◇ part II · applications

Coulomb-limit atomic physics has a reduced Bohr-scale foothold; the anomaly package still has one branch datum open

The current source stack supports the localized Maxwell reduction, the Coulomb-limit bookkeeping needed for atomic bound states, a reduced hydrogenic Bohr-scale/binding-scale derivation, and a sharply narrowed g−2 branch-selection problem. It does not currently justify older site wording that treated the full hydrogen spectrum or the leading anomaly term as finished derived results of the published 4D stack.

Coulomb-limit reductionHydrogenic precision still provisionalg−2 branch datum still live4D · Maxwell 4D · 1PN bridge

◇ Coulomb limit

The atomic sector starts from the controlled brane Maxwell reduction

In the far-field zero-mode limit, an opposite-charge pair inherits the usual attractive Coulomb potential energy on the brane:

◇ static Coulomb sector · controlled reduction

This is the attractive Coulomb coefficient read through the localized Maxwell zero mode.

\begin{aligned} V_C(r) &\;=\; -\frac{g_C}{r}, \\ g_C &\;=\; \frac{Z_{\rm nuc}\,e_{\rm eff}^2}{4\pi\varepsilon_0}, \\ e_{\rm eff} &\;=\; e_\star/\sqrt{Z_{\rm int}} \end{aligned}

If one then takes the non-relativistic Coulomb limit for a slow probe and suppresses the inherited finite-localization and GNLS corrections, the leading effective equation is the standard Schrödinger problem

◇ Coulomb-limit target equation

This is the brane-effective non-relativistic limit. The current site should treat it as a reduced-sector result, not as a finished exact theorem of the full moving-throat stack.

i\hbar\,\partial_t \chi \;=\; \left[-\frac{\hbar^2}{2m_e}\nabla^2 - \frac{Z_{\rm nuc}\,e_{\rm eff}^2}{4\pi\varepsilon_0\,r}\right]\chi

◇ hydrogenic target

The current source record supports the Bohr-scale reduction, not the full spectrum

The clean one-body variational reduction gives the familiar Bohr radius and binding scale

◇ hydrogenic Bohr scale · reduced sector

This is the clean decoupling-sector result. Excited-state and precision-spectrum structure still need a clean current-paper derivation.

\begin{aligned} a_\star &\;=\; \frac{4\pi\varepsilon_0\hbar^2}{m e_{\rm eff}^2}, \\ E_\star-E_\perp &\;=\; -\frac{m e_{\rm eff}^4}{2(4\pi\varepsilon_0)^2\hbar^2} \end{aligned}

The same note upgrades the fixed-source result to a two-defect reduction, replacing

m

by the reduced mass

\mu

, and shows that the orbital scale is thickness-controlled through

e_{\rm eff}=e_\star/\sqrt{Z_{\rm int}}

. What is still not closed is the complete excited-state/angular structure, the full finite-size interaction, and precision Rydberg matching from the full moving-throat branch.

Bohr-scale reduced consequenceFull precision spectrum still open

◇ g-factor and anomaly

The current g−2 package narrows the miss to one branch-selection datum

The site should present the anomaly story in the same way it presents 2.5PN and 4PN: most of the algebraic packaging is in place, but one sharply identified physical datum is still open.

◇ anomaly bookkeeping · current status

The current

g-2

summary does not force the final electron-point quartic sliver. It localizes the remaining freedom to one narrow microscopic branch-selection / outgoing-normalization datum.

\begin{aligned} g &\;=\; 2\,[1+a_{\rm geom}], \\ a_{\rm geom} &\;=\; a_{\rm carried} + \delta a_{\rm branch} \end{aligned}

That is the right technical stance for this page: no claim that

a_e = \alpha/2\pi

has already been derived here from a declared

Z(w)

profile, and no claim that the full QED series has been matched. What the current package does claim is that the open residue has collapsed to one narrow branch datum rather than a broad coefficient fit space.

◇ open targets

What still belongs on the ledger

Document a clean current-paper derivation of the full hydrogenic spectrum and excited-state structure if one exists in the stack.
Close the remaining g−2 branch-selection datum on the realized moving-throat branch.
Translate the same branch data into Lamb-shift and muon-anomaly observables.

◇ forward reference

What uses this

Topic 10 shares the same style of open bridge problem in the outgoing sector. Topic 11 is where the branch data behind both the PN and the anomaly stories are meant to be realized microscopically.

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