Time as an Operational Construction: Implicit Clicks in Distributed, Physical, and Biological Systems

This work introduces a minimal operational framework for internal temporal order in distributed, physical, and biological systems. Rather than assuming time as a primitive global parameter, the paper formalizes temporal order as an operational reachability structure induced by internally distinguishable and irreversibly stabilized state updates (“implicit clicks”). The framework separates: operational temporal order (reachability-based precedence), temporal metric (clock representations), and thermodynamic orientation. It provides structural results concerning: order–metric separation, granularity bounds under resource constraints, and applicability to asynchronous distributed systems and biologically constrained systems. This version (v1.0) establishes the original formulation and priority of the framework.