What the observable and Born-probability page adopts, proves and does not claim

Adopted, not restated. The physical content of this page adopts the state and composition postulate P1 (State and composition postulate P1: Hilbert spaces, preparations and composite systems), assigning unit rays and density operators as preparations, and the observable and probability postulate P2 (Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes), read in the vocabulary of Assigned sharp observables and the outcome law of the framework: assigning to a declared system a self-adjoint operator with a declared unit, its spectral projection-valued measure, and the outcome law ${\mu }_{\rho }^A(B)=\mathrm{tr}(\rho E_A(B))$. The general outcome-law, moment and state-separation statements are conditional on P1 and P2; the choice hypotheses of the spectral theorem are carried explicitly, and the position and momentum statements additionally adopt the massive-particle model postulate P5 (Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators), assigning the spinless state space $L^2({\mathbb{R}}^3;\mathbb{C})$ and its position and momentum operators, as read through the magnetic-and-detection operator definitions. No additional physical postulate is introduced on this page.

What is proved. Atoms of the spectral measure and eigenvalues of a self-adjoint operator identifies the atoms of the spectral measure with the eigenspace projections, so that an exact value has zero probability unless it is an eigenvalue. Support of the spectral measure, the spectrum and the values of the scalar law identifies the support of the spectral measure with the spectrum, giving the precise reading of the statement that an outcome lies in the spectrum. Affinity and trace-norm continuity of the scalar spectral law proves affinity under statistical mixtures and uniform trace-norm continuity. Mean, second moment and variance of the scalar spectral law assembles the mean, second moment and variance formulas with their exact domain criteria, and Distinct density operators are separated by the law of a bounded rank-one projection proves that distinct states are separated by the law of a rank-one projection. The physics theorems read these results as conditional statements about assigned observables: The Born outcome law: probability measure, mixtures, atoms and spectral support (the law is a probability measure, affine in mixtures, atomic only at eigenvalues, supported in the spectrum and not assumed to have a density), Mean, second moment and variance of an assigned observable in a preparation (moments exist exactly under the stated domain hypotheses), Position and momentum outcome laws and the Born density of the massive particle model (the joint position density $\mathrm{\mid }\psi {\mathrm{\mid }}^2$ and the momentum-coordinate density obtained from the frequency density $\mathrm{\mid }{\mathcal{F}}_2\psi {\mathrm{\mid }}^2$ by taking its marginal and the Jacobian factor $(2\pi \mathrm{\hslash }{)}^{-1}$) and Distinct preparations are separated by the statistics of a bounded sharp observable (distinct preparations differ in some bounded observable's statistics; all such exact laws uniquely determine the preparation within the model).

No complete eigenbasis. Following the correction recorded in the source audit, the page never asserts that a general self-adjoint operator has a complete discrete system of eigenvectors. The spectral projection-valued measure replaces the eigenbasis; discrete sums are the special case of an atomic law, and continuous parts are handled through measures. The expectation formula $⟨\widehat{O}⟩=⟨\psi ,\widehat{O}\psi ⟩$ is proved here only for $\psi$ in the domain of the operator and through the spectral integral, not through an assumed eigenfunction expansion.

What is not claimed. No item of this page asserts an apparatus realization of any observable, a joint measurement of two observables, a state update after an outcome, a density for an arbitrary outcome law, the existence of a mean or variance for an arbitrary preparation, a measured value, determination of a preparation from finitely many measurements, or a comparison with data. The finite unsharp-effect extension of P2(iv) and all instrument content belong to the measurement-and-instruments page, the componentwise spin extension of the position and momentum laws belongs to the kinematics page, and the identification of concrete laboratory procedures with assigned observables is a tested identification under P7, not a consequence of the postulates. Agreement of the conditional predictions with data would test the combined model and apparatus assumptions; it is not established or claimed here.