What the angular momentum examples establish and what they do not

Recorded content. The companion page contains exact model computations: Spin-half outcome probabilities along an arbitrary direction: the Bloch parametrization and worked cases evaluates the Born probabilities of the directional spin components for an arbitrary unit spinor and works the coordinate cases; Orbital angular momentum eigenfunctions: the low spherical harmonics of the massive model computes the low spherical harmonics, their orthonormality and their orbital eigenvalues; The two-level Zeeman toy model: sector energies, level splitting and spin precession records the sector expressions of the uniform-field Zeeman operator on product vectors in $D(H_0)$, its factorized flow on all product vectors, an explicitly labeled two-level truncation with its level splitting, and the explicit precession matrices; and The idealized Stern-Gerlach splitting: transverse momentum separation of the two spin sectors under a window hypothesis derives, under an explicitly hypothetical containment premise, the change in the difference of the two sector mean transverse momenta.

Status. All four items are conditional statements inside the model of The massive spin-s particle model of the framework: state space, operators, units and frame readings with the readings of Orbital, spin and total angular momentum of the massive spin-s model: assigned sharp observables, units and readings and, in the last two, with the adopted magnetic-moment coupling. None of them reports a measurement; the thought experiment's containment premise is a hypothetical assumption, and the toy truncation is a toy model assumption; the numerical values appearing in them ($\mathrm{\hslash }$-multiples, probabilities and frequencies) are model expressions, not measured numbers.

Empirical qualifications. The identification of any of these preparations and observables with a laboratory setup is a tested identification with uncertainty under Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions; detector resolution, background, calibration, sampling and selection are not modeled here. The directional-probability example specializes Spin spectra and outcome laws of the massive spin-s model: directional components, the Casimir law and the spin-half probabilities, while the Zeeman toy example specializes the uniform static-field result Spin precession and the Zeeman splitting in a uniform magnetic field: Larmor angular frequency and the conservation of one component, retaining its isolated evolution and its separate truncation assumption. The gradient-field thought experiment instead retains its own hypotheses from The idealized Stern-Gerlach splitting: transverse momentum separation of the two spin sectors under a window hypothesis: real bounded $B_z\in C^1(\mathbb{R})$ with bounded derivative, an initial unit vector in $D(H_Z^2)$, a constant gradient on the declared interval and the hypothetical containment premise throughout $[0,\tau ]$; normalized sector means require positive sector weights. It does not specialize the uniform-field precession theorem. The orbital example retains its harmonic-polynomial and normalization hypotheses. Each computation inherits the qualifications of the suppliers it actually uses.

What is deliberately absent. No experiment item, no measured frequency, no measured gyromagnetic parameter, no source-backed Stern-Gerlach dataset, no spin-orbit or fine-structure computation and no radiation or relaxation model appears on the examples companion; those belong to other pages of the category, with their own sources and qualifications.