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We examine magnetic-free photonic isolation using a time-periodic, real, Hermitian three-resonator Hamiltonian, a Gaussian-smoothed three-dimensional toroidal Maxwell supercell, periodic T 3 Bloch conditions, and a possible M¨obius polarization-sewing operator. An exactly soluble resonant three-port reduced model yields unit forward same-carrier power and zero reverse same-carrier power, with reverse photons routed to a frequency-shifted auxiliary channel. Exact detuning and phase-error formulae and a resolvent-based deterministic robustness theorem are proved. A non-rotating-wave Floquet calculation with Ω/κ = 20, including all spatial output ports and harmonics, yields 0.004073 dB same-carrier forward insertion loss, 44.06765 dB carrier-only isolation and 37.08825 dB all-sideband isolation, in a zero-intrinsic-loss model. The infinite-harmonic truncation error is bounded a posteriori by a dissipative resolvent argument. The loss sensitivity, not-yet-calibrated electrical power, Gaussian fabrication uncertainty, Maxwell mesh dependence and polarization-sewing frequency shift are separately reported. A travelling-wave single-guide phase modulator can eliminate the backward carrier while transmitting all its energy to sidebands; it is therefore not a broadband full-frequency two-port isolator without an additional spatial or spectral dump. Published 2023 and 2026 broadband magnet-free optical-isolator experiments already establish strong prior art. New R03 results establish an exact N-arm full-harmonic spatial-routing law, factorial leakage bound, optimal phase-code uniqueness, Gaussian RF phase-covariance formula, a conditional dispersion/power tradeoff and an exact polarization-exchange echo for unequal TE/TM modulation coefficients. A fourth smooth-torus 3D Maxwell … [source excerpt; see full PDF]
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