Recursion, Coherence, and the Lowest Global Mode

This work presents an operator-based framework in which physical recurrence is identified as a necessary precondition for the existence of coherent structure, and coherence is shown to emerge as a selective outcome of nonlinear recurrence rather than as a fundamental assumption. Time, geometry, and dynamical laws are treated as operational constructs that become meaningful only within coherence-stabilized regimes. Using an operational spectral definition of coherence, the paper demonstrates that global coherence exists only within a finite admissible regime and is characterized by the presence of a lowest stable global coherence mode. Below this mode, no global coherence can exist; above it, coherence fragments into locally phase-locked but globally unaligned structures. Empirical analyses across atomic clock networks and related datasets identify a candidate value of approximately 31 nHz for this lowest global coherence mode. This mode is interpreted not as a driving force or background oscillation, but as the most persistent phase reference selected by nonlinear recursive dynamics—the final coherent structure to disappear as spacetime coherence collapses. The article deliberately focuses on the minimal physical mechanism and selection principle underlying global coherence, rather than reproducing empirical pipelines or numerical fits, which are addressed in companion works. The framework has implications for the operational emergence of time and geometry, coherence collapse in extreme regimes (including black holes), and cross-scale structural transitions in physical systems.