What does this research mean for the field?

For a non-principal prime of a number field, the obstruction to defining the affine Bost-Connes-type monoid over the ring of integers is the invariant itself, with the globally definable scalings forming exactly the subsemigroup hN of the local valuation semigroup. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

Resolve the breakdown of affine Bost-Connes-type monoids over rings of integers for non-principal primes by constructing a minimal repaired dynamical system.

September 8, 2026Open Access

The ray-class repair for affine systems: the class group as the index of definable scalings

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Authors

RCRuqing ChenZhejiang Normal University

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Implication

Mathematical analysis reveals how class group obstructions repair affine systems in number fields, indicating that arithmetic ideal structures manifest as torsion in operator K-theory.

Key Points

Resolve the breakdown of affine Bost-Connes-type monoids over rings of integers for non-principal primes by constructing a minimal repaired dynamical system.
Characterized globally definable scalings as the subsemigroup generated by the order of the non-principal ideal class in the class group.
Applied exterior-tower Pimsner-Voiculescu exact sequences with prime norm powers to compute the topological K-theory of the resulting C*-algebras.
Evaluated the explicit K-groups and unit torsion orders in the imaginary quadratic field Q(sqrt(-5)) for the non-principal prime over 3.
Proved that the obstruction defines a minimal repaired system scaled by an h-th power generator, with the unit class in K_0 exhibiting an exact torsion order of N(p)^h - 1.
Demonstrated that ideal class group structures appear in operator K-theory strictly as torsion rather than rank, contributing cyclic groups determined by field traces.
Calculated explicit K-theory invariants for Q(sqrt(-5)), yielding K_0 containing Z + Z/8 with unit order 8 and K_1 containing Z + Z/2.

Cite This Study

Ruqing Chen (2026) studied this question.

synapsesocial.com/papers/6a9fd78a58e84d0ff5b464ec https://doi.org/10.5281/zenodo.22432978

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Implication

Mathematical analysis reveals how class group obstructions repair affine systems in number fields, indicating that arithmetic ideal structures manifest as torsion in operator K-theory.

The ray-class repair for affine systems: the class group as the index of definable scalings

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Implication

Key Points

Cite This Study

The ray-class repair for affine systems: the class group as the index of definable scalings

Discussion

Member takes

Implication

Key Points

Cite This Study

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