This work provides the full rigorous derivation of the suppression of even‑order (even‑k) parametric resonances in sinusoidally driven spherical electromagnetic cavities, completing the analysis initiated in the companion paper: C. R. Singleton, "Parametric Instabilities and Mode Coupling in Oscillating Spherical Cavities: Theoretical Framework, " Phys. Rev. A 112, 063533 (2025), https: //doi. org/10. 1103/4dc7-hx5b That earlier work derived exact coupling coefficients λₙₙ'⁽ˡ⁾ (t) and identified instability tongues at Ω = 2ωₙₗ/k (k = 1, 2, 3, …), with numerical Floquet evidence suggesting even‑k resonances are suppressed by a factor ∼ε^k-1 relative to odd‑k. The analytical proof was deferred. Here we deliver it explicitly, establishing four principal results: 1. Mechanism 2 (inter‑mode coupling) contributes identically zero to all even‑k resonances at leading order in ε. The O (εᵏ) instability is carried entirely by Mechanism 1 (Mathieu frequency modulation). The proof reveals a structural zero in the Floquet coupling matrix that operates identically for k=2 and all higher even k, independent of geometry. 2. Exact growth rate at k=2: γ₂ = (√5/6) ε² ω₀ ≈ 0. 3727 ε² ω₀. This irrational coefficient arises from the competition of two off‑resonant Floquet mediators (D₋₁ = +4, D₊₁ = –12). Standard single‑mediator treatments underestimate the correct value by a factor 2√5/3 ≈ 1. 491. 3. Drive contamination characterisation: Only a sin (2Ωt) component in the boundary position can seed the k=2 instability. The exact contamination growth rate is γcontam = (1/2) δ ω₀, and the dominance threshold is δ < (√5/3) ε² (e. g. –111 dBc at 2Ω for ε = 2×10⁻³). 4. All‑orders, all‑modes suppression theorem: The ε^k-1 suppression of even‑k relative to odd‑k resonances holds to all orders in ε, for any number of coupled modes, and for any resonator whose inter‑mode coupling has the form λₙₙ' (t) ∝ sin Ωt (velocity coupling). The governing principle is the temporal ℤ₂ symmetry t → t + π/Ω, which forces the minimum contributing perturbative order at even k to be εᵏ. The proofs rest on: a closed‑form, phase‑independent evaluation of the overlap integral Iₖ = ∫₀^2π/Ω sin (Ωt) cos (kΩt/2) dt over one period, vanishing for even k; a complete trigonometric expansion showing the λ̇ term vanishes for all k ≥ 2; a first‑principles Floquet recurrence with exact integer weights (m±2) ; explicit Schur complement analysis; an all‑orders suppression proof via Fourier parity of interaction vertices; and universality: the suppression holds for any resonator with λ (t) ∝ sin Ωt time dependence, independent of geometry or Bessel‑function factors. The results predict an experimentally decisive growth‑rate hierarchy in high‑Q cavities (e. g. 4. 77 GHz superconducting with Q=10⁹, ε=2×10⁻³), enabling mode‑selective control and suppression of instabilities in cavity optomechanics, quantum state engineering, and dynamical Casimir setups. The temporal ℤ₂ symmetry identified here suggests broader analogies in Floquet‑driven systems. All results are classical; the quantum generalisation—photon statistics, squeezing spectra, and the role of the angular‑momentum degeneracy factor 2l+1—is deferred to future work.
C.R. Singleton (Sun,) studied this question.