What the examples companion records and what it does not

One exact model computation. The Gaussian family attains the position-momentum bound in the massive model computes the position and momentum densities and spreads of the Gaussian family in the spinless massive model and verifies that the product equals $\mathrm{\hslash }\mathrm{/}2$ with the centered vectors linearly dependent and their pairing purely imaginary, realizing the equality case of The position-momentum preparation relation of the massive spin-s model. It is a statement about one family of preparations of the model; it contains no dynamics, no detection model and no measured number, and it does not characterize all equality states. The spectral outcome-law and moment steps use the Axiom of Choice, as recorded in the example's assumptions.

One finite internal-factor deduction. Spin one half: spread identity, incompatible component outcomes and saturated Robertson bounds derives the spread identity ${\sum }_i(\mathrm{\Delta }{S}_i{)}^2=3{\mathrm{\hslash }}^2\mathrm{/}4-\mathrm{\mid }⟨\mathbf{S}⟩{\mathrm{\mid }}^2$ for the spin-$\frac{1}{2}$ internal factor, the lower bound ${\mathrm{\hslash }}^2\mathrm{/}2$, the failure of spectral compatibility for every pair of components, and the saturation of the Robertson bound in the $S_z$ eigenstates. It is a conditional deduction inside the model's internal factor; it asserts no Stern-Gerlach apparatus, no dynamics and no measurement.

One purely mathematical counterexample. A commutator vanishing on a dense invariant domain does not imply strong spectral commutation is a statement of pure mathematics about two self-adjoint extensions of the interval momentum operator: it refutes the implication from a commutator identity on a dense invariant domain to strong spectral commutation. It carries no physical premise and makes no physical claim.

Three remark items. Time is not an observable of the framework: the conditional timescale bound records that time is a parameter and not an observable, so no energy-time Robertson pair exists here and only the conditional timescale bound is available; Preparation spread, apparatus resolution and disturbance are different objects separates preparation spread, apparatus resolution and disturbance. Both are readings of other items, not new theorems. This item is the third remark item and audits the companion's inventory and evidence scope.

No experiment and no data. The companion records no preparation procedure, apparatus, calibration, trial count, count rate, histogram, uncertainty budget or statistical analysis, and it uses no empirical premise. Every quantity computed here is a model quantity. Any comparison with data would test the combined model and apparatus assumptions through the identification of a preparation and an apparatus with the model's objects, a tested identification with uncertainty under the preparation and apparatus postulate P7 (Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions), and would not turn the conditional results into unconditional statements about nature.

No universal claims. The Gaussian family is not all preparations; the spin deduction concerns the internal factor of the model; the counterexample concerns two specific operators. In particular the companion does not claim that every state attaining $\mathrm{\hslash }\mathrm{/}2$ is Gaussian, that every pair of incompatible components excludes all forms of joint measurement, or that the timescale bound is a universal energy-time relation.

• Reading order. The mathematical items precede the model computations on the main page; the companion consumes the main page's items and adds no new supplier to them.
• Status. As everywhere in this category, agreement of the conditional predictions with a laboratory result would be evidence for the combined model and apparatus assumptions, not a proof of the framework; the companion reports no such result.