What does this research mean for the field?

The KMS states of the S-local Bost-Connes system are unconditionally classified for every inverse temperature, revealing that uniqueness depends on the divergent beta-weighted sum of non-splitting primes, which refutes the expectation that divergence of the partition function forces uniqueness. Novelty: ClaimNovelty.CONTRADICTORY. Consensus alignment: ConsensusAlignment.CHALLENGES_CONSENSUS.

What question did this study set out to answer?

Classify the KMS equilibrium states of localized Bost-Connes systems unconditionally for all positive inverse temperatures and identify the exact mechanisms driving state uniqueness and phase transitions.

August 30, 2026Open Access

KMS state uniqueness and phase transitions in localized Bost-Connes systems: from Chebotarev density to the Hardy-Littlewood conjecture

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Authors

RCRuqing ChenZhejiang Normal University

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Implication

Theoretical analysis reveals exact conditions for KMS state uniqueness in localized Bost-Connes systems, indicating that partition function divergence does not guarantee state uniqueness.

Key Points

Classify the KMS equilibrium states of localized Bost-Connes systems unconditionally for all positive inverse temperatures and identify the exact mechanisms driving state uniqueness and phase transitions.
Analyzed the algebra of observables for an arbitrary number field and prime subset across inverse temperatures using class field theory and Galois group representations.
Identified the extremal KMS state simplex via affine isomorphism to probability measures on quotient symmetry groups.
Employed finite algorithmic verification routines to test specific number-theoretic configurations and character obstruction bounds.
Proved the KMS state simplex is affinely isomorphic to the probability space of the symmetry quotient, where uniqueness requires divergent beta-weighted sums over non-splitting primes.
Disproved the assumption that partition function divergence forces uniqueness by constructing prime sets where the state simplex remains a segment despite infinite zeta values.
Established that localized Bost-Connes systems can realize any prescribed finite number of phase transitions, with closed symmetry-broken phases emerging under Brun-summable prime sets.

Cite This Study

Ruqing Chen (2026) studied this question.

synapsesocial.com/papers/6a941ce46c1a8fb52e79e7cf https://doi.org/10.5281/zenodo.22152101

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Implication

Theoretical analysis reveals exact conditions for KMS state uniqueness in localized Bost-Connes systems, indicating that partition function divergence does not guarantee state uniqueness.

KMS state uniqueness and phase transitions in localized Bost-Connes systems: from Chebotarev density to the Hardy-Littlewood conjecture

Discussion

Member takes

Implication

Key Points

Cite This Study

KMS state uniqueness and phase transitions in localized Bost-Connes systems: from Chebotarev density to the Hardy-Littlewood conjecture

Discussion

Member takes

Implication

Key Points

Cite This Study

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