What the angular momentum and spin page adopts, proves and does not claim

Adopted assumptions. This page adopts P1, P2, P4 and P5 in the readings of the framework and the spin assignment P6 of Identical-particle and spin assignments P6: permutation sectors and finite-spin representations clause (ii) (The massive spin-s particle model of the framework: state space, operators, units and frame readings, Assigned sharp observables and the outcome law of the framework). The effective magnetic-moment coupling with its gyromagnetic parameter ${\gamma }_s$ and bounded field profile is the additional model assumption Effective magnetic-moment coupling of the finite-spin model: gyromagnetic parameter, magnetic moment and Zeeman energy operator; like the spin label, mass and charge, the parameter ${\gamma }_s$ is a species parameter with uncertainty and is not derived on this page.

What is proved. The local mathematics Self-adjointness and the flow of a self-adjoint operator perturbed by a bounded operator commuting with its flow and The SU(2) lift of the spatial rotation action on the component-valued wave-function space and its coordinate generators supplies the bounded-perturbation flow and the lifted SU(2) rotation action with its coordinate generators with the explicit transported physical-axes cover (the literal quaternion-coordinate cover is denoted ${\rho }_0$ there), and The orbital Casimir identity: the restrictions of the orbital generators to the degree-l blocks are spin-l models identifies the orbital Casimir on the harmonic blocks. On that basis: Spin spectra and outcome laws of the massive spin-s model: directional components, the Casimir law and the spin-half probabilities gives the spectra and Born laws of the spin components; Orbital angular momentum eigenstates of the massive model: spherical harmonics, the l,m quantum numbers and the Casimir value describes the orbital eigenstates with the quantum numbers $l,m$ and the Casimir value; Conservation of total angular momentum for a spin-independent rotation-invariant bounded-potential dynamics proves the conservation of the total angular momentum for a spin-independent rotation-invariant bounded-potential dynamics; Spin precession and the Zeeman splitting in a uniform magnetic field: Larmor angular frequency and the conservation of one component gives the Larmor precession of the transverse spin expectations and the conservation of the field component in a uniform static field; Composition of two spin labels: total spin operators, the Clebsch-Gordan block decomposition and the spin-half singlet supplies the internal-factor composition with the Clebsch-Gordan decomposition and the spin-half singlet; and The conditional Stern-Gerlach force prediction of the spin-half Zeeman model: transverse momentum rate in a smooth bounded gradient field transfers the conditional smooth-field force identity to the model.

What is not claimed. No result of this page derives any value of ${\gamma }_s$, $s$, $m$, $g$-factor or $\mathrm{\hslash }$; no spin-orbit, fine or hyperfine structure, relativistic spinor, anomaly or radiative correction is included; no spin-statistics theorem is asserted; no atomic, molecular or solid-state model is constructed; and no statement about the compositeness or internal structure of any particle is made. The internal factor is a finite-dimensional bookkeeping device, not a rotating body.

Experiments. This page contains no experiment item: it records no apparatus, calibration, sampling, count or uncertainty. The historical Stern-Gerlach setup appears only as the name of the conditional model of The conditional Stern-Gerlach force prediction of the spin-half Zeeman model: transverse momentum rate in a smooth bounded gradient field, whose clause (b) is a prediction of the model conditional on an explicit preparation hypothesis. Identifying the operators and parameters of this page with a laboratory preparation and apparatus is a tested identification with uncertainty under Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions; the measurement statistics of sequential spin filters are treated on the measurement pages of the category, and agreement or disagreement with data there constrains the combined model and apparatus assumptions without proving them.

Consumers. Composites, Bell and identical-particle pages of the category consume the internal-factor results of Composition of two spin labels: total spin operators, the Clebsch-Gordan block decomposition and the spin-half singlet and the lifted rotation action of The SU(2) lift of the spatial rotation action on the component-valued wave-function space and its coordinate generators; their empirical qualifications are added on those pages and are not inherited as certainty from this one.