The Representational Status of the Linear Hilbert-Space Formalism

This paper distinguishes the predictive use of the linear Hilbert-space formalism from claims about its ontological completeness. A reconstruction of the bounded periodic sector of Heisenberg’s Umdeutung identifies the role of a uniform affine classical action rule in his quantum condition on transition amplitudes. Semiclassical refinements clarify the limits of that starting rule. The derivation examined does not establish universally exact physical linearity; spectral equivalence between matrix and wave mechanics within their common domain supplies no independent justification of this stronger claim. The analysis also distinguishes quantum-state representation from its reification as the complete physical condition of an enduring object. Measurement is examined through the different roles of unitary correlations, conditional projection, and definite events. Fourier representations and Bohm’s concept of metamorphosis motivate interpreting the evolving phase structure as an organization of event possibilities. On this reading, formal superposition does not describe simultaneously actual, mutually incompatible macroscopic properties of an object. The proposal retains the predictive formalism while reframing the measurement problem in terms of event actualization. It does not derive individual outcomes or the Born rule from a deeper dynamics. Keywords: Quantum foundations; Hilbert space; Linearity; Measurement problem; Reification; Event ontology.