Structural Medicine v3.4: Phase-Locked Control, Amplitude Optimization, and the Controllable Window of Neurodegenerative Instability

This repository accompanies the preprint: "Structural Medicine v3. 4: Phase-Locked Control, Amplitude Optimization, and the Controllable Window of Neurodegenerative Instability" This work extends the Structural Medicine framework by introducing a minimal control-theoretic description of neurodegenerative instability under non-stationary conditions. Previous versions established that fixed-frequency control fails due to time-varying structural dynamics. In this work, adaptive control is reformulated under three fundamental constraints: 1. Phase alignment (anti-phase condition) 2. Amplitude optimization (existence of optimal A*) 3. A finite controllable window prior to critical transition The adaptive control signal is defined as: U (t) = A (t) sin (ω (t) t + φ (t) ) Control effectiveness is quantified as: ΔR = − Four figures illustrate the complete structure: - Fig17: Adaptive control suppresses instability (empirical behavior) - Fig18: Control depends on phase alignment (mechanism) - Fig19: Non-linear amplitude dependence with optimal A*- Fig20: Finite controllable window Wcontrol = t | R (t) < Rcrit The central conclusion is: Adaptive control of neurodegenerative instability is fundamentally constrained by phase alignment, amplitude optimization, and a finite controllable window prior to critical transition. This work does not claim clinical efficacy. Instead, it defines the structural conditions under which control is theoretically possible. All figures are generated from reproducible Python scripts included in this repository. Empirical foundations are based on the Alzheimer's Disease Neuroimaging Initiative (ADNI). This preprint represents the control-theoretic completion of the Structural Medicine framework (v3. 0–v3. 4), connecting prediction, failure of fixed control, adaptive recovery, and fundamental limits.