What the worked checks of the companion page establish

This remark audits the companion page. It records what each item computes, and what none of them claims. It proves nothing and reports no observation.

Elementary normalization and phase. A two-dimensional normalization and phase computation computes, on the explicit two-dimensional data $\psi =(1,i)$ and $\phi =(1+i)\psi$, the norms, the normalized representatives, the phase relating them and the normalized rank-one operators; it verifies the normalization and phase class lemma of the main page and exhibits the representative independence of the pure state on a worked vector pair.

Finite spin and the two-level state space. The state space of a two-level finite-spin system is the Bloch ball proves that the states of the model's two-level finite-spin system are exactly the operators $\frac{1}{2}(I+r\cdot \sigma )$ with $\mathrm{\mid }r\mathrm{\mid }\le 1$, that the pure states are the boundary points $\mathrm{\mid }r\mathrm{\mid }=1$, and that each boundary state has one-dimensional range and is the state of the spin-up ray defined there by the eigenvalue $+1$ of $r\cdot \sigma$. Pure and mixed states of a fixed two-level internal space repeats the corresponding elementary computation for an abstract two-level internal space carrying a pair of polarization labels, showing the pure state of the diagonal vector and the equal mixture of the basis rays. Neither item asserts the identification of the abstract two-level space with photon polarization; that reading belongs to the radiation assumptions of the category, and no statement about photon number, energy or momentum is made here.

Superposition versus mixture, and non-unique ensembles. A coherent two-level preparation and the equal statistical mixture are different states shows that the relative-phase superposition $(h+iv)\mathrm{/}\sqrt{2}$ and the equal statistical mixture of the rays $[h]$ and $[v]$ are assigned different states: the first has one-dimensional range and purity $1$, and the second has two-dimensional range and purity $1\mathrm{/}2$. One two-level state from two different preparation ensembles shows the converse direction of freedom: one and the same mixed state is determined by two genuinely different ensembles, an orthonormal pair with unequal weights and a uniform pair of non-orthogonal states. Both computations stop at the level of the assigned states; which outcome probabilities distinguish states, and with which apparatus, is the subject of the framework's probability law and measurement assumptions.

Zero-vector exclusion. The zero vector determines no pure preparation and no state refutes the informal claim that every vector determines a pure preparation with the zero vector as an empty preparation: the zero vector has no unit representative, the normalization map is undefined at it, and the zero operator has trace zero and is not a state. The refutation concerns the model's definitions and assignment; it is not an experiment and it makes no claim about nature.

Empirical neutrality. Every item of this page is a finite computation or the refutation just described. None of them uses measured data, none adopts a law of nature, none predicts the outcome of any measurement, and none compares a model prediction with an observation. Agreement or disagreement with data would test the combined framework and apparatus assumptions, and is not established or claimed here.