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Abstract Scattering theory is the basis of all linear optical and photonic devices, whose spectral response underpins wide-ranging applications from sensing to energy conversion. Unlike the Shannon theory for communication channels, or the Fano theory for electric circuits, understanding the limits of spectral wave scattering remains a notoriously challenging open problem. We introduce a mathematical scattering representation that inherently embeds fundamental principles of causality and passivity into its elemental degrees of freedom. We use this representation to reveal strong constraints in the mathematical structure of scattered fields, and to develop a general theory of the maximum radiative heat transfer in the near field, resolving a long-standing open question. Our approach can be seamlessly applied to high-interest applications across nanophotonics, and appears extensible to general classical and quantum scattering theory.