What the uncertainty and compatibility page adopts, proves and does not claim

Adopted, not restated. The physical content of this page is the state and composition postulate P1 and the observable and probability postulate P2 in the vocabulary of Assigned sharp observables and the outcome law of the framework (Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes, Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators for the position and momentum statements). The compatible-sharp-observables item also adopts P3 solely for its stipulated ideal finite-bin Lüders instrument and its conditional updates and repeatability when the joint-bin probability is positive. P3 does not assert that a laboratory device realizes that ideal instrument. The position and momentum claims use P5 and P6 to name $Q_j$ and $P_j$; the choice hypotheses of the spectral theorem are carried explicitly in the mathematical items.

What is proved. Preparation variance and spread of an assigned sharp observable fixes the reading of preparation variance and spread. The Robertson relation for preparation spreads of assigned sharp observables proves the weak Robertson bound for unit vectors in both domains, the Robertson bound and its anticommutator refinement on the product domains, and the automatic validity for bounded observables. The position-momentum preparation relation of the massive spin-s model proves $\mathrm{\Delta }{Q}_j\mathrm{\Delta }{P}_j\ge \mathrm{\hslash }\mathrm{/}2$ on the full maximal domain of the pair, with the exact imaginary part $-\mathrm{\hslash }\mathrm{/}2$ and the equality characterization. Strong spectral commutation of self-adjoint operators, Strong spectral commutation, joint projection-valued measures and bounded Borel calculus and Compatible sharp observables, their joint law and the ideal joint instrument fix strong spectral commutation, prove its equivalence with the existence and uniqueness of a joint sharp observable and give the joint law with its marginals; Position and momentum of the massive model have no joint sharp observable proves that the model's position and momentum are incompatible. All of these are conditional consequences of the postulates and the cited mathematics.

What is not claimed. The position-momentum incompatibility item rules out a joint sharp PVM with the exact spectral marginals of $Q_j$ and $P_j$; no item asserts that an unsharp joint measurement or unsharp joint law exists or does not exist. The compatible-observables item does derive conditional finite-bin updates and repeatability inside the ideal P3 model for joint outcomes of positive probability. No universal disturbance relation or disturbance claim for a concrete apparatus is asserted. The inequalities concern the spreads of the outcome laws of one preparation; they are not statements about the precision of a laboratory procedure, and the equalities of the position and momentum item are properties of specific preparations, not of every state. The identification of a preparation and an apparatus with the model's states and operators is a tested identification with uncertainty under the preparation and apparatus postulate P7; neither the postulates nor the theorems here report a measured value, a trial count or a statistical inference. Agreement of the conditional predictions with data would test the combined model and apparatus assumptions; it is not established or claimed here.

Distinctions preserved. The page keeps preparation spread, apparatus resolution and disturbance separate throughout; it also keeps the mathematical identity $[A,B]u=0$ on a dense invariant domain separate from strong spectral commutation. The examples companion records a domain-sensitive counterexample, the Gaussian family that saturates the position-momentum bound, the spin-$\frac{1}{2}$ spread account, the energy-time scope remark and the error-disturbance distinction.