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

==== To go from discrete shells + events → macroscopic Maxwell (D, H, ε, μ), use a standard multi-scale homogenization / ensemble average: ====
# Define microscopic scale: individual shells, event timescales τevent\tau_{\rm event}τevent, area element ΔA\Delta AΔA. On this scale fields fluctuate due to discrete absorption/emission.
# Average over many events and many atoms inside a macroscopic volume VVV (volume >> atomic spacing, but << macroscopic experiment scale): define macroscopic polarization

P(x,t)=⟨dipole moment per unit volume from boundary displacements⟩,\mathbf{P}(\mathbf{x},t)=\langle \text{dipole moment per unit volume from boundary displacements}\rangle,P(x,t)=⟨dipole moment per unit volume from boundary displacements⟩,
and magnetization M\mathbf{M}M from loop currents. These are computed from averages of the surface charge σ\sigmaσ and surface current j∥\mathbf{j}_\parallelj∥ on the many shells inside VVV.
# Insert P\mathbf{P}P, M\mathbf{M}M into Maxwell’s constitutive relations:

D=ε0E+P,H=1μ0B−M.\mathbf{D}=\varepsilon_0\mathbf{E}+\mathbf{P},\qquad \mathbf{H}=\frac{1}{\mu_0}\mathbf{B}-\mathbf{M}.D=ε0E+P,H=μ01B−M.
The macroscopic Maxwell equations for D,H\mathbf{D},\mathbf{H}D,H (with continuous ρfree\rho_{\rm free}ρfree, Jfree\mathbf{J}_{\rm free}Jfree) follow by definition. The microscopic QAT currents are the microscopic origin of P\mathbf{P}P and M\mathbf{M}M. Again, the macroscopic equations are unchanged — they arise from averaging the exact microscopic Maxwell + boundary currents.

Remark: the permittivity and permeability become emergent, geometry-dependent response functions determined by the statistics of photon-electron events and shell geometry (this is a direct QAT prediction: ε(ω), μ(ω), and dispersion link to surface geometry and event rates).