Localized Bost-Connes systems

This book assembles, as thirty-nine chapters in eight parts, the foundational paper and Papers I-XXV and XXVII-XXXIX of the author's series on localized Bost-Connes systems: the Bost-Connes construction over a number field K with the set of all primes replaced by an arbitrary set S of finite primes. Each chapter is one paper of the series, unchanged in its mathematics; citations between papers have been converted to chapter references, the notation and macros have been unified, and a preface, a reading guide with a chapter-to-paper table and a dependency table, a page of standing notation, a closing chapter of open problems, and a single merged bibliography have been added. The foundational result (Part 1) is that the KMS_beta simplex of the S-local system A_{K,S} is affinely isomorphic to Prob(G_S / Xi_beta^perp), where G_S is the symmetry group, an extension of the subgroup of the narrow class group generated by S by the S-units modulo the closure of the totally positive global units, and Xi_beta is the group of characters chi of G_S for which the series of N(p)^{-beta} |1 - chi(Frob_p)|^2 over the primes p in S converges. Uniqueness of the equilibrium state is therefore a Chebotarev-type equidistribution criterion and not a consequence of the divergence of the partition function: there are sets S with divergent zeta_{K,S}(beta) and dense parameter subgroup whose KMS_beta simplex is a segment at every temperature. The phase transition is a tail event, a Kakutani dichotomy in which every local factor is harmless and only the infinite product degenerates. Part 2 (Chapters 2-14) works out the structure of the abelian systems: the factor type and the spectrum of transitions, including Cantor transition loci and the unattainable devil's staircase; the ideal lattice, boundary quotient, K-theory and index theory; the identification of the obstruction group with the measurable first cohomology of the tail relation and the turbulence of that cohomology; dynamical and relative entropy; the compact quantum metric structure and the Wasserstein separation of the broken phases, with its finite-dimensional reduction, chaos bounds and adaptive couplings; the K-theory of the affine boundary quotient, the ray-class repair, and the Galois descent through which the class group emerges; and the free Toeplitz-Cuntz variant with its free critical exponent. A recurring conclusion is that every invariant which tensorises, such as K-theory, the type of the factor, the spectral distance and dynamical entropy, is blind to the arithmetic, which lives in the measure class alone. Parts 3 and 5 (Chapters 15-19 and 25-29) document the search for a genuinely noncommutative arithmetic thermodynamics: the matrix Kakutani dichotomy and the Tannakian obstruction, the Galois-twisted arithmetic groupoid, non-abelian Bost-Connes groupoids built from the ordered Frobenius cocycle, the abelian scaling bottleneck, the Hurwitz quaternion monoid and the angular Hurwitz system, and the localized GL_2 system. The search ends in a no-go theorem: for every admissible arithmetic Ore monoid the orbit relation is hyperfinite, and the Ramanujan bound for the Hecke operator on the Bruhat-Tits tree is the tempered bound, the signature of amenability rather than an obstruction to it. Part 4 (Chapters 20-24) treats rigidity and reconstruction: the von Neumann algebras of the systems are injective, so W*-superrigidity fails maximally, while the C*-dynamical system recovers the set of primes with their norms, and under an intertwining hypothesis the symmetry group, because the modular flow pins down the Cartan subalgebra; quantum optimal transport is shown not to see the transition. Part 6 (Chapters 30-34) transfers the theory to arithmetic topology, where primes are knots and the Kakutani dichotomies become linking dichotomies, and to global function fields, where the commensurability of the log-norms forbids type III_1, the factor recovers the temperature, the Weil zeta function is the partition function, and abelian reconstruction stops at the isogeny class plus a torsor coordinate, with twin curves already in genus one. Part 7 (Chapters 35-36) proves that there are no p-adic KMS states and locates where Leopoldt's conjecture actually enters. Part 8 (Chapters 37-39) applies the criterion to sieve-theoretic and Sato-Tate families of primes: confinement to a residue class obstructs uniqueness exactly when that class lies in the kernel of a character, so the Landau and Friedlander-Iwaniec primes are permanently obstructed while the twin, Sophie Germain and Chen primes are not; and the free twin-prime system is shown, unconditionally, to be supercritical. Status. This is an unrefereed monograph assembled from unrefereed preprints, and the book proves nothing about the classical distribution of primes. Each chapter closes with an assessment recording what was and was not established at the time of writing; where a later chapter withdraws or settles an earlier claim, the later chapter is authoritative, and the correspondence is recorded in the closing chapter of open problems. Conditional results are labelled as such in the text. A priority check against the literature on KMS states of groupoid C*-algebras for infinite sparse S, and an independent verification of the bibliography, are both outstanding. The LaTeX sources, the assembly script, the verification code and the data are in the repository linked below.