Scope and empirical status of the central-potential and hydrogen page

What the page establishes. The pair declares a central-potential model (The declared central-potential model: state space, assigned energy operator, spherical reading and bound-state vocabulary) and a Coulomb and two-body hydrogen model (The declared Coulomb and two-body hydrogen model: one-centre operator, relative problem, reduced mass and scope exclusions), proves the spherical reduction of smooth compactly supported vectors, with the radial equation read almost everywhere and pure-preparation readings only for nonzero vectors after normalization (The declared central-potential model: spherical reduction of smooth compactly supported radial states), and derives the energy-observable and flow reading (The one-centre Coulomb model: energy observable, flow, stationary states and spectral support), the bound energies $E_n=-\frac{m{\kappa }^2}{2{\mathrm{\hslash }}^2{n}^2}$ with $n^2$-fold model degeneracy and the completeness of the bound states in the negative spectral subspace with a nonzero continuum (The one-centre Coulomb model: bound energies, n^2-degeneracy, completeness of the bound states in the negative spectral subspace and the continuum), the radial and ground-state position readings (The one-centre Coulomb model: position and radial distributions of the bound states, with the ground-state density and its scales), and the two-body reduced-mass decomposition with the relative levels as thresholds and the absence of stationary total states (The declared two-body Coulomb model: reduced-mass decomposition, relative levels as thresholds and the absence of stationary total states). Every item is conditional on the cited postulates, the declared model and the cited mathematics; none establishes that the model or its postulates describes nature.

What the page does not contain. No experiment, no reported spectral line, no count, no calibration, no detector model, no sampling assumption and no statistical inference appears on the page, and no numerical value of $\mathrm{\hslash }$, $m$, $\kappa$, $q$, $m_1$, $m_2$ or ${\epsilon }_0$ is fixed by any item. The transition frequencies, line intensities and selection rules of a hydrogen-like system require the radiation extension and the light-matter coupling of later items, and the scattering, ionisation and continuum processes require the three-dimensional scattering pair; neither is built or assumed here. The dipole matrix elements and their angular selection rules are reserved to the light-matter pair, and the numerical comparison with measured spectra to a future experiment item with its own preparation, calibration, sampling and uncertainty record; no such item exists in this batch.

Excluded physics. The gross-structure model excludes spin and magnetic couplings, relativistic fine structure and the splitting of the $n^2$-fold degeneracy, finite nuclear size and nuclear structure, recoil corrections beyond the two-body reduced-mass formulation, QED corrections including the Lamb shift, hyperfine structure, shielding and many-body modifications of the $1\mathrm{/}r$ potential, external fields, and radiation reaction. The source treatments describe several of these corrections as small compared with the gross-structure energies; no such magnitude comparison is a premise of any item on this page, and none of the excluded effects is claimed to be negligible by this page.

Empirical status of the model's numbers. Statements such as "the hydrogen ground-state binding energy is about $13.6$ electronvolts" are conditional evaluations of the model's formula with measured constants (such as the fine-structure constant, the electron-to-proton mass ratio or the Rydberg constant) as inputs; they carry the uncertainties, conditions and conventions of those inputs and are not measurements of hydrogen by this page. Agreement of a computed transition frequency with a measured line would test the combined model, apparatus and constant assumptions; it would not be a deductive proof of the model.

Physical-interface note. The model declarations, the readings and the scope exclusions above are the public physical interface of the pair; changing them requires the ordinary direct and indirect consumer review under the workflow rules, and the empirical qualifications carried here must be preserved by every later item that consumes these results.