TITLE: The Lighthouse, the Spiral, and the Phase-Locked Ring: A Finite-Stiffness Effective Model of Radial Accommodation, Torque Transfer, and Relativistic Rotation

DESCRIPTION: This paper addresses the Ehrenfest rotating-disk problem by constructing a conditional effective model for a relativistic ring endowed with a conserved phase winding and a preferred proper wavenumber. Instead of imposing a rigid proper-circumference constraint, the work introduces a minimal finite-stiffness energy functional that penalizes deviations in the proper circumference, wavenumber, and winding compatibility. The hard-locking kinematic map r(ρ, ω) = ρ / sqrt(1 + ω²ρ²/c²) is recovered as the leading-order strong-locking limit, while finite stiffness predicts a calculable outward radial correction and a positive proper-circumference shift. A local stability analysis yields a sufficient dimensionless criterion Λ > 1/4 for the axisymmetric branch. By promoting the rotational angle to a dynamical coordinate, the analysis reveals a turning point in the orbital angular momentum at β = 1/sqrt(3). An independent wound "Tick" phase is introduced as a necessary internal angular-momentum channel, providing J = J_orb + N Q_Θ and a quantitative condition for monotonic torque-driven spin-up. The model is presented as a mathematically explicit, testable effective theory and a roadmap toward a gauge-invariant finite-width field model, with all distinctions between derived results, model-dependent hypotheses, and open microscopic tasks clearly demarcated.