Geometry-to-Observable Mapping in Quantum Systems: A Measurement-First Control Framework

This preprint formalizes a measurement-first framework for treating quantum geometry as an operational control variable rather than a purely descriptive construct. Building on recent experimental demonstrations that the quantum metric can be directly inferred from electronic transport and optical response, the work extracts a general geometry-to-observable mapping principle: when internal degrees of freedom are constrained onto a structured state manifold, intentional symmetry breaking couples geometric quantities into measurable response channels. The framework introduces no new physical assumptions, remains fully compatible with standard quantum mechanics, and yields explicit falsifiable predictions across material platforms. It provides a disciplined methodological basis for geometry-aware system characterization and design.Keywordsquantum metricquantum geometryquantum geometric tensorgeometry-to-observable mappingstate-space curvaturespin-momentum lockingtransport asymmetryoptical responsesymmetry breakingmeasurement-first frameworkgeometry-aware materialscoherence structure