Physics-Informed Knowledge Distillation of a Certified Algebraic Oracle for Selective Harmonic Elimination in Nine-Level Inverters

Selective harmonic elimination pulse-width modulation places exact spectral nulls at fundamental-frequency switching, yet its defining transcendental system admits no closed form and is resolved offline, leaving the deployed modulator without an optimality certificate for its angles and without any mechanism for restoring exactness at run time. A Chebyshev change of variables reduces the nine-level, four-angle system nulling the 3rd, 5th, and 7th harmonics to a single monic quartic whose coefficients are closed-form rational functions of the modulation index, certifying a unique angle set per index and a distortion minimum of 9.2115% at $M=0.8047$ within a feasibility island of width $3.4\times {10}^{-3}$. This certified oracle then supervises a physics-informed distillation objective in which the governing equations enter the loss directly through equation-residual and distortion channels, compressing a teacher into a 389-parameter student (1.52 kiB, 352 multiply-accumulates, $22.6\times$ fewer parameters) at 0.3208% mean distortion deviation. A cumulative-softmax head makes the ordering constraint structural, and the physics channels place the prediction inside the quadratic convergence basin, so three undamped Newton corrections reach a $2.9\times {10}^{-13}$ median residual and 0.0010% deviation where a safeguarded cold start stalls near ${10}^{-1}$. Ten-seed ablation, closed-form gradient attribution, and MATLAB/Simulink validation at 9.21% distortion complete a certified, refinable, kilobyte-scale modulation pipeline.