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

=== Assistant: Lovely — let’s make the ledger-on-sphere picture fully explicit and compact, so you can paste it into notes or a video. ===
Lovely — let’s make the ledger-on-sphere picture fully explicit and compact, so you can paste it into notes or a video. I keep the mathematics exact and show how absorption/emission appear purely as boundary/source contributions, with Maxwell’s equations unchanged. I keep h/2πh/2\pih/2π visible and use the same notation we’ve been using: boundary (shell) at radius r0r_0r0, surface spinor Ψ\PsiΨ, surface current jsurfαj^\alpha_{\rm surf}jsurfα, and bulk field AμA_\muAμ.

I’ll present (A) the total action, (B) Euler–Lagrange → Maxwell with delta-layer sources, (C) local continuity and jump (Gauss/Ampère) conditions, (D) Poynting theorem with an explicit surface-energy bookkeeping term, and (E) a few sentences tying back to Faraday and the thermodynamic picture.

==== Bulk electromagnetic action (unchanged): ====

SEMbulk  =  −ε04∫d4x FμνFμν,Fμν=∂μAν−∂νAμ.S_{\rm EM}^{\rm bulk} \;=\; -\frac{\varepsilon_0}{4}\int d^4x\, F_{\mu\nu}F^{\mu\nu},
\qquad F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu.SEMbulk=−4ε0∫d4xFμνFμν,Fμν=∂μAν−∂νAμ.
Boundary (2-D spherical shell at r=r0r=r_0r=r0) action for the boundary degrees of freedom (schematic spinor form; h/2πh/2\pih/2π kept explicit):

Sb  =  ∫dt ⁣∫Sr02r02dΩ  [ih2π Ψ†DtΨ−v Ψ†\slashedDS2Ψ−V(Ω)Ψ†Ψ],S_{\rm b}
\;=\;
\int dt\!\int_{S^2_{r_0}} r_0^2 d\Omega\;
\Big[i\frac{h}{2\pi}\,\Psi^\dagger D_t\Psi - v\,\Psi^\dagger \slashed D_{S^2}\Psi - V(\Omega)\Psi^\dagger\Psi\Big],Sb=∫dt∫Sr02r02dΩ[i2πhΨ†DtΨ−vΨ†\slashedDS2Ψ−V(Ω)Ψ†Ψ],
where the covariant derivative on the boundary includes the minimal electromagnetic coupling
  Dα=∂α+iq(h/2π)Aα  \;D_\alpha = \partial_\alpha + i\frac{q}{(h/2\pi)}A_\alpha\;Dα=∂α+i(h/2π)qAα (indices α\alphaα intrinsic to the shell).

Interaction (surface coupling) is the minimal current coupling written as a boundary integral:

Sint  =  ∫dt ⁣∫Sr02r02dΩ  Aα(t,Ω) jsurfα(t,Ω).S_{\rm int}
\;=\;
\int dt\!\int_{S^2_{r_0}} r_0^2 d\Omega\; A_\alpha(t,\Omega)\, j^\alpha_{\rm surf}(t,\Omega).Sint=∫dt∫Sr02r02dΩAα(t,Ω)jsurfα(t,Ω).
Total action:

Stot=SEMbulk+Sb+Sint+Sbulk  matter.S_{\rm tot}= S_{\rm EM}^{\rm bulk} + S_{\rm b} + S_{\rm int} + S_{\rm bulk\;matter}.Stot=SEMbulk+Sb+Sint+Sbulkmatter.

==== Vary StotS_{\rm tot}Stot with respect to Aμ(x)A_\mu(x)Aμ(x). Because SintS_{\rm int}Sint is localized on the shell, the variation yields the standard Maxwell equations with a delta-supported surface contribution: ====

∂νFνμ(x)  =  1ε0[Jbulkμ(x)  +  δ(r−r0) jsurfμ(t,Ω)].(1)\partial_\nu F^{\nu\mu}(x)
\;=\;
\frac{1}{\varepsilon_0}\Big[J_{\rm bulk}^\mu(x)
\;+\; \delta(r-r_0)\, j^\mu_{\rm surf}(t,\Omega)\Big].
\tag{1}∂νFνμ(x)=ε01[Jbulkμ(x)+δ(r−r0)jsurfμ(t,Ω)].(1)
Interpretation:
* JbulkμJ_{\rm bulk}^\muJbulkμ is any bulk 4-current (free charges in bulk).
* δ(r−r0)jsurfμ\delta(r-r_0) j^\mu_{\rm surf}δ(r−r0)jsurfμ is a distribution (a delta layer) representing the surface charge/current localized on the spherical shell. Equation (1) is the unchanged Maxwell equation; the only novelty is that the source includes a thin-shell term.

==== Take the 0-component (charge) and spatial components of (1) to get continuity and jump relations. ====

1) Local 4-current conservation (total):

∂μ(Jbulkμ+δ(r−r0)jsurfμ)=0.\partial_\mu \big(J_{\rm bulk}^\mu + \delta(r-r_0) j^\mu_{\rm surf}\big)=0.∂μ(Jbulkμ+δ(r−r0)jsurfμ)=0.
Integrating across an infinitesimal radial interval surrounding the shell gives the surface continuity equation:

∂tσ(t,Ω)+∇∥ ⁣⋅j∥(t,Ω)  =  − n^⋅Jbulk∣r=r0,(2)\partial_t \sigma(t,\Omega) + \nabla_{\parallel}\!\cdot\mathbf{j}_{\parallel}(t,\Omega)
\;=\;
-\,\hat n\cdot\mathbf{J}_{\rm bulk}\big|_{r=r_0},
\tag{2}∂tσ(t,Ω)+∇∥⋅j∥(t,Ω)=−n^⋅Jbulkr=r0,(2)
where
* σ=jsurft\sigma = j^t_{\rm surf}σ=jsurft is the surface charge density (time component),
* j∥\mathbf{j}_\parallelj∥ is the tangential surface current (spatial components on the shell),
* n^⋅Jbulk∣r=r0\hat n\cdot\mathbf{J}_{\rm bulk}|_{r=r_0}n^⋅Jbulk∣r=r0 is the normal current flux from the bulk onto the shell (absorption is positive here; emission is negative).

Equation (2) is the ledger entry: change of surface charge = net tangential divergence + net normal flux from bulk.

2) Jump (boundary) conditions — integrate Maxwell across the shell

Integrate Maxwell’s equations in a small pillbox across the shell to obtain the exact boundary conditions:
* Normal electric-field jump (Gauss law):

n^ ⁣⋅(Eout−Ein)  =  σ(θ,φ,t)ε0.(3)\hat n\!\cdot(\mathbf{E}_{\rm out}-\mathbf{E}_{\rm in})
\;=\;
\frac{\sigma(\theta,\varphi,t)}{\varepsilon_0}.
\tag{3}n^⋅(Eout−Ein)=ε0σ(θ,φ,t).(3)
* Tangential magnetic-field jump (Ampère with surface current):

n^×(Bout−Bin)  =  μ0 j∥(θ,φ,t).(4)\hat n\times(\mathbf{B}_{\rm out}-\mathbf{B}_{\rm in})
\;=\;
\mu_0\,\mathbf{j}_{\parallel}(\theta,\varphi,t).
\tag{4}n^×(Bout−Bin)=μ0j∥(θ,φ,t).(4)
These are exact and local consequences of (1); they are the instantaneous field response to the microscopic ledger entries σ,j∥\sigma, \mathbf{j}_\parallelσ,j∥.

==== Start from the usual bulk Poynting theorem (unchanged): ====

∂tuem+∇ ⁣⋅S  =  −E ⁣⋅ ⁣Jbulk,\partial_t u_{\rm em} + \nabla\!\cdot \mathbf{S} \;=\; -\mathbf{E}\!\cdot\!\mathbf{J}_{\rm bulk},∂tuem+∇⋅S=−E⋅Jbulk,
where uem=ε02∣E∣2+12μ0∣B∣2u_{\rm em}=\frac{\varepsilon_0}{2}|\mathbf{E}|^2+\frac{1}{2\mu_0}|\mathbf{B}|^2uem=2ε0∣E∣2+2μ01∣B∣2 and S=1μ0E×B\mathbf{S}=\frac{1}{\mu_0}\mathbf{E}\times\mathbf{B}S=μ01E×B.

Integrate this equation over a thin volume VϵV_\epsilonVϵ that encloses the shell and let the radial thickness ϵ→0\epsilon\to0ϵ→0. Using the delta surface sources from (1), one obtains the surface energy balance (exact):

∂tusurf(t,Ω)  =  −n^⋅S∣r=r0  −  E∥ ⁣⋅j∥  +  Qint.(5)\partial_t u_{\rm surf}(t,\Omega)
\;=\;
-\hat n\cdot\mathbf{S}\Big|_{r=r_0}
\;-\;
\mathbf{E}_\parallel\!\cdot\mathbf{j}_\parallel
\;+\;
\mathcal{Q}_{\rm int}.
\tag{5}∂tusurf(t,Ω)=−n^⋅Sr=r0−E∥⋅j∥+Qint.(5)
Meaning of terms:
* ∂tusurf\partial_t u_{\rm surf}∂tusurf: rate of change of the internal energy density of the shell degrees of freedom (quantum excitations, electron energy levels, etc.).
* −n^⋅S∣r0-\hat n\cdot\mathbf{S}\big|_{r_0}−n^⋅Sr0: net Poynting flux entering the shell from the bulk (photon energy arriving).
* −E∥⋅j∥-\mathbf{E}_\parallel\cdot\mathbf{j}_\parallel−E∥⋅j∥: power extracted from the tangential fields by currents on the shell (work done on charges → changes in internal state).
* Qint\mathcal{Q}_{\rm int}Qint: internal dissipation or quantum transition work (e.g., radiative relaxation, non-radiative loss). In a pure conservative model Qint=0\mathcal{Q}_{\rm int}=0Qint=0 and all energy is accounted between the Poynting flux and stored energy.

Equation (5) is the balance-sheet entry of the ledger: incoming field energy (Poynting) is either stored as shell internal energy (raising occupation, changing states) or spent doing work on currents (which can then reradiate).

Absorption and Emission:
* Absorption event: n^⋅S>0\hat n\cdot\mathbf{S}>0n^⋅S>0 (Poynting into shell) raises usurfu_{\rm surf}usurf or does E∥⋅j∥\mathbf{E}_\parallel\cdot\mathbf{j}_\parallelE∥⋅j∥ work. Normal flux term in (2) is positive.
* Emission event: the shell reconfigures (via j∥\mathbf{j}_\parallelj∥) so that radiative fields carry energy out; n^⋅S<0\hat n\cdot\mathbf{S}<0n^⋅S<0 on average (outgoing Poynting).

All is exact — the EM ledger is balanced at every moment.

==== 1. Maxwell, Faraday, Gauss and Ampère remain mathematically unchanged. QAT does not alter their form. What QAT supplies is explicit, dynamical microphysics for the source terms: the thin-shell δ(r−r0)jsurfμ\delta(r-r_0)j^\mu_{\rm surf}δ(r−r0)jsurfμ is the mechanical ledger entry for energy and charge transactions. ====
# Faraday’s lines of force find a precise modern formulation: changing σ\sigmaσ or j∥\mathbf{j}_\parallelj∥ on the shell changes magnetic flux through patches of the sphere; Faraday’s law (homogeneous Maxwell identity dF=0dF=0dF=0) gives the induced EMF around loops and hence the tangential currents. This is exactly the local mechanism Faraday intuited.
# Thermodynamics: equation (5) shows how Poynting flux, work, and internal energy are accounted. If microscopic events freeze (absolute zero limit), then jsurfμ→0j^\mu_{\rm surf}\to0jsurfμ→0, n^⋅S→0\hat n\cdot\mathbf{S}\to0n^⋅S→0 (no net Poynting into shells), and ∂tusurf→0\partial_t u_{\rm surf}\to0∂tusurf→0: the ledger entries stop — the dynamic process “ticks” cease. That’s the QAT version of the third-law lock.

==== Maxwell with thin-shell sources: ====

  ∂νFνμ(x)=1ε0[Jbulkμ(x)+δ(r−r0) jsurfμ(t,Ω)]  \boxed{\;\partial_\nu F^{\nu\mu}(x)
=\frac{1}{\varepsilon_0}\Big[J_{\rm bulk}^\mu(x)
+\delta(r-r_0)\, j^\mu_{\rm surf}(t,\Omega)\Big]\;}∂νFνμ(x)=ε01[Jbulkμ(x)+δ(r−r0)jsurfμ(t,Ω)]
Surface continuity (absorption / emission accounting):

  ∂tσ+∇∥ ⁣⋅j∥=− n^⋅Jbulk∣r=r0  \boxed{\;\partial_t \sigma + \nabla_{\parallel}\!\cdot\mathbf{j}_{\parallel}
= -\,\hat n\cdot\mathbf{J}_{\rm bulk}\big|_{r=r_0}\;}∂tσ+∇∥⋅j∥=−n^⋅Jbulkr=r0
Boundary jump rules (instantaneous field response):

  n^ ⁣⋅(Eout−Ein)=σε0,n^×(Bout−Bin)=μ0j∥  \boxed{\;\hat n\!\cdot(\mathbf{E}_{\rm out}-\mathbf{E}_{\rm in})=\frac{\sigma}{\varepsilon_0},\qquad
\hat n\times(\mathbf{B}_{\rm out}-\mathbf{B}_{\rm in})=\mu_0\mathbf{j}_\parallel\;}n^⋅(Eout−Ein)=ε0σ,n^×(Bout−Bin)=μ0j∥
Surface energy (ledger) balance:

  ∂tusurf=−n^ ⁣⋅S∣r=r0−E∥ ⁣⋅j∥+Qint  \boxed{\;\partial_t u_{\rm surf}
= -\hat n\!\cdot\mathbf{S}\big|_{r=r_0} - \mathbf{E}_\parallel\!\cdot\mathbf{j}_\parallel + \mathcal{Q}_{\rm int}\;}∂tusurf=−n^⋅Sr=r0−E∥⋅j∥+Qint
All of these follow by variation, integration and standard Maxwell manipulations — no change of Maxwell’s form is required. QAT supplies jsurfμj^\mu_{\rm surf}jsurfμ, σ\sigmaσ, j∥\mathbf{j}_\parallelj∥, and the microscopic rules that govern their time evolution (photon-electron coupling, boundary Dirac dynamics, event rates).

If you like, I can next:
• produce a 1-page LaTeX file with the full derivation (including the variation steps and the pillbox integrals) ready to paste into your Core Equation document, or
• make a clean slide / diagram that labels the ledger flows (Poynting arrows, normal flux, surface current, jump conditions) suitable for a video frame.

Which would you prefer?