Performance Aware Residual-Space Resonant Control: A Two-Part Computational Study of the Matthews Residual Framework Part I: Wireless Power Transfer (Two Resonators) Part II: Tesla-Style Branch Management (Four Resonators)

This preprint presents two independently validated computational studies of adaptive resonant control, unified by a shared mathematical framework for identifying and holding the power-optimal operating state of a coupled resonant system, as distinct from the phase-locked state, which this work shows is frequently not the same thing. Part I develops closed-form residual-space mathematics for pairs and triads of coupled phases and validates a shifted-residual, search–identify–hold–reacquire controller on a two-resonator wireless-power-transfer plant. All electrical results were produced by an independent computational reconstruction after the original implementation became unavailable; a settling-time methodological error discovered during that reconstruction is documented in full, including its effect on every affected result, rather than silently corrected. Part II extends the same architectural principle to a four-resonator plant patterned on resonant-transformer topology, with a three-component residual vector, degradation- and opportunity-triggered reacquisition, and an extensive robustness and nonideality study. Every headline numerical claim in this part, including a 60-case robustness matrix and four separate nonideality studies, was independently reproduced to within Monte Carlo precision. A head-to-head comparison against standard perturb-and-observe control (the actual industry-standard method) is included, along with the correction of a bug found during that comparison’s own construction. A bridging section proves that the two parts’ residual constructions are not independent choices but the same covector/Gram-matrix formalism at different scales, with a general theorem covering a chain of any length. The central finding common to both parts: a controller can hold a branch with near-perfect phase coherence while delivering only a small fraction of the power actually available, demonstrated here as a reproducible failure mode, not a hypothetical one, in two structurally different plants. This is a computational study. No hardware has been tested, and the paper states explicitly, throughout, what is and is not established by the results presented. A brief note on the name: this framework is named in memory of my sister, Tracey Matthews, who encouraged me toward this area. This is a personal dedication and implies nothing about the correctness or importance of what follows, that should be judged on the mathematics, code, and results alone.