Paper α–ΩΩE: Autonomous Clocks and Shared Mechanical Time in Holosphere Theory Collective Rotation, Passive Loading, and a Closed Common-Phase Model

Different clock mechanisms can share a reference, a common mean rotation, or an entirecalibrated phase evolution. These statements require different mechanical conditions. Startingfrom the autonomous real cosine-contact action, we obtain an exact collective angular clockand a finite-window bound on every local rotation. Below a relative-energy barrier, local clocksshare one asymptotic mean rate while their internal motion remains nonlinear. On an explicitlyoccupied thirteen-site FCC cuboctahedron, a collective relative libration has an exact nonlinearperiod and a uniform transverse ellipticity certificate. A real dipole-wave readout has a controlledfinite-amplitude phase and tick error. An additional occupied angular probe gives a fourteen-siteclosed apparatus whose local mean rate changes by the derived factor 13/14 at the specifiedpreparation, with no imposed source motion or dissipative settling. These native results establishautonomous reference clocks and their loading, while the collective rotation itself decouplesfrom internal clocks. A separate closed Hamiltonian with a common multiplier of completeclock energies demonstrates exact shared calibrated evolution for a rotor, a nonlinear libratorand a modal clock, including reciprocal source backreaction and finite-energy source turningpoints. Different single-cycle rates can occur even in this exact common-phase model becausethey average different time windows; matched-window and count-resolution bounds resolve thatambiguity. The required kinetic coupling is identified explicitly and is not inferred from thenative potential-only contact law. Complete real-number computational audits are embeddedand printed.