What does this research mean for the field?

The localized Bost-Connes system attached to the prime family of Q47(n) has a fractional critical inverse temperature of 1/46, exhibits unique KMS states in the high-temperature phase (conditional on the Bateman-Horn conjecture), and demonstrates 'congruence invisibility' where deterministic congruences do not cause symmetry breaking. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

Characterize the thermodynamic phase transitions, KMS state uniqueness, and arithmetic symmetries of the localized Bost-Connes system associated with the degree-46 prime-generating polynomial Q47.

August 31, 2026Open Access

The localized Bost-Connes system attached to the prime family of Q47(n) = n^47 - (n-1)^47: fractional critical temperature, KMS uniqueness, and a congruence that the dynamics cannot see

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RCRuqing ChenZhejiang Normal University

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Implication

Theoretical study demonstrates fractional critical temperature and KMS uniqueness in localized Bost-Connes systems, revealing congruence invisibility in prime-generating dynamics.

Key Points

Characterize the thermodynamic phase transitions, KMS state uniqueness, and arithmetic symmetries of the localized Bost-Connes system associated with the degree-46 prime-generating polynomial Q47.
Constructed the localized C*-algebraic Bost-Connes system and its partial Euler product partition function for primes represented by Q47(x) = x^47 - (x-1)^47.
Established bounds on the critical inverse temperature using the Selberg sieve unconditionally and the Bateman-Horn conjecture conditionally.
Calculated character sums using Weil's bound for multiplicative characters and a Hensel step to evaluate obstruction group vanishing for KMS states.
Determined the critical inverse temperature to be beta_c = 1/46, with the partition function diverging at the doubly logarithmic rate (A/46) log log X at the critical point.
Demonstrated that the obstruction group vanishes for all beta <= 1/46 under Bateman-Horn, establishing KMS state uniqueness across the high-temperature half-line.
Identified congruence invisibility, proving that the deterministic congruence p = 1 mod 282 leaves symmetry breaking absent because the modulus cannot be represented by the prime family.

Cite This Study

Ruqing Chen (2026) studied this question.

synapsesocial.com/papers/6a95427bf20e493292a74dfb https://doi.org/10.5281/zenodo.22177596

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Implication

Theoretical study demonstrates fractional critical temperature and KMS uniqueness in localized Bost-Connes systems, revealing congruence invisibility in prime-generating dynamics.

The localized Bost-Connes system attached to the prime family of Q47(n) = n^47 - (n-1)^47: fractional critical temperature, KMS uniqueness, and a congruence that the dynamics cannot see

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The localized Bost-Connes system attached to the prime family of Q47(n) = n^47 - (n-1)^47: fractional critical temperature, KMS uniqueness, and a congruence that the dynamics cannot see

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