Central characters and commutative sectors in a proposed observable algebra for fractional quantum anomalous Hall systems

This independent mathematical preprint examines the observable-algebra presentation in the author-hosted manuscript of Sati and Schreiber, “Identifying anyonic topological order in fractional quantum anomalous Hall systems” (Applied Physics Letters 128, 023101, 2026; DOI: 10.1063/5.0305441). Taking that presentation as given, it proves that all 2|C|-th roots occur as central eigenvalues, establishes commutativity at C = ±1, and recalls with a proof the finite-dimensional irreducible-sector dimension ord(zeta²). Explicit unitary examples and supplementary checks of 312 sectors are included. Scope: the specific source examined is the author-hosted manuscript dated 30 December 2025, linked by an author as the published version. The corresponding passages in the publisher’s typeset PDF have not been verified. The note acknowledges that the authors’ October 2026 revision of arXiv:2505.22144 already states the precise dimension formula and distinguishes exchange-phase signs. The note makes no claim of a new dimension theorem or a refutation of the entire physical proposal. This is an independent preprint, not a publisher-issued erratum. The PDF includes the author’s affiliations and ORCID. AI disclosure: AI tools were used for English improvement, numerical checks, and assistance in constructing counterexamples and developing theorem and proof arguments. The author is responsible for all mathematical statements, lemmas, theorems, proofs, interpretations, references, and final verification.