RTLI Variance–Cycle Duality: An Operational Framework for Dimensional Structure and Observer Dependence

This paper develops an operational framework in which physical context and dimensional quantities arise from the coupling of two minimal structures: variance partition (magnitude, accumulation, change) and cycle partition (phase, periodic reference, ordering). Neither structure alone suffices to define rate quantities, spatial extension, or temporal duration; their coupling is necessary and minimal. Within this framework, time is characterized as the count of cycles of a local periodic reference, space as accumulated variance orthogonal to the time-defining cycle, and energy and momentum as variance–cycle ratios. Observer dependence in relativistic physics is rephrased as dependence on the choice of cycle partition, while the invariant spacetime interval corresponds to the integrated quantity preserved under all such repartitionings. The framework does not introduce new physical laws or predictions. Instead, it provides a unified operational grammar that clarifies the relational structure of dimensional quantities, the origin of observer-dependent decompositions, and the role of resolution limits. In discrete and coarse-grained systems, fine structure may persist while becoming unobservable due to projection and aliasing rather than physical disappearance. Connections to operationalism, information geometry, coarse-graining, and measurement theory are discussed.