Ontology of Transition—Part III: The Thermodynamic Price of External Time: Rate–Distortion Bounds for Physical Clock Records

This article is Part III of the three-part scientific work Ontology of Transition. The work’s central idea is that transition, rather than the static thing, is the primary unit of description, while operational time is realized through physical records whose production, transmission, interpretation, storage, and reuse have distinct informational and thermodynamic limits. Part I established that external time is accessible only through a physical record of a selected clock process. Part II quantified the reconstruction error caused by information loss in the delivery channel and the precision limit imposed by dissipation in the reading register. Part III adds a lower bound on the work required to renew a reusable clock-recording system. For a finite classical information-bearing device operating isothermally, the minimum renewal work is bounded by the amount of information required to reconstruct the external clock at the prescribed accuracy. This information requirement depends on the actual physical delivery channel and is described by indirect rate–distortion theory. If the desired accuracy exceeds what the channel permits, no amount of reset work can make the target reconstruction attainable. The result applies to correlated finite histories without requiring independent and identically distributed observations. Exact solutions are obtained for uniform cyclic clocks transmitted through symmetric channels and through channels combining erasures with substitutions. For sequential ring clocks, the asymptotic renewal cost is determined by the entropy of unpredictable innovations rather than by the total number of ticks. The complete framework therefore identifies three mutually irreducible limits: information destroyed during transmission, precision paid for through register dissipation, and reuse paid for through renewal work. These limits constrain different physical quantities through different mechanisms. The result does not equate time with work; it determines the minimum ideal work required to make a specified resolution of external time reusable at a chosen system boundary. Numerical computations confirm the analytical results to machine precision.