A Diophantine Non-Resonance Ansatz for Recurrent Vortex-Ring Phase Flows on T²: A Conditional Extremal Phase Partition

We consider a particle-like vortex ring in a three-dimensional incompressible continuous medium and ask how a localized recurrent excitation might be modeled so as to reduce low-order self-commensurability. The claim of the paper is intentionally limited. We do not derive vortex stability from first-principles Euler or Navier-Stokes analysis, nor do we prove a KAM theorem for an actual vortex-ring Hamiltonian. Instead, we formulate an effective non-resonance ansatz in which recurrent phase flows with stronger resistance to low-order rational approximation are treated as preferred arithmetic candidates under the adopted modeling assumptions. At the level of effective phase kinematics, we model a recurrent vortex-ring excitation on the torus T², with two angular sectors corresponding to toroidal transport and core rotation. Writing ΦT for the toroidal phase advance and ΦC for the core-rotation phase advance, we define the winding number ν = ΦT / ΦC. To sharpen the non-resonance heuristic, we introduce a Diophantine non-resonance selection ansatz: among admissible recurrent phase partitions, those with stronger resistance to low-order rational approximation are regarded as preferred non-resonant candidates. Using Hurwitz's theorem, continued fractions, and the extremal role of the Diophantine class represented by φ = (1 + √5) /2, we identify that class as the distinguished arithmetic candidate under this ansatz. Coupled with the cycle normalization ΦT + ΦC = 2π, and choosing the representative value ν = φ, we obtain the conditional extremal phase partition ΦT = 2π / φ, ΦC = 2π / φ². Under the normalization convention that the intrinsic core variable is the core-rotation phase advance per toroidal cycle, the associated dimensionless core phase increment is ωcore = 2π / φ². The arithmetic input is rigorous, but the bridge from Diophantine extremality to actual vortex-ring dynamics remains an effective-medium ansatz rather than a theorem of vortex dynamics. This is version M4 (corrected) of the preprint. Full mathematical details, including rendered equations, are in the uploaded PDF.