What the worked items of the companion page establish

The Gaussian packet. The Gaussian position and momentum packet of the massive model computes, in the spin-zero massive model, the normalized Gaussian ${\psi }_{\sigma }(x)=(\pi {\sigma }^2{)}^{-3\mathrm{/}4}{e}^{-\mathrm{\mid }x{\mathrm{\mid }}^2\mathrm{/}(2{\sigma }^2)}$, shows that its momentum representative is again a normalized Gaussian with width ${\sigma }_p=\mathrm{\hslash }\mathrm{/}\sigma$, and evaluates the position and momentum outcome densities of the model together with their variances ${\sigma }^2\mathrm{/}2$ and ${\mathrm{\hslash }}^2\mathrm{/}(2{\sigma }^2)$ and the product $\mathrm{\hslash }\mathrm{/}2$. This verifies the model's sharp laws of Position and momentum outcome laws of the massive spin-s model on a family of Schwartz preparations; it computes no dynamics, no measurement and no general uncertainty statement.

The plane wave as a generalized functional. The plane wave as a generalized momentum eigenfunctional defines the conjugate-linear functional $u_k(\phi )=\int e^{ik\cdot x}\bar{\phi }$ on Schwartz classes, proves the eigenrelation $u_k(P_j\phi )=\mathrm{\hslash }{k}_j{u}_k(\phi )$, and proves that no $L^2$ class represents $u_k$. This is the precise sense in which the plane wave is a generalized momentum eigenfunction: it diagonalizes the momentum action on the test space while carrying no state or probability density. The framework's state assignment excludes it as a preparation, in agreement with the model's atomless sharp laws.

The boundary warning. A confined model with Dirichlet walls: the naive interval momentum is not a sharp observable shows, in a hypothetical confined-model extension, that the formal interval momentum $-i\mathrm{\hslash }D$ on the Dirichlet domain is symmetric but not self-adjoint, that no self-adjoint extension keeps the Dirichlet domain, and that the Dirichlet functions are not momentum eigenfunctions of any self-adjoint extension. Its explicit twisted-periodic extensions are $\mathrm{\hslash }{P}_{\theta }=-i\mathrm{\hslash }D$ on the domains $u^{\ast }(L)=e^{i\theta }{u}^{\ast }(0)$. The companion counterexample No bounded operators satisfy the canonical commutation relation with a nonzero commutator shows that the canonical commutation relation cannot hold as an identity of bounded operators at all, so the core relation of Canonical commutation relations on the component-valued core and commutation with the internal factor is genuinely a relation on a dense core of unbounded operators rather than a bounded operator identity.

None of the items is an experiment: no counts, calibration, resolution, efficiency, background, uncertainty or data comparison appears anywhere on the page, and no value of $m$, $\mathrm{\hslash }$ or any model parameter is measured. The Gaussian computation is a single family of preparations; the plane-wave item is a distributional statement without a state; the boundary warning and the bounded-commutator counterexample are conditional mathematical statements about hypothetical or bounded models, not claims about apparatuses. In particular, no item asserts that confined models, twisted-periodic momenta or any self-adjoint realization describe a laboratory device, and no item draws conclusions about decoherence, measurement disturbance or relativistic effects, which belong to other model pages.