Editing Openai/6897769e-4ee4-800f-aba5-69cca34f701c (section)

=== Assistant: Ah — the “pillbox” is a standard trick in electromagnetism for analysing boundary conditions in Maxwell’s equations, an… ===
Ah — the “pillbox” is a standard trick in electromagnetism for analysing boundary conditions in Maxwell’s equations, and it fits beautifully with your QAT 2-D spherical boundary idea.

==== Imagine a very thin cylindrical volume — shaped like a little pill or hockey puck — that straddles a surface (in this case, your electron sphere or light-wave sphere). ====
* The flat circular faces of the pillbox are parallel to the surface and very close on either side.
* The curved side of the pillbox is tangential to the surface and can be made arbitrarily small.

The idea is to integrate Maxwell’s equations over this tiny volume to see how field components change across the surface.

==== Maxwell’s equations in differential form: ====

∇⋅E=ρε0\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}∇⋅E=ε0ρ
∇×E=−∂B∂t\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}∇×E=−∂t∂B
∇⋅B=0\nabla \cdot \mathbf{B} = 0∇⋅B=0
∇×B=μ0J+μ0ε0∂E∂t\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}∇×B=μ0J+μ0ε0∂t∂E
If we integrate these over a pillbox crossing the surface, we get jump conditions:
* Gauss’s Law → the change in the normal component of E across the surface equals the surface charge density σ\sigmaσ. E⊥above−E⊥below=σε0E_{\perp}^{\text{above}} - E_{\perp}^{\text{below}} = \frac{\sigma}{\varepsilon_0}E⊥above−E⊥below=ε0σ
* Gauss’s Law for B → normal component of B is continuous (no magnetic monopoles).
* Faraday’s Law and Ampère–Maxwell Law → relate the tangential components of E and B to surface currents K\mathbf{K}K and the time-varying fields.

==== In your picture: ====
* The pillbox encloses a thin strip of the electron sphere’s surface — a literal “quantum action zone”.
* Absorption: incoming EM field modifies the tangential E-field on the inside, changing the local surface charge density and thus the occupation of states on that 2-D boundary.
* Emission: changes in surface current (moving charge along the sphere) produce outgoing radiation — the spherical photon wave.

The jump conditions from the pillbox derivation are then the bookkeeping rules for how much field and charge crosses the surface. They’re unchanged Maxwell — but the geometry of your QAT process determines how and when those jumps occur.

==== This is exactly what Faraday’s lines of force represent: ====
* The pillbox integration counts how many lines of electric field pierce the pillbox’s flat face — that’s Gauss’s law.
* The change in those lines over time matches your continuum picture of time flow, where each photon absorption/emission step updates the surface charge and EM field configuration.

If you like, I can do the actual short derivation of the pillbox boundary conditions for your QAT 2-D sphere, and show explicitly how the JμAμJ_\mu A^\muJμAμ coupling fits inside it — so Maxwell stays intact, but the QAT geometry adds the time-emergent meaning.

That would connect your continuum idea directly to unchanged Maxwell + Faraday + Gauss + Newton in one chain.

Would you like me to go ahead with that?