Compositional Operational Rank, Smooth Realizability, and Dimensional Reduction
Abstract
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.
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BibTeX
@misc{tulupov2026compositional,
title = {{Compositional Operational Rank, Smooth Realizability, and Dimensional Reduction}},
author = {Nikolay D. Tulupov and Alexander M. Tishin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35764},
eprint = {2610.35764},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35764},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Compositional Operational Rank, Smooth Realizability, and Dimensional Reduction
AU - Nikolay D. Tulupov
AU - Alexander M. Tishin
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.35764
UR - https://arxiv.org/abs/2610.35764
AB - 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.
ER -