Compositional Operational Rank, Smooth Realizability, and Dimensional Reduction
Higher-dimensional theories distinguish the dimensionality of a microscopic description from the number of directions accessible to a physical observer. Relating these notions requires not only dimensional reduction, but also a consistent preservation of operational independence across different levels of description. We establish sufficient conditions under which a compositional operational rank is preserved first as the dimension of a smooth integral leaf and then as an observable Fisher-information rank. The key requirement is closure-faithfulness: the smooth realization must preserve the dependency relations of the operational closure, rather than merely its total rank. Together with real representability, constant rank, involutivity, and observer/scale compatibility, this yields a local integral leaf whose dimension equals the operational rank. A further primary-identification condition is required at the measurement level. If an independently specified observer image coincides locally with the primary integral distribution and the admissible protocols provide local resolution, finite labeled-mixture closure, and locally constant outcome support, then the same rank can be realized by a finite composite Fisher-information protocol. The construction holds for arbitrary finite rank. At r = 4 it provides a conditional operational criterion for dimensional reduction in higher-dimensional theories: a viable reduction must preserve the relevant operational dependency structure, not merely produce four effective coordinates. Quantum-hardware negative controls further show that the same restricted Fisher rank can conceal different larger ranks under a broader pre-specified measurement protocol. The framework therefore constrains the operational outcome of dimensional reduction without imposing a universal four-dimensional bound on microscopic state spaces.