Axiomatic Emergence T; The Physical Origin of Subfactors and Planar Algebraic Structures

changzheng zhou, ZIQING zhou

Published Jan 24, 2026• 0 citations

Abstract

Based on the two fundamental principles—Information Conservation (A1) andFinite-Step Computability (A2)—this paper derives the core structures of operatoralgebras, namely subfactors, the Jones index, and planar algebras. The researchdemonstrates that the factor structure of the observable algebra in quantum systems, the finite index of subfactor inclusions, and its discrete set of possible values(the Jones index) are not pure mathematical abstractions, but rather the inevitablealgebraic forms of organization for physical systems under information constraintsand realizability requirements. Information Conservation necessitates the existenceof a unique finite faithful trace on the algebra, forcing the observable algebra tomanifest as a type II factor. Computability requires the algebraic description tobe finitely generated, which locks the index of a subfactor inclusion to be finite andquantizes its spectrum into the classical Jones index set. This discrete spectrum directly corresponds to planar algebras (e.g., Temperley–Lieb algebras) which encodethe braiding statistics of anyons in topological quantum systems and the algebraicproperties of fault-tolerant quantum error-correcting codes. Furthermore, this paper proposes two operable experimental verification schemes: validating Jones polynomial evaluations by measuring the statistical phases of anyon braiding in quantumHall systems, and indirectly determining the effective dimension parameter relatedto the Jones index by analyzing the scaling behavior of logical error rates in surface codes implemented on superconducting quantum processors. These schemesestablish a falsifiable, unified bridge between abstract algebra, topological order,and quantum information.

DOI: 10.5281/zenodo.18359667

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