Article 14: Closure of Ten Open Problems in the PEW Series: KAM Stability of R/r=φ, Bekenstein Entropy, Spin Remnant χ_f=1/φ, Nuclear Magic Numbers, Schrödinger Emergence, H-O-H Bond Angle, Scattering Length, and Spin-Orbit Coupling

This article 14 closes ten open problems of the PEW (Primordial Energy Waves) framework identified in Articles 1–13. The central new result is the exact identity ⟨κ_g⟩ = −√φ for the mean geodesic curvature on the golden Hopf torus (R/r = φ), from which ten results follow. PO-φ: The ratio R/r = φ of the proton Hopf vortex is derived — not postulated — as the unique KAM-stable fixed point under quasi-periodic perturbations of the Mother Wave (Hurwitz theorem: C(φ) = 1/√5 is maximal among all irrationals). The Mother Wave vacuum fluctuation spectrum is dense, so all other torus ratios are destroyed. Status [A]. Auxiliary result: ⟨κ_g⟩ = −√φ on the golden torus (exact, from φ²−1 = φ). Status [A]. PO-γ: The beta function rate γ = φ is derived from ⟨κ_g⟩² = φ (torsion energy). Complete beta function: β(ρ) = −φ·ρ·(ρ − 1/√(8π))·(ρ − φ). Status [A]. PO-BH-1: Bekenstein entropy S = A/(4l_P²) reproduced via the Hopf torus area A_Hopf = 4π²φξ² and the geometric factor φ/(4π). Status [B]. PO-BH-2: Remnant spin χ_f = 1/φ ≈ 0.618 derived topologically. Test on GWTC-3+O4a gives tension +11.7σ; prediction reformulated with spin correction Δχ(χ_eff, q). Status [D] (reformulation required). Nuclear magic numbers 2, 8, 20 from harmonic closure and 28, 50, 82, 126 from phinaire spin-orbit coupling E_SO = √φ·⟨l·s⟩. Status [B]. Schrödinger equation emerges from the PEW Lagrangian with ℏ = √(2mα_c). Status [A]. H-O-H bond angle: exact formula cos(θ_HOH) = −1/L₃ = −1/4, giving θ = 104.48° (vs experimental 104.45°, error 0.027°). Corrects Article 9 formula. Status [A]. n-p scattering length a_sc = ξ_p^coh · φ^10 = 5.15 fm (agreement 4.9% with experimental 5.419 fm). Status [B]. Phinaire spin-orbit coupling E_SO = 8π√φ/M · ⟨l·s⟩ from |⟨κ_g⟩| = √φ. Falsifiable prediction: SO splitting factor √φ ≈ 1.272× the Mayer-Jensen value, testable on NUBASE2020 doublets. Status [B].