A new theoretical framework establishes the conditions under which a compositional operational rank is preserved when passing from a high-dimensional microscopic description to an effective description accessible to a physical observer. The central result is that a viable dimensional reduction is not enough to produce four effective coordinates: it must preserve the operational dependency relations of the original theory. This requirement, which the authors call closure-faithfulness, ensures that the smooth realization of the theory maintains the dependency structure of the operational closure, and not merely its total rank.

The work addresses an open problem in higher-dimensional theories: how to relate the number of directions accessible to an observer with the dimensionality of the underlying microscopic description. Along with closure-faithfulness, the authors impose real representability, constant rank, involutivity, and observer/scale compatibility. Under these hypotheses, there exists a local integral leaf whose dimension exactly matches the operational rank. At the measurement level, a further primary-identification condition is required: if the observer image coincides locally with the primary integral distribution, and the admissible protocols allow 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 r. At r = 4 it provides a conditional operational criterion for dimensional reduction applicable to higher-dimensional theories: a physically viable reduction must preserve the relevant operational dependency structure, not merely generate four effective coordinates. Negative controls on quantum hardware further show that the same restricted Fisher rank can conceal different larger ranks under a broader pre-specified measurement protocol, underscoring the need to specify the set of accessible observables.

The framework therefore constrains the operational outcome of dimensional reduction without imposing a universal four-dimensional bound on microscopic state spaces. The theory does not rule out higher-dimensional microscopic descriptions, but requires that any effective reduction be faithful to the operational structure from which it derives. The concrete application of the criterion to specific models of high-energy physics and quantum gravity remains open, as does the search for experimental protocols that could discriminate between realizations with the same apparent Fisher rank.