The Transformation-Response Framework: An Operational Reformulation of Quantum Mechanics

We present the transformation-response framework, an operational reformulation of quantum mechanics. A quantum state is not a Hilbert space object but the catalog of a system's responses to all physical transformations: for each operation g from the system's local group G, an interference experiment gives a complex value χ(g). The collection \{χ(g): gin G \} is the characteristic function and defines the state. The only postulate is that χ is positive-definite, encoding the requirement that no superposition of transformations yields negative probability. From this single assumption, the entire standard formalism is derived: Hilbert space via GNS construction, Born rule via Bochner theorem, Schrödinger equation from group automorphisms, and especially Feynman path integral as a Trotter limit. The framework is background-independent and time-neutral: time is a coordinate along a one-parameter subgroup of G. It also reveals a new physical constraint, product order positivity, which may lead to testable predictions. The framework provides a unified, economical, and falsifiable foundation for quantum theory rooted in operational primitives.