Scope audit of the state-geometry page: what is mathematics, what is adopted, and what is deferred

This remark audits the status of the items of the pair. It records which statements are conditional mathematics, which are readings of an adopted model assumption, and which questions are deliberately left to other pages. It proves nothing and asserts no empirical result.

Mathematical layer, independent of any physical assumption. The items State vectors, unit vectors, rays, normalization and phases of a Hilbert space, Normalization, ray equivalence and the phase orbits of the unit sphere, Rays, unit vectors and rank-one Hilbert projections, The state space: density operators, pure and mixed states, and purity, The state space is convex, closed under countable mixtures, and complete in the trace norm, Purity, spectral weights and the criterion for a pure state and The extreme points of the state space are exactly the pure states are theorems and definitions of mathematics. They use only published or previously authored mathematical items, assume the stated choice hypotheses of those suppliers, and would be true of any complex Hilbert space regardless of any physical interpretation. In particular, the convexity of the state space, the criterion that purity is trace of the square, and the identification of the extreme points with the pure states carry no physical premise.

The adopted state postulate is homed on the framework page. The model assumption that a system is assigned a nonzero separable complex Hilbert space, that pure preparations are its unit rays and that general preparations are its density operators is clause-level content of State and composition postulate P1: Hilbert spaces, preparations and composite systems, authored on the framework page of this category. The present page does not restate it as a second postulate; The state space of the framework: preparations, pure rays and density operators only fixes the readings of its clauses in the geometry vocabulary, and the two physical theorems Pure preparations are well defined by their rays, and phase factors do not change their states and Statistical mixtures, spectral decompositions and the extremal role of pure preparations are conditional consequences of it together with the mathematical layer. No new physical assumption is adopted anywhere on this page.

No observable consequence of phase is claimed here. The two physical theorems use the redundancy of the unit representative and of phase factors only at the level of the assigned density operator. That a global phase has no consequence for any outcome probability is a statement about the framework's probability law and is not asserted on this page; the probability law, the moment conditions and the role of the phase in interferences belong to the observable and Born-probability items of this category.

Ensembles are not determined by states. The state postulate states, and the companion page exemplifies, that distinct weighted ensembles of pure preparations can determine the same density operator. This is a feature of the model's state assignment and not a claim that the model is incomplete, that nature is classical, or that some ensemble is the physically realized one. The mixture weights are classical preparation weights; they are not amplitudes, and no coherence between the alternatives of a mixture is asserted.

Deliberate deferrals and exclusions. The page adopts no superselection restriction, no identical-particle symmetry sector, no external field or radiation content and no measurement model; those are separate assumptions of the framework adopted on the pages that name them. The two-level vector space of the fixed-mode examples is treated purely as a two-dimensional Hilbert space; the identification of its vectors with photon polarization states, and any statement about photon number, energy or momentum, is deferred to the radiation assumptions of the category and is not made here.

Empirical status. No item of this page reports, predicts or compares a measured outcome. The worked checks of the companion page are finite computations verifying the definitions and the mathematical lemmas; they are not experiments, they use no data, and agreement of the model with any dataset is not established by them.