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In this article, we introduce a new viewpoint on electromagnetic scattering. Tailoring spectral electromagnetic response underpins important applications ranging from sensing to energy conversion, and is flourishing with new ideas from non-Hermitian physics. There exist excellent theoretical tools for modeling such responses, particularly coupled-mode theories and quasinormal-mode expansions. Yet these approaches offer little insight into the outer limits of what is possible when broadband light interacts with any designable nanophotonic pattern. We show that a special scattering matrix, the "$\mathbb{T}$" matrix, can always be decomposed into a set of fictitious Drude--Lorentz oscillators with matrix-valued (spatially nonlocal) coefficients. For any application and any scatterer, the only designable degrees of freedom are these matrix coefficients, implying strong constraints on lineshapes and response functions that had previously been "hidden." To demonstrate the power of this approach, we apply it to near-field radiative heat transfer, where there has been a long-standing gap between the best known designs and theoretical limits to maximum energy exchange. Our new framework identifies upper bounds that come quite close to the current state-of-the-art, and explains why unconventional plasmonic materials should be superior to conventional plasmonic materials. More generally, this approach can be seamlessly applied to high-interest applications across nanophotonics -- including for metasurfaces, imaging, and photovoltaics -- and may be generalizable to unique challenges that arise in acoustic and/or quantum scattering theory.