal persistent memories and edge strengths. Equal-norm inputs have equal stationary supplied power under the declared actuator contract, despite sustaining different amplitudes, memory and stored energy. A continuing-work result makes the external energy requirement explicit. The conditional coherent and dissipative generators remain linked through the same carrier. Freely responding memory nevertheless produces a causal kernel and distinct static and frequency responses. The paper derives these differences and specifies the calibration, phase-sensitive access and controlled-memory resources needed to test them. Complete proofs are supported by matched controls, independent numerical formulations, a high-precision scalar calculation and reproducible verification. The accompanying public package contains executable code, specifications, retained outputs, generated figures and tables, integrity checks and reproduction instructions. This study extends the adaptive-carrier programme from transient relaxation to sustained externally powered states, distinguishing imposed preparation from the quantitative effects of reciprocal feedback. Author: Jeremy RodgersDOI: 10.5281/zenodo.23927855Website: https://everythingequation.com