Compositional Clock Theory: Temporal State Machines and Multitime

Time is often treated as a single scalar, yet modern socio-technical systems evolve under multiple independent clocks with radically different rates, propagation speeds, and reversal costs. This paper introduces Compositional Clock Theory (CCT) through Temporal State Machines (TSMs): a transition-first foundation in which time is defined as accounting of state change, not as a privileged axis inherited from any particular physical paradigm. The central claim is that multitime is not optional — it emerges whenever system evolution depends on multiple transition families (e.g., execution, verification, commitment, repair, legal accountability, institutional adaptation) that cannot be collapsed into one total order without losing governance-relevant structure. To model real deployments, we treat space as a set of local machines over a domain D: a system is a family of coupled local TSMs, where locality makes concurrency native and “now” becomes a partial order of events rather than a global instant. We define an Operational Temporal Regime (OTR) for each entity (human, machine, organization) as its multiaxial temporal signature across rates, propagation, and commit slope. From OTR and environmental change rates, we derive Realtime Obsolescence Score (ROS) as a field over entities and regions, capturing when systems silently fall behind before stakeholders notice. We then formalize Causal Capture (CC) as the regime where execution outruns verification under high commit slope, allowing machine-speed loops to commit trajectories “as soon as possible” before human authorization or comprehension closes the loop. Moreover, we connect multitime dynamics to managed irreversibility by integrating Time-to-Okay (TTO) horizons and the Rollback Ratio (1:N) as enforceable recovery budgeting: for each unit of committed exposure, systems must provision sufficient rollback capacity to recover to an explicitly accepted “okay” within TTO. We show how these constructs turn reversibility into a measurable asset, and how governance becomes an executable constraint on the reachable-futures graph through admissibility corridors, receipts, hesitation, and uncommit — aligning TSMs with field-specific jurisdiction and overfield-level runtime control.