Closure Forces Spherical Geometry: Genuinely Closed Three-Dimensional Systems Are Diffeomorphic to S³

Geometric rigidity at the manifold stage: closed 3-systems are S³
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First page of Closure Forces Spherical Geometry
Research Paper

Chast K. Wolfe · v3 · June 13, 2026

Manuscript Record

Work code
CRI-CFSG
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 Not peer reviewed unless otherwise stated.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/cfsg/. DOI pending.
Canonical page
closureresearchinitiative.org/cfsg/
Files
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Rights
All rights reserved
Citation record
Use the canonical page and BibTeX block below for the current version.
At the faithful smooth-realization stage, a genuinely frame-complete closed three-dimensional system is forced to the spherical case. The theorem is stated through isometry-induced frame transport, frame-bundle transitivity, constant curvature, and the compact simply connected space-form classification.
Method
The paper works after faithful smooth realization has been obtained. It identifies admissible frame transport with isometry-induced transport and formulates genuine closure as frame-completeness on the orthonormal frame bundle.
Main result
Frame-completeness is equivalent to transitivity of the isometry action on the frame bundle. That transitivity makes sectional curvature constant; compactness and simple connectivity then select the S³ space-form case.
Scope
The theorem is a manifold-stage rigidity result. It does not rederive the continuum from raw comparison data; that interface is supplied by the monograph’s faithful-realization analysis and recorded separately on Logical Status.

Scope: binding statements are the definitions, hypotheses, and theorem in the paper. The result is stated at the faithful smooth-realization interface; see Notation and Logical Status for condition (T), axiom (D), and the canonical realization, and Overview for the linked differential-geometric background sources.

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Version History

VersionDateStatusFileDetails
v3June 13, 2026Current versionPDF·Source
Frame-completeness theorem release

Recasts the manifold-stage theorem through frame-complete closed realization, isometry-induced frame transport, frame-bundle transitivity, constant curvature, and the compact simply connected space-form classification. DOI pending for the current release.

v2June 10, 2026Superseded versionPDF·Source
CRI metadata release

Affiliation, correspondence, and manuscript format updated to the common Closure Research Initiative article style. Mathematical content unchanged from v1.

v1June 5, 2026Superseded versionPDF·Source
Initial preprint

Initial preprint; superseded archival release.

BibTeX Citation
@misc{wolfe2026closure, author = {Chast K. Wolfe}, title = {Closure Forces Spherical Geometry: Genuinely Closed Three-Dimensional Systems Are Diffeomorphic to S\textsuperscript{3}}, year = {2026}, version = {3}, howpublished = {Closure Research Initiative preprint}, note = {Version 3; DOI pending for the current release}, url = {https://closureresearchinitiative.org/cfsg/}, license = {All Rights Reserved. See https://closureresearchinitiative.org/license/} }