Residual-Budget Stability for Hysteretic Witness-Directed Adaptive Channels

This manuscript develops a residual-budget framework for hysteretic witnessdirected corrective-channel selection in projected dynamical systems. A lowdimensional witness is treated as an observable defect indicator, while unobserved transverse modes, coefficient variation, and switching costs are kept in explicit accounts. The main stability result is a dwell-time multiple-Lyapunov theorem with a paid switching account and an explicit geometric residual-floor amplifier. The paper also proves a witness-blind obstruction: if the zero-witness transverse dynamics contain an unstable invariant component, then no feedback law depending only on the witness can stabilize the full residual state. The framework separates theorem-level core budget estimates from diagnostic monitoring terms, gives computable Lie-derivative and one-step scores, records nominal / robust / adaptive disturbance-scoring regimes, and instantiates the definitions in a two-channel benchmark. The results are finite-horizon residual-certification statements unless additional locking or convergence hypotheses are imposed. The manuscript does not claim a complete BBGKY closure theorem, trace-norm propagation of chaos, universal nonlinear optimal control, or multi-witness fusion; it isolates an E5 gate condition for safe witness-directed feedback under explicit residual-budget, switchingbudget, and transverse-stability assumptions.