What the controlled-approximations page adopts, proves and does not claim: gaps, branch populations, turning points and the static-Stark limitation

Adopted model clauses. The page declares three models: the variational trial model The declared variational trial reading: assigned energy observable, form domain, trial value, bottom and gap (a self-adjoint energy observable bounded below, its form domain and trial value), the perturbed family The declared perturbed-family model: bounded self-adjoint perturbation, isolated gapped label and followed spectral cluster (a bounded self-adjoint perturbation of a gapped isolated label) and the WKB regime model The declared WKB regime model: nondimensionalised line equation, turning points, phase integrals and the declared barrier transmission reading (the nondimensionalised one-dimensional stationary equation with a declared semiclassical parameter and a compactly supported $C^5$ potential), plus the static finite-level dipole coupling Static finite-level electric-dipole coupling: compressed dipole of the declared two-body Coulomb model in a uniform field. Each declaration is a model assumption; none is derived from P1-P7.

What is proved. Conditional on those declarations the page proves: the trial lower bound, the gap-overlap control and the min-max characterisation of the discrete ladder (Variational trial bounds, gap-overlap control and the min-max characterisation of the declared model's discrete eigenvalues); the nondegenerate perturbation expansion with explicit cubic remainder (Nondegenerate stationary perturbation of a simple gapped label: first- and second-order coefficients and an explicit cubic remainder); the degenerate first-order splitting localisation within $4{\lambda }^2{v}^2\mathrm{/}g$ (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); the allowed-region WKB Gronwall error (WKB approximation on an allowed interval of the declared regime model: explicit Gronwall error bound away from turning points); the single simple-turning-point Airy connection with envelope errors (Airy connection across a single simple turning point of the declared regime model, with envelope error control); and the separated smooth-barrier transmission asymptotics (Separated smooth-barrier tunnelling: the declared transmission coefficient is exp(-2S/h) up to explicit small-h and barrier-thickness errors). Every one of these statements carries its hypotheses explicitly: the gap $g>0$, the boundedness of the perturbation, the smoothness of $W$, the separation of the turning points, and the value of the semiclassical parameter.

No attained ground state is assumed. The variational statements use the form domain $Q(H)$ and the declared spectral lower bound; they do not assume that the bottom $E_0$ is an eigenvalue, that a ground eigenvector exists, or that the trial infimum is attained. The gap-overlap clause is the only place where an isolated bottom is used, and it states exactly the overlap forced by a trial value within $\epsilon$ of the bottom.

No branch population or equilibration. The perturbation theorems locate spectral labels of the declared families; they assert no instantaneous transition of a preparation to the branch with the smallest first-order shift, no relaxation to a lowest branch and no equilibration. The motivating textbook conjecture of a jump to the lowest branch is not imported. For each fixed coupling $\lambda$, P4 assigns the autonomous flow $e^{-itH(\lambda )\mathrm{/}\mathrm{\hslash }}$; it commutes with the spectral projections of that fixed operator, so their populations are fixed by the initial preparation. The textbook conjecture instead concerns a time ramp $\lambda (t)$: P4 supplies no propagator for that ramp and gives no conservation law for its instantaneous spectral-branch populations (Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution clauses (ii), (iv)). Such a question requires separately specified driven dynamics; an equilibration statement would require a separately declared environment or dissipation model, which this page does not supply.

Turning points and barriers. The allowed-region error excludes turning points by hypothesis; the connection theorem treats a single simple turning point and does not produce a transmission coefficient; the barrier theorem assumes exactly two separated simple turning points of a compactly supported $C^5$ barrier and excludes rectangular walls, coalescing turning points, barrier-top uniformity and resonant multiwell interference. The declared transmission coefficient is a matching datum of the model, not a detector response.

The static-Stark limitation. For ${\mathbf{E}}_{\mathrm{e}\mathrm{x}\mathrm{t}}\mathrm{\ne }0$, the uncompressed static perturbation of a Coulomb model by a uniform field is the unbounded multiplication term $-{\mathbf{E}}_{\mathrm{e}\mathrm{x}\mathrm{t}}\cdot \mathbf{D}$. The common-test-domain calculation in Static finite-level electric-dipole coupling: compressed dipole of the declared two-body Coulomb model in a uniform field translates a normalized fixed $\phi \in C_c^{\mathrm{\infty }}(B(0,\rho ))$ to ${\psi }_R(\mathbf{r})=\phi (\mathbf{r}+R\widehat{e})$, where $\widehat{e}={\mathbf{E}}_{\mathrm{e}\mathrm{x}\mathrm{t}}\mathrm{/}\mathrm{\mid }{\mathbf{E}}_{\mathrm{e}\mathrm{x}\mathrm{t}}\mathrm{\mid }$. Its kinetic expectation is constant, its Coulomb integral is bounded by $1\mathrm{/}(R-\rho )$, and the linear term contains $-qR\mathrm{\mid }{\mathbf{E}}_{\mathrm{e}\mathrm{x}\mathrm{t}}\mathrm{\mid }$; hence the operator expectations tend to $-\mathrm{\infty }$ on the explicit common test domain. No quadratic form bounded below there can agree with those values, although no particular closed-form domain is asserted. No bounded-perturbation theorem of this page applies to that nonzero-field interaction. At ${\mathbf{E}}_{\mathrm{e}\mathrm{x}\mathrm{t}}=0$, the maximal interaction is the bounded zero operator on $L^2({\mathbb{R}}^3;\mathbb{C})$; $\widehat{e}$ is not defined, and the preceding translation argument and bounded-perturbation exclusion do not apply. This page covers only the compressed finite-level operator of Static finite-level electric-dipole coupling: compressed dipole of the declared two-body Coulomb model in a uniform field, and it states no eigenvalue theorem for the uncompressed model.

Empirical status. No experiment, reported observation, measured shift, splitting, transmission or count appears on this page; no apparatus, calibration, sampling or uncertainty record is supplied. The statements are conditional predictions of the declared models. A comparison with data would test the combined assumptions -- preparation, isolation, parameter identification, truncation, field description and detection response -- under the preparation and apparatus discipline P7 (Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions); agreement would be evidence about that combination, not a deductive proof of the model, and a discrepancy would require the auxiliary assumptions to be examined before the model is rejected.