What the controlled-approximations examples establish and what they do not

What the page contains. Four conditional computations of the declared models of the companion A page: the exact two-level finite-gap family with its second-order coefficient and a finite check of the cubic remainder bound (A two-level finite-gap model: exact eigenvalues, the second-order correction and a finite check of the cubic remainder bound); the finite level static Stark model of the declared Coulomb pair with vanishing first-order shift, the exact square-root eigenvalue separation, its leading quadratic small-field correction and a finite check of the remainder bound (A finite-level static Stark model: vanishing first-order shift, exact splitting with leading quadratic shift and a finite check of the remainder bound); the spin-one multiplet in a uniform field with its three exact eigenvalues counted with multiplicity, distinct splitting only for nonzero coupling, and the degenerate first-order check (A spin-one multiplet in a uniform field: the exact Zeeman spectrum and degenerate first-order check); and the two tunnelling asymptotic limits, the smooth-barrier semiclassical limit and the exact rectangular thick-barrier prefactor (Tunnelling asymptotic limits: the smooth-barrier semiclassical limit and the exact rectangular prefactor comparison). Each computation states its model and its sources and cites the page's theorem it checks.

What the checks are. The first two examples verify the quantitative remainder bound of Nondegenerate stationary perturbation of a simple gapped label: first- and second-order coefficients and an explicit cubic remainder on exactly solvable two-state families, one with a finite gap and a second-order shift, one with the compressed dipole coupling of Static finite-level electric-dipole coupling: compressed dipole of the declared two-body Coulomb model in a uniform field; the third verifies the first-order localisation of Degenerate stationary perturbation: the perturbed eigenvalues stay within 4 lambda^2 v^2/g of the first-order shifts, and no branch relaxation is asserted in the exactly diagonalizable Zeeman family; the fourth compares two asymptotic limits, one covered by Separated smooth-barrier tunnelling: the declared transmission coefficient is exp(-2S/h) up to explicit small-h and barrier-thickness errors and one explicitly excluded by it. These are mathematical checks inside declared models, not tests of the models against data.

What the page does not contain. No experiment, no measured Stark or Zeeman shift, no splitting value, no gyromagnetic ratio, no measured matrix element or polarizability value, no transmission count, no detector response, no apparatus, no calibration, no sampling model and no uncertainty record appears; the radial dipole parameter $D_{\mathrm{r}\mathrm{e}\mathrm{l}}$ of the Stark example is computed exactly from the declared charge and Coulomb length scale, and is not measured. The examples assert no transition rate, no lifetime and no branch-relaxation process; see What the controlled-approximations page adopts, proves and does not claim: gaps, branch populations, turning points and the static-Stark limitation for the corresponding scope record of the A page.

Deferred content. Empirical numerical matrix-element comparisons for concrete Coulomb systems, time-dependent driving, transition rates, adiabatic evolution, and any comparison with measured spectra belong to the driven-dynamics and light-matter pairs of the track; the uncompressed static-field model is outside the present scope by the postulate's exclusion clause.