Spectral-Operator Calculus (Part A): Trace-Form Evaluators and Spectral Growth Taxonomy

Abstract: We develop a spectral-operator calculus on separable Hilbert space that treats self-adjoint operators and their bounded spectral transforms as the basic objects. On a class of such ``spectral geometries'' we introduce abstract evaluators, required to satisfy natural invariance, locality, extensivity, and dominated-convergence continuity conditions. Our first main result is a trace-form representation theorem: on the natural trace-class envelope every such evaluator is given by the trace of a single nondecreasing profile applied through the functional calculus. Thus, once that profile and a scalar normalization are fixed, all admissible scalar values are determined by the underlying spectra, yielding a rigidity principle for evaluators in this setting. The second main result is a spectral growth taxonomy: we classify self-adjoint operators by counting-function asymptotics and show that the polynomial growth class is stable under the basic constructions of the calculus. Together these results provide an arithmetic-neutral analytic backbone for later parts of the series and for applications to concrete spectral models.