Electromagnetism: The Classical Continuum Limit of the EM Constraint Subnetwork

This document establishes the 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 -modulated Maxwell equations are derived with complete -factor self-consistency verification: () = ₀/ (), () = ₀/ (), c () = c₀ (), recovering standard Maxwell theory at =1. The dual-channel damping framework (ₖ = ₀ (1-) + ₀ (1-e^- C/C_{}) ₖ/₁) is instantiated for electromagnetic waves, with the Cauchy interlacing theorem (GT v3. 0 III. 4, CLOSED) guaranteeing that each photon emission necessarily increases the spectral gap ₁ and each absorption necessarily degrades it. The U (1) ₄₌ gauge symmetry is derived from Type~II edge phase consistency on the constraint network, with the discrete graph connection ₈₉ = e^i₈₉ as its mathematical foundation. Five core academic contributions are established. (i) Complete -modulated Maxwell theory: the full set of Maxwell equations in media, including -dependent constitutive parameters, covariant form _ (F^) = ₀ J^, Poynting theorem with -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). (ii) Five-channel EM thermodynamics and degradation corridor: the five-channel complete constraint Second Law (CND v4. 0 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, -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 (R₅₌, R₌ₓ, R₂₅, R₌₅, R₈ₑₑ), cross-matrix with all EM processes, degradation corridor with five EM-domain physical signatures, and vacuum baseline calibration are provided. (iii) Faraday law, Noether theorem, and conservation laws: Faraday's law is derived as a topological identity on the constraint network (₂ₘ₂₋₄₈₉ = -ₜ₂ₘ₂₋₄₈₉), 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 charge conservation on the constraint network. -modulated Poynting theorem, EM momentum density, radiation pressure, and angular momentum conservation are derived. (iv) Waveguides, transmission lines, cavities, and scattering theory: TE/TM mode classification for rectangular waveguides, quality factor Q (), Ohm's law constraint-network derivation ( (, T₄₅₅) ), skin effect, eddy currents, impedance matching, and unified scattering theory framework (Thomson, Compton, Mie, Rayleigh with constitutional distinction by a/₄₅₅ ratio). (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 _ = _), classical electron 4/3 problem (resolved by Ef vs Ec distinction), hidden momentum and Shockley-James paradox, electromagnetic knots (Ra\~nada 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 Eb^melt (L) L^). 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 weak Euler representation (Vacuum v2. 1 2. 5) with Ec=0, m=0, and Ṡ₌₀₈₍ₓ^EM=0 (four-path constitutional convergence). The wave impedance ₀ = 377\, is shown to be -invariant ( () / () = ₀ for all) --- 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), 1 evolutionary attractor (CND v4. 0 III-E. 2, CLOSED), and U (1) Abelian nature (no gauge boson self-coupling 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; -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).