Electromagnetism: The Classical Continuum Limit of the EM Constraint Subnetwork

This document establishes the \textbf{comprehensive classical continuum limit of electromagnetism} on the EET constraint network, providing the L4 structural realization of the constitutional definitions established by Interaction v2.1 and all 28 companion mother texts. The $\eta$-modulated Maxwell equations are derived with complete $\Gamma$-factor self-consistency verification: $\epsilon(\eta) = \epsilon_0/\Gamma(\eta)$, $\mu(\eta) = \mu_0/\Gamma(\eta)$, $c(\eta) = c_0 \cdot \Gamma(\eta)$, recovering standard Maxwell theory at $\eta=1$. The dual-channel damping framework ($\gamma_k = \gamma_0(1-\Gamma) + \gamma_0(1-e^{-\gamma C/C_{\max}})\lambda_k/\lambda_1$) is instantiated for electromagnetic waves, with the Cauchy interlacing theorem (GT v3.0 \S III.4, CLOSED) guaranteeing that each photon emission necessarily increases the spectral gap $\lambda_1$ and each absorption necessarily degrades it. The $U(1)_{\mathrm{EM}}$ gauge symmetry is derived from Type~II edge phase consistency on the constraint network, with the discrete graph connection $\Gamma_{ij} = e^{i\theta_{ij}}$ as its mathematical foundation. \textbf{Five core academic contributions} are established. \textbf{(i) Complete $\eta$-modulated Maxwell theory:} the full set of Maxwell equations in media, including $\eta$-dependent constitutive parameters, covariant form $\partial_\mu(\Gamma F^{\mu\nu}) = \mu_0 J^\nu$, Poynting theorem with $\eta$-correction terms, Fresnel equations, Cherenkov radiation, static EM (Poisson/Laplace, uniqueness theorem, multipole expansion, method of images), magnetostatics (magnetic dipole moment, magnetic energy, magnetic scalar potential, magnetic circuit theory), and superconducting EM (London equations, Meissner effect). \textbf{(ii) Five-channel EM thermodynamics and degradation corridor:} the five-channel complete constraint Second Law (CND v4.0 \S III-C.2, CLOSED) is instantiated for all EM processes --- formation (photon emission --- Bremsstrahlung, transition radiation, Larmor formula, multipole radiation, photon statistics), meltdown (photon absorption and scattering theory --- Thomson, Compton, Mie, Rayleigh, optical theorem), maintenance (constitutional zero for photons --- four-path convergence proof), transient (virtual photons --- Casimir effect, dynamic Casimir effect, Schwinger critical field, $\theta$-vacuum, instantons, chiral anomaly, trace anomaly, Unruh effect), and capture (stimulated emission --- laser constitutional mechanism, Schawlow-Townes linewidth, Purcell effect, Dicke superradiance, photon BEC, coherent and squeezed states). The five-ratio diagnostic dashboard ($R_{\mathrm{FM}}, R_{\mathrm{MT}}, R_{\mathrm{CF}}, R_{\mathrm{MF}}, R_{\mathrm{irr}}$), cross-matrix with all EM processes, degradation corridor with five EM-domain physical signatures, and vacuum baseline calibration are provided. \textbf{(iii) Faraday law, Noether theorem, and conservation laws:} Faraday's law is derived as a topological identity on the constraint network ($\sum_{\mathrm{cycle}}\Delta\mathcal{U}_{ij} = -\partial_t\sum_{\mathrm{cycle}}\theta_{ij}$), with the flux rule limitations and Faraday disc paradox resolved through the constitutional distinction between induced electric field and motional EMF. Noether's theorem is instantiated for the EM subnetwork --- $U(1)$ gauge symmetry $\to$ charge conservation on the constraint network. $\eta$-modulated Poynting theorem, EM momentum density, radiation pressure, and angular momentum conservation are derived. \textbf{(iv) Waveguides, transmission lines, cavities, and scattering theory:} TE/TM mode classification for rectangular waveguides, quality factor $Q(\eta)$, Ohm's law constraint-network derivation ($\sigma(\eta,T_{\mathrm{eff}})$), skin effect, eddy currents, impedance matching, and unified scattering theory framework (Thomson, Compton, Mie, Rayleigh with constitutional distinction by $a/\lambda_{\mathrm{eff}}$ ratio). \textbf{(v) Constitutional resolution of major unsolved problems:} ten academic priority registrations (PRI-EM-001 to PRI-EM-010) address the Abraham-Minkowski controversy (photon momentum in media resolved via force decomposition theorem), Abraham-Lorentz force runaway solutions (constitutionally eliminated by finite formation rate and causal cone), QED Landau pole (resolved by constraint network natural UV cutoff $\omega_{\max} = \sqrt{\lambda_{\max}}$), classical electron $4/3$ problem (resolved by $E_f$ vs $E_c$ distinction), hidden momentum and Shockley-James paradox, electromagnetic knots (Ra\~{n}ada solutions expressed via constraint network cycle-space topology), fast magnetic reconnection (congestion/decongestion framework), EM arrow of time (formation/meltdown asymmetry as constitutional root), photon non-localizability (constitutional positive explanation of the Newton-Wigner theorem), and magnetic helicity topological selectivity of dissipation (Arrhenius hierarchical scaling law $E_b^{\mathrm{melt}}(L) \propto L^\delta$). A systematic EET response to 38 major unsolved problems and controversies in electromagnetism is provided, categorized as 10 PRI-body, 12 body, 4 mother-text reference, 7 constitutional open problems, and 5 appendix treatments. The photon is constitutionally anchored as a pure $e^{i\theta}$ weak Euler representation (Vacuum v2.1 \S2.5) with $E_c=0$, $m=0$, and $\dot{S}_{\mathrm{maint}}^{\mathrm{EM}}=0$ (four-path constitutional convergence). The wave impedance $\eta_0 = 377\,\Omega$ is shown to be $\eta$-invariant ($\sqrt{\mu(\eta)/\epsilon(\eta)} = \eta_0$ for all $\eta$) --- a core constitutional prediction. The ``cleanest force'' status of electromagnetism is traced to three constitutional roots: zero maintenance burden (categorical distinction from all other fundamental interactions), $\eta\approx1$ evolutionary attractor (CND v4.0 \S III-E.2, CLOSED), and $U(1)$ Abelian nature (no gauge boson self-coupling $\to$ no confinement). Astrophysical and cosmological EM applications are provided, including CMB as degradation calibration, primordial magnetic field origin, coronal heating, dynamo problem, photon sphere, and Hawking EM radiation. Forty bridge declarations (BR-01 to BR-40) register all cross-layer connections to the complete companion ontology network, including three cross-REAL bridges to Nuclear REAL v1.0. Seven falsifiable predictions (P-1 to P-7) with complete PM-9 statistical specifications are presented. {Keywords:} Electromagnetism; Constraint Network; $\eta$-Modulated Maxwell Equations; Dual-Channel Damping; Five-Channel Second Law; $U(1)$ Gauge Emergence; Graph Connection; Photon Non-Localizability; Abraham-Minkowski Controversy; Abraham-Lorentz Force; Magnetic Reconnection; Magnetic Helicity; Electromagnetic Arrow of Time; Degradation Corridor; Faraday's Law; Noether's Theorem; Waveguides; Resonant Cavities; Scattering Theory; Nonlinear Optics; High-Harmonic Generation; Casimir Effect; Schwinger Critical Field; Quantum Electrodynamics Correspondence; Energy-Efficiency Theory (EET).