What question did this study set out to answer?

The research aims to characterize bounded pair algebras and their unique properties within the EB carrier framework. It seeks to formalize the underlying mathematical structures and applications.

April 29, 2026Open Access

A Characterization Theorem for Bounded Pair Algebras: The Unique O(1) Enclosure-Preserving Algebra over the EB Carrier (r4)

Key Points

The research aims to characterize bounded pair algebras and their unique properties within the EB carrier framework. It seeks to formalize the underlying mathematical structures and applications.
Included structural reviews and external verification of the planar carrier-set theorem.
Incorporated extensive bibliographic references to strengthen prior-work comparisons across mathematical models.
Verified the rigidity proof for the characterization stages (Stage 1/2/3).
Established classification across the Cayley-Dickson tower with relevant empty cell proof by Hurwitz.

Abstract

Manuscript revision r4. RSOS-260797 planar carrier-set theorem; this revision succeeds r2 (the ScholarOne submission currently under RSOS initial assessment) and incorporates external structural review findings plus tonight's verification work. Changelog r2 → r3 → r4: r2 → r3 (intermediate revision; not separately deposited on Zenodo): Bibliography additions: Dawood (2019), Skowron abstract now describes the planar theorem only. §8.4 reproducibility discussion expanded; engagement with Boldo et al. (2023) floating-point certification framework added. r3 → r4 (post-Manusights-corrected-review revision; this deposit): Seven essential revisions and five recommended revisions from the Manusights structural review incorporated. Thirteen additional references added: Atanassov, Boldo et al. (2023), Bustince, Dawood, de Figueiredo richer methods that retain auxiliary state are outside this class rather than incorrect." Backup-venue list catalogued (Reliable Computing, J. Symbolic Computation, J. Logic and Algebraic Methods in Programming, Numerical Algorithms, Soft Computing). Page count: ~35 pp. Tonight's (2026-04-26 / 2026-04-27) line-by-line verification work on the broader Memoirs monograph (10.5281/zenodo.19822813) verified the base rigidity proof (Stage 1/2/3 of the three-stage characterisation), the four-cell classification across the Cayley-Dickson tower (Cell 1 ℂ, Cell 2 ℍ, Cell 3 provably empty by Hurwitz, Cell 4 𝕆), Definition 4.3 (native O(1) two-way criterion), and the universality theorem. Several specification tightenings identified by that verification work apply to the RSOS planar theorem as well; they are scheduled for a future r5 revision and are not yet incorporated in r4 deposited here. Status: r4 is the rapid-hardening revision prepared after the Manusights structural review surfaced four legitimate findings on the originally-deposited r2 (ScholarOne r2 currently under RSOS initial assessment by Editor Brandrick). Restricted-access priority deposit; access requests evaluated case-by-case.

A Characterization Theorem for Bounded Pair Algebras: The Unique O(1) Enclosure-Preserving Algebra over the EB Carrier (r4)

Key Points

Abstract

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