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

Abstract

For a number field K and an arbitrary set S of finite primes we classify the KMS states of the S-local Bost-Connes system A_{K,S}, unconditionally, for every inverse temperature beta > 0. The KMS_beta simplex is affinely isomorphic to Prob(G_S / Xi_beta^perp), where the symmetry group G_S is an extension of the subgroup of the narrow class group generated by S by U_S / closure(O_{K,+}^x), and Xi_beta is the group of characters chi with sum_{p in S} N(p)^{-beta} |1 - chi(Frob_p)|^2 finite. Uniqueness thus holds exactly when, for every nontrivial finite abelian L/K unramified outside S and infinity, the primes of S not splitting completely in L have divergent beta-weighted sum. For S the set of all primes this is Chebotarev density, and at beta = 1 it is the non-vanishing L(1,chi) != 0 of Hecke and Landau; the theorems of Bost-Connes and of Laca-Larsen-Neshveyev are the case S = P_K.\n\nThe criterion refutes the natural expectation that divergence of the partition function forces uniqueness. We exhibit S with zeta_{K,S}(beta) infinite, and with the parameter subgroup dense in G_S, whose KMS_beta simplex is a segment at every beta > 0, the two extreme states separated by an explicit pointwise invariant whose almost-everywhere definedness is a Borel-Cantelli condition. Density asks for one prime detecting a character; uniqueness asks for their beta-weighted abundance, and the two separate by a Kakutani dichotomy.\n\nTwo further mechanisms are isolated. Over a field of narrow class number greater than 1 the Hilbert class field supplies obstructing characters that are everywhere unramified, impossible over Q. And beta -> Xi_beta is nondecreasing, so systems with any prescribed finite number of phase transitions exist. Finally, for Brun-summable S of critical exponent 1 the partition function is finite at the critical point, so the symmetry-broken phase is closed, the opposite of the classical system.\n\nThis record archives the LaTeX source of the paper together with the verification suite, the data it produces, and the figure-generation code. The verification scripts perform finite checks: they can refute a claim but never prove one.\n\nUnrefereed preprint. Theorem 5.5(2),(3),(5) are conditional on a Hardy-Littlewood-strength lower bound for the Sophie Germain primes and are labelled as such in the text; Remark 4.18 notes that the description of G_S as a topological group, though not the criterion itself, is entangled with Leopoldt's conjecture. No priority check against the literature on KMS states of semigroup and groupoid C*-algebras has been completed, and the bibliography has not been independently verified.