What the autonomous-dynamics page adopts, proves and does not claim

What the page adopts. The single physical assumption materialized here is the autonomous-dynamics postulate Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution (P4), read in the vocabulary of Assigned energy observable, unitary flow and the two pictures of autonomous evolution for an isolated system of the framework with state space of the type fixed by The state space of the framework: preparations, pure rays and density operators. In particular the assigned energy observable $(H_{\mathrm{o}\mathrm{p}},u_H)$, its energy unit and the reduced Planck constant entering the phase are model specifications in the sense of Dimensions, SI unit conventions and quantity types of the framework; no numerical value of any of them is produced by this page.

What is proved. The Schroedinger-picture flow: unitarity, domain invariance and the domain-qualified equation draws, from P4 and the local mathematics Flow calculus of a self-adjoint generator: commutation, invariance and differentiability, the unitary flow, the invariance of the energy domain, the domain-qualified Schroedinger equation and the evolution of density-operator preparations. The Heisenberg picture: equivalent predictions and the domain-qualified equation of motion, using Heisenberg flow of a bounded operator: conjugates, covariance and the equation of motion, identifies the Heisenberg representative of a bounded assigned sharp observable and proves that the two pictures assign the same outcome laws, together with the domain-qualified Heisenberg equation and the freezing of energy-compatible bounded observables. Energy conservation, conserved sharp observables and the stationary eigenrays reads the conservation of the energy law in every preparation, its finite first and second moments for pure preparations represented by nonzero vectors in the energy domain (using unit representatives), the conservation of any bounded energy-compatible observable, and the equivalence of stationary pure preparations with the eigenrays of the energy; the mathematical core of that equivalence is Stationary rays of a self-adjoint flow are exactly the eigenrays. All statements are conditional consequences of the adopted postulates and the cited mathematics; none of them is an empirical claim.

What is not claimed. (i) The page treats only the autonomous case: a prescribed time-dependent family $t\mapsto H(t)$ and the propagator problems it raises are not covered by P4(ii) and are deferred to the driven-dynamics development, and an open or reduced non-unitary evolution is an extension requiring an environment model, in the sense of clauses (iv) and (v) of P4. (ii) No measurement, collapse, conditioning or apparatus model is introduced; the outcome law of Assigned sharp observables and the outcome law of the framework is used only to read statistics of a preparation, and the state after an outcome is not treated. (iii) No relaxation, equilibration or convergence of the law of a general preparation is asserted. (iv) The converse of the conservation criterion — that a bounded observable whose law is constant in every preparation must be strongly spectrally compatible with the energy — is not proved for a general possibly unbounded $H_{\mathrm{o}\mathrm{p}}$ and is therefore not claimed. (v) The equation of motion is asserted exactly on the operator domain; for vectors outside the domain only the unitary flow, not the differential equation, is available.

Empirical status. Whether a concrete laboratory system realizes P4's isolated-system idealization, and whether a concrete apparatus realizes a preparation of the type used here, are tested identifications with uncertainty, in the sense of the framework's preparation and apparatus assumptions; they are not consequences of P4. This pair records no measurement, apparatus, calibration or data, and the worked companion page contains finite model computations only.