Geometric Subsystem Quantization of the Sine-Gordon Breather

We quantize the moduli space of the sine--Gordon breather by the geometric subsystem method. The correct canonical internal coordinate is identified as the phase evaluated at the breather centre, \(θ = φ - ωγ v a\). Using translation invariance and the explicit symplectic form on the static slice, together with a rigorous global extension via the Hamiltonian action of Lorentz boosts, we prove that the pulled‑back 2‑form is globally \[ ωbr= da dP + dθ dI, \] with \(P = {16v√{1-ω²}}{√{1-v²}}\) and \(I = 16ω\). Global Darboux coordinates are therefore \((Q,P,θ,I)\). The Moyal product in these coordinates gives a rigorous formal deformation quantization. All gaps of previous versions are closed, and the derivation is largely self‑contained.