Adaptive Effective-Rank Selection for Time-Dependent Tensor Correctors: Hysteretic Switching and Residual-Budget Certification

We develop an adaptive effective-rank selection framework for a finite family of time-dependent projected tensor correctors. The problem is to let the effective rank respond to a time-varying residual channel without producing frequent rank oscillations, uncontrolled re-certification cost, or loss of a prescribed residual budget. The construction uses a finite certified rank window, an integrable budget density, a deterministic score, a two-threshold hysteresis rule, and an explicit obstruction flag for times at which the admissible rank set is empty. Under bounded-variation assumptions on admissibility margins and score gaps, the number of rank switches and the cumulative switching cost are bounded. The residual estimate is formulated as a measure-driven or impulsive Gronwall inequality with additive atomic switching increments, and the proof is given by an explicit interval-by-interval iteration over switching times. A discrete-time robustness corollary shows how score uncertainty margins prevent false switches in sampled implementations, while the algorithmic summary specifies conservative behavior when a candidate fails metric certification. Metric certification is preserved under rank updates by direct re-certification or by a conservative perturbative Gram-condition criterion. The construction is positioned between rank-adaptive reduced modelling, hysteretic switching, certified residual control, and impulse-type stability estimates. It is not a general optimal-control theory, does not claim full trace-norm propagation of chaos, and does not remove a three-dimensional Coulomb cutoff; it provides a certified adaptive decision layer for a predefined family of projected corrector channels.