From Rotating-Frame Time Holonomy to Dynamical Phase Memory A Finite-Stiffness Phase-Locked Ring Model with an Operational Clock Readout

This technical research note develops a structured bridge between rotating-frame time holonomy, finite-stiffness phase-locking dynamics, and an operational definition of time based on net phase advance. The first part presents a self-contained standard-relativistic derivation of the slow-clock-transport limit for a clock completing one directed equatorial loop on the rotating Earth. In the limit of arbitrarily slow transport, the traveller’s additional quadratic speed contribution vanishes while the rotation–transport cross term remains finite. For an ideal eastward equatorial loop, the result is [ \Delta\tau_{\mathrm{slow,east}} -\gamma_{\oplus} \frac{2\pi\Omega R^2}{c^2} \simeq -207.386\ \mathrm{ns}. ] The opposite sign is obtained for westward transport. This is the standard Sagnac/slow-clock-transport holonomy. It is additive under repeated winding and introduces no new physics. The second part constructs a logically separate finite-stiffness effective model of a relativistic phase-locked ring. The collective variables include a proper circumference, an independent proper wavenumber, a material rotation angle, a wound Tick phase, and a second dynamical reference or lighthouse phase. The strong-locking branch recovers the radial-accommodation relation [ r_{\mathrm{lock}}(\omega) \frac{\rho} {\sqrt{1+\omega^2\rho^2/c^2}}, ] while finite stiffness produces controlled corrections and an explicit local-stability condition. An operational clock count is defined from the relative advance of the material, Tick, and reference phases. The central dynamical result is the exact logarithmic Tick-rate equation [ \frac{\mathrm d\ln F}{\mathrm dn} \frac{2\pi}{\omega} \left[ \frac{\dot Q_{\Theta}}{Q_{\Theta}} \frac{\dot I_{\Theta}}{I_{\Theta}} \right], ] where F is the Tick-phase rate, Q_{\Theta} is its conjugate phase charge, I_{\Theta} is the effective phase inertia, and n is the completed material-rotation count. A closed, conservative, uniformly rotating branch with constant phase inertia reduces to a phase-pendulum system and produces no secular phase pumping. A persistent multiplicative phase drift therefore requires additional dynamics, such as changing phase inertia, non-adiabatic driving, an open phase reservoir, radiation, a persistent phase lag, or a structural transition. The paper also introduces illustrative, non-microscopic closure forms for the phase inertia, distinguishes closed and open lighthouse sectors, formulates a differential–algebraic system for numerical evolution, and imposes explicit energy, angular-momentum, stability, causality, and falsification requirements. The terrestrial loop interval is additionally expressed as a caesium-133 phase-count equivalent: [ N_{\mathrm{Cs,loop}}^{\mathrm{eq}} \Delta\nu_{\mathrm{Cs}} \left|\Delta\tau_{\mathrm{loop}}\right| \simeq 1906.42 ] caesium cycles. This number is a metrological restatement of the 207.386\ \mathrm{ns} interval, not an independent model parameter and not proof of a stationary-clock drift. The central open quantity is [ \overline{\epsilon}_{\phi} \left\langle \frac{\mathrm d\ln F}{\mathrm dn} \right\rangle. ] The proposed multiplicative phase-memory regime exists only if this quantity is dynamically non-zero while all conservation, stability, and causality conditions remain satisfied. Cosmological applications are intentionally deferred until this coefficient is derived and connected to dimensionless observables. Author: S. M. H. Emamifar Spokesperson, Independent Research Collaboration on Black Hole and Cosmology Concepts (IRCBHC) ORCID: 0009-0007-6257-0163 Technical Research Note, Version 1.2, July 2026