Paper XXI - Operational Origins of Time Delay in Bounded Dynamical Systems

DOI: 10.5281/zenodo.18009276 Operational Origins of Time Delay in Bounded Dynamical Systems Paper XXI of the Ordered-Dynamics Reconstruction Program Paul Cooneya aIndependent Researcher, Innisfil, Ontario, Canada E-mail: paul.co[email protected]to.ca Contents 1 Introduction 1 2 Operational Constraints 2 3 Sources of Update Cost 2 4 Capacity-Based Origin of Processing Delay 3 4.1 Queueing model as a saturating example 3 5 Screening and Stability 3 6 Operational Consequences 3 7 Relation to Gravity and Emergence 4 8 Conclusion 4 A Illustrative Classical Clock Model 4 B Quantum Channel Formulation 4 ime is not assumed as a primitive observable in the Ordered-Dynamics Reconstruction Program, but reconstructed operationally from correlations between finite clocks and records. Paper XI showed that bounded clocks cannot generically identify their reconstructed time with the ordering parameter governing reversible microscopic dynamics. Consistency under locality forces a regulated mapping between ordering time and operational time, introducing a dimensionless processing delay αeff . In this paper we identify the dynamical origin of this delay. We show that clocks which reconstruct time from locally stabilized records under bounded capacity cannot generically avoid additional ordered updates per operational tick. Environmental informational load increases update cost without modifying fundamental dynamics or influence propagation. The resulting delay is local, probe-dependent, and naturally suppressed in strongly bound systems. Operational time delay is sharply distinguished from geometric propagation delay responsible for gravity. Operational time dilation is thus shown to decompose into a universal geometric contribution and a probe-dependent computational contribution. 1 Introduction In the Ordered-Dynamics Reconstruction Program, time is derived operationally from correlations between finite physical systems rather than assumed as a primitive parameter. Reversible microscopic dynamics are indexed by an abstract ordering parameter λ, while physical clocks reconstruct time from locally stored records. Paper XI established that finite clocks cannot generically identify their reconstructed operational time ˜ twith the ordering parameter λ. Chronological consistency under locality – 1 – forces a regulated mapping d˜ t dλ =1+αeff ,(1.1) where αeff quantifies the excess number of ordered updates required per operational tick. The present paper identifies the physical origin of this processing delay. We do not claim that all conceivable clock constructions incur such delay. Rather, we show that clocks which reconstruct time from locally stabilized records under bounded capacity cannot generically avoid it. This analysis does not modify propagation, spacetime geometry, or microscopic reversibility. Path-dependent propagation delay responsible for gravity is derived independently in Paper XIII and encoded by the universal influence-delay factor Z(x). 2 Operational Constraints Any admissible mechanism for operational time delay must satisfy: •Bounded clocks: finite internal state space and update capacity. •Locality: dependence only on the operational causal past. •Order preservation: 1+αeff >0. •Reversibility: no microscopic dissipation. •Stability: suppression in strongly bound regimes. Remark 1 (Scope).The analysis applies to clocks reconstructing time from locally stabilized records. Ephemeral correlation-based clocks without record persistence fall outside this scope. Remark 2 (No-go for delay-free record clocks).Any record-based clock avoiding processing delay entirely would require unbounded internal capacity, nonlocal record stabilization, or irreversible microscopic dynamics, all of which violate the operational assumptions adopted here. 3 Sources of Update Cost Finite clocks incur additional ordered updates due to: •record formation and stabilization, •environmental correlation load, •state-space crowding, •temporal buffering and update backlog. Each mechanism increases update cost without modifying influence propagation or fundamental dynamics. – 2 – 4 Capacity-Based Origin of Processing Delay The origin of operational time delay can be formulated independently of any specific model by appealing to information-theoretic capacity limits. Let Rrec(x) denote the local rate (per unit ordering time λ) at which record-relevant information must be stabilized for a clock to register reliable ticks. Let CCdenote the clock’s effective local processing and recording capacity, determined by its finite architecture and internal dynamics. Stability requires Rrec(x)≤CC.(4.1) [Capacity lower bound on processing delay] For any clock reconstructing time from stabilized records under bounded capacity, the processing delay satisfies a lower bound of the form αeff (x)≳Rrec(x) CC ,(4.2) with divergence as Rrec →CCexcluded by stability. Remark 3.This bound follows directly from finite information-processing capacity and does not depend on specific clock dynamics. It may be interpreted as a consequence of limits on accessible classical information (e.g. Holevo-type bounds) and finite local channel capacity. 4.1 Queueing model as a saturating example A queueing model provides an explicit realization that saturates the capacity bound above. Let record-relevant updates arrive at rate a(x) and be resolved at finite service rate µ(x). In the stable regime, 1+αeff (x) = µ(x) µ(x)−a(x), a(x)< µ(x).(4.3) This model does not introduce new physics; it makes explicit the general capacity-based structure implied by bounded clocks. Remark 4.The queueing picture is illustrative rather than fundamental. Any physical realization that saturates local processing capacity yields the same qualitative behavior. 5 Screening and Stability In strongly bound systems, internal dynamics dominate environmental load, implying αeff →0.(5.1) This suppression is local, automatic, and required for the stability of precision clocks. 6 Operational Consequences Operational time delay produces: •probe-dependent clock comparisons, •sensitivity to informational environment, •no modification of null propagation or Lorentz structure. Geometric propagation delay Z(x) cancels in local comparisons and remains operationally distinct. – 3 – 7 Relation to Gravity and Emergence Operational time dilation decomposes into two distinct contributions: 1. Geometry: path-dependent propagation delay Z(x) (Paper XIII), 2. Computation: internal processing delay αeff . Because αeff does not affect null propagation or photon exchange, it cannot mimic or renormalize gravitational redshift. 8 Conclusion Operational time delay arises as a forced consequence of reconstructing time from bounded records under minimal operational assumptions. Time dilation in the reconstruction therefore admits a dual origin: geometry governs propagation, while computation governs clock reconstruction. A Illustrative Classical Clock Model A discrete cycle with stabilization probability pyields 1+αeff =1 p. B Quantum Channel Formulation Repeat-until-success stabilization under a local CPTP channel yields the same update overhead. References [1] P. Cooney, Operational Time Regulation from Ordered Dynamics, Paper XI of the Ordered-Dynamics Reconstruction Program (2025). [2] P. Cooney, Gravity as Inhomogeneous Influence Propagation, Paper XIII of the Ordered-Dynamics Reconstruction Program (2025). [3] A. S. Holevo, Bounds for the Quantity of Information Transmitted by a Quantum Communication Channel, Probl. Inf. Transm. 9, 177 (1973). [4] J. Watrous, The Theory of Quantum Information, Cambridge University Press (2018). [5] H. Salecker and E. P. Wigner, Phys. Rev. 109, 571 (1958). [6] A. Peres, Am. J. Phys. 48, 552 (1980). – 4 –