What the worked items of the symmetry companion page establish

The free model: translation symmetries and conserved momentum in the momentum representation is the exact free-model computation: with $V\equiv 0$ every translation is a symmetry, the free flow in the momentum representation is the unimodular multiplier $M_{f_t}$, $f_t(p)=e^{-it\mathrm{\mid }p{\mathrm{\mid }}^2\mathrm{/}(2m\mathrm{\hslash })}$, the translations are the multipliers $M_{e^{-ia\cdot p\mathrm{/}\mathrm{\hslash }}}$, and the momentum components are conserved sharp observables with the law, moment and generator statements of Translation invariance of bounded potentials and conservation of momentum; the modulus of the momentum representative is invariant in time almost everywhere. A bounded nonconstant potential with no nontrivial translational symmetry shows that translations need not remain symmetries for nonconstant potentials: for the bounded measurable potential energy $E_{\ast }{1}_{B(0,R)}$, with $E_{\ast }>0$ in J and $R=1\mathrm{m}$, no nonzero translation is a symmetry, because the shifted potential differs from the original on a set of positive measure, so the symmetry criterion of the translation theorem is not met. For a specified translation $a$, the criterion is exactly $V(\cdot -a)=V$ almost everywhere; nonconstancy in a direction alone does not exclude discrete translations along that direction. Definite parity of nondegenerate eigenstates and the position-matrix-element selection rule derives, under its stated model and choice hypotheses for a real bounded measurable potential even almost everywhere, that every energy eigenspace is invariant under the parity action, that a nondegenerate eigenstate has definite parity, and that the position matrix element $⟨Q_j\psi ,{\psi }^{\mathrm{\prime }}⟩$ vanishes for unit vectors $\psi ,{\psi }^{\mathrm{\prime }}\in H_s=L^2({\mathbb{R}}^3;{\mathbb{C}}^{2s+1})$ of equal definite parity with $\psi \in D(Q_j)$, as in clause (c). Here $j\in \{1,2,3\}$, $Q_j\psi (x)=x_j\psi (x)$ on the maximal domain $D(Q_j)=\{f\in H_s:x_{j}f\in H_s\}$, and equal definite parity means $\mathrm{\Pi }\psi =\sigma \psi$ and $\mathrm{\Pi }{\psi }^{\mathrm{\prime }}=\sigma {\psi }^{\mathrm{\prime }}$ for a common $\sigma \in \{-1,1\}$, with $\mathrm{\Pi }f(x)=f(-x)$. No position-domain condition is required on ${\psi }^{\mathrm{\prime }}$; evenness of the potential supplies none on $\psi$. A radial bounded potential: rotational and parity symmetry and conservation of the rotation generators verifies the almost-everywhere rotational and parity invariance of a bounded radial potential and instantiates Rotational and parity invariance of bounded-potential models: symmetries and conservation of angular momentum and parity laws: all rotations are symmetries, the rotation action is a strongly continuous unitary symmetry group, and the coordinate-rotation generators $L_i$ are conserved sharp observables, with the underlying conservation mechanism of Conservation of a symmetry generator: commuting flows, energy compatibility and conserved outcome laws.

None of the four items is an experiment: no apparatus, preparation map, calibration, detection model, efficiency, resolution, background model, sampling model, uncertainty budget, count or dataset appears, and no measured value of any model parameter is asserted. The objects $m$, $q$ when present, $s$, the potentials and the conserved laws are model data and model statements in the units of Dimensions, SI unit conventions and quantity types of the framework; deciding that a laboratory system realizes the model, that its potential is approximately radial or even, or that a transition amplitude is the matrix element computed here, is a tested identification with uncertainty under the preparation and apparatus assumptions, and it is external to this page. In particular the parity selection rule is an identity about the model's assigned position observables: turning it into a statement about observed intensities requires a time-dependent coupling model, the corresponding transition-rate analysis, a detection model and a sampling model, none of which is supplied here.

The free computation is conditional on $V\equiv 0$; the counterexample concerns one potential inside the bounded measurable class and refutes only the universal claim that every bounded nonconstant potential has all nonzero translations as symmetries (potentials constant along a direction keep the corresponding translations). The central-potential example assumes the radial identity almost everywhere and a bounded radial function; it asserts no eigenvalue, bound state, degeneracy, radial solution or spectral line, and its boundedness hypothesis excludes the Coulomb and other unbounded radial models. The parity example requires the evenness hypothesis and, for definite parity, the nondegeneracy of the eigenvalue; at a degenerate eigenvalue only the eigenspace invariance is asserted for arbitrary eigenvectors. The equal-parity matrix-element identity separately requires the displayed $D(Q_j)$ condition and directly assumed parity signs; it requires neither an eigenvector hypothesis nor nondegeneracy. No item of this page makes a claim about antiunitary or time-reversal symmetries, about superselection, about spinorial rotations or the internal factor, or about relativistic and field-theoretic structures.