Using differential calculus, we derive an exact dipolar condition for the extinction peak position of a nanocrystal, reducing the problem of locating the extinction maximum to a differentiable state equation. This leads to a characteristic polynomial for the plasmon frequency, whose coefficients are set by the dielectric function, environment, particle volume, static depolarization factor, and an effective retardation scale. After calibrating this scale with electrodynamic simulations, we construct geometro-optical depolarization manifolds that offer a compact space for tracking plasmonic changes during growth, reshaping, or etching. The resulting polynomials accurately reproduce fifty-six experimental wavelengths of gold (Au) spheres, rods and bipyramids in aqueous solution, and two experimental plasmonic transitions, from rods to spheres and from bipyramids to spheres. In addition to providing a general organizing principle for plasmonic transitions, this framework enables accurate variational analyses of extinction peaks, facilitates the inference of nanoparticle morphology from optical responses through data-driven approaches, and provides a robust characterization of plasmonic systems and experiments even in the presence of spurious or non-ideal effects that are difficult to capture analytically.