Theorem of Temporal Resolution Limitation and the Phase Coordination Principle (v1.1)

This paper formalizes a fundamental limitation of long-horizon temporal coordination in distributed systems with autonomous local clocks and no continuous physical coupling. Using a variance identity for integrated frequency deviations under mild wide-sense-stationary assumptions, we show that inter-node clock-offset uncertainty grows without bound over time. Consequently, protocols that target fixed absolute timestamps cannot provide time-uniform, fixed-tolerance coordination guarantees without either continuous coupling or periodic recalibration. In response, the paper states the Phase-Coordination Principle: replace absolute time targets with a shared cyclic phase convention \(\Phi(t)\in\mathbb{S}^1\), and trigger actions within wrap-around-safe phase acceptance windows. Under explicit assumptions—cycle-anchored triggering and bounded short-horizon relative clock variation—the paper proves a Bounded Coordination result: inter-node execution skew is bounded by the chosen window duration plus a short-horizon stability term, and does not grow with total elapsed time since initialization. The result provides a theoretical foundation for window-triggered coordination layers relevant to distributed quantum networks and classical distributed systems. Keywords phase coordination, cyclic phase, distributed systems, clock drift, synchronization, quantum networks, window triggering, \(\mathbb{S}^1\).