Rectangular Completeness Encompasses Standard Physical Closure

Research Paper

Chast K. Wolfe · v3 · June 13, 2026

Manuscript Record

Work code: CRI-RC
Work type: Research paper
Status: Public preprint; versioned research manuscript. Not peer reviewed unless otherwise stated.
Author identifier: Chast K. Wolfe, ORCID 0009-0008-8846-2539
Publisher: Closure Research Initiative
Version: v3, June 13, 2026
Citation target: Current version: closureresearchinitiative.org/rc/. DOI: 10.5281/zenodo.20695177.
Canonical page: closureresearchinitiative.org/rc/
Files: PDF · LaTeX source
Rights: All rights reserved
Citation record: Use the canonical page and BibTeX block below for the current version.

Rectangular completeness is exactly equivalent to the full-product state-space condition for classical Hamiltonian systems under separating cross-subsystem observables and realization normalization. The thermodynamic preparation-space analogue follows as a corollary. In quantum mechanics the same comparison theorem holds on the product-state sector, while the full density-matrix state space contains correlation data not determined by subsystem marginal profiles.

Method: The paper compares rectangular completeness with standard classical, thermodynamic, and quantum two-subsystem formalisms at the level of product-profile availability.
Main results: Classically, rectangular completeness is equivalent to the full-product state-space condition under the stated observable and realization normalizations. The thermodynamic preparation-space analogue is immediate. Quantum mechanically, the equivalence holds on the product-state sector, while full density-matrix states contain correlation data not fixed by subsystem marginals.
Scope: The paper is a comparison theorem. It does not replace Hamiltonian isolation, thermodynamic no-exchange conditions, or unitary quantum dynamics; it identifies the product-structure condition those settings instantiate where such a condition is present.

Scope: binding statements are the definitions, hypotheses, and theorems in the paper. Related internal sources: CCW for rectangular completeness, Overview for the standard-closure comparison, and Reading Guide for linked classical and quantum background sources.

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Version History

Version	Date	Status	File	Details
v3	June 13, 2026	Current version	PDF·Source·DOI	Product-structure precision release States the theorem as a comparison with the full-product component of standard two-subsystem closure, separates that condition from dynamics and exchange laws, updates the rigid-constraint and quantum-entanglement passages, and cites current CCW and CSM records. DOI assigned to v3 on June 15, 2026.
v2	June 10, 2026	Superseded version	PDF·Source	CRI metadata release Affiliation, correspondence, and manuscript format updated to the common Closure Research Initiative article style. Mathematical content unchanged from v1.
v1	June 5, 2026	Superseded version	PDF·Source	Initial preprint Initial preprint; superseded archival release.

BibTeX Citation

@misc{wolfe2026rectangular, author = {Chast K. Wolfe}, title = {Rectangular Completeness Encompasses Standard Physical Closure}, year = {2026}, version = {3}, howpublished = {Closure Research Initiative preprint}, doi = {10.5281/zenodo.20695177}, note = {Version 3; current version at https://closureresearchinitiative.org/rc/}, url = {https://closureresearchinitiative.org/rc/}, license = {All Rights Reserved. See https://closureresearchinitiative.org/license/} }