QSG000: Quantum Spectral Geometry (QSG) Manifesto

This document presents the manifesto of Quantum Spectral Geometry (QSG), an operator-spectral research program aimed at developing a common mathematical language for structures that are usually treated separately: spectra, spectral measures, geometry, symmetries, weights, correlations, and dynamics. The central idea of QSG is to place a self-adjoint operator and its associated spectral data at the organizing level of the theory. Geometry is not assumed as a fixed background in all cases, but may be treated, in suitable classes of models, as a reconstructible or induced structure arising from operator-spectral data. The manifesto introduces the conceptual architecture of QSG, including the minimal operator core, the extended spectral carrier, the hierarchy of five operator levels, the distinction between primitive and derived structures, and the modular use of partial and full constructions. It also clarifies the relation of QSG to functional analysis, spectral geometry, noncommutative geometry, quantum theory, global analysis, and analytic number theory. QSG is not presented here as a completed axiomatic theory, a physical model, or a replacement for existing mathematical frameworks. Rather, this volume fixes the coordinates of a research program and outlines the framework in which further rigorous volumes of the Quantum Spectral Geometry Notes series will develop the formal definitions, axioms, reconstruction mechanisms, arithmetic-spectral constructions, measures, symmetries, and composition rules.