Ruqing Chen

Localized Bost-Connes systems V: the free variant and the free critical exponent

The free-probabilistic variant of the localized Bost-Connes construction is built, replacing the tensor independence of the prime coordinates by free independence. The free critical exponent is identified and compared with the classical one, and the extent to which the phase-tran...

Leopoldt and the p-local Bost-Connes groupoid: the rank of the isotropy, not its triviality, and why log_p(Np) vanishes

The isotropy of the p-local Bost-Connes groupoid at an intermediate stratum Y_T is identified exactly: it is the kernel of the map from the lattice of T-supported principal divisors to the quotient of the complementary unit factors by the projected closure of the global units. Co...

Quantum optimal transport on Bost-Connes systems: the twist is harmful, the generator has a sign error, and the transition is invisible to local inequalities

Quantum optimal transport is set up on localized Bost-Connes systems. The GKLS generator must be taken in symmetrized form, the proposed twisted transport metric is shown to be harmful (cross-prime cancellation makes the twisted couplings degenerate), and tensorisation reduces mo...

Localized Bost-Connes systems IV: noncommutative metric geometry

The localized Bost-Connes algebra is made a compact quantum metric space. Lipschitz seminorms adapted to the prime coordinates are constructed, the tensorization of spectral gaps across primes is controlled (Lemma 6.1), the exclusivity theorem shows the tail observables lie outsi...

The Galois-twisted arithmetic groupoid: its KMS classification, and why the Tannakian obstruction does not compute it

The Galois-twisted groupoid model, in which the Frobenius elements act through a homomorphism of the abelian ideal group (hence with commuting lifts), is classified: its KMS_beta simplex is Prob(E_beta \ G) for the abelian essential-value group E_beta, which agrees with the Tanna...

Leopoldt and the p-local Bost-Connes groupoid: the rank of the isotropy, not its triviality, and why log_p(Np) vanishes

The isotropy of the p-local Bost-Connes groupoid at an intermediate stratum Y_T is identified exactly: it is the kernel of the map from the lattice of T-supported principal divisors to the quotient of the complementary unit factors by the projected closure of the global units. Co...

Circle actions on function field Bost-Connes systems: the compact group advantage is illusory, and the Weil zeta function is thermodynamic

The circle action coming from the degree grading on a function field Bost-Connes system is analyzed: the apparent advantage of a compact acting group is shown to be illusory for classification purposes, while the Weil zeta function of the underlying curve is exhibited as a thermo...

Non-abelian Bost-Connes groupoids: two no-go theorems, and a construction that realizes Prob(G/N_beta) exactly when the Frobenius lifts mix

Two routes to a non-abelian Bost-Connes system are closed: KMS states of B x J_S are traces on B, so noncommutative coefficient algebras are invisible; and isotropy produces the dual data Irr(N) with a symmetry-fixed point, so it can never yield Prob(G/N) with a transitive action...

The universal multiplicity-norm mismatch, and the reduction of (C4) to the norm-one subgroup

In every localized Bost-Connes system over a global field the number of sublattices or ideals of a given norm exceeds the norm: d_p = (q^d - 1)/(q - 1) > q, a universal multiplicity-norm mismatch (Theorem 1.2) that empties boundary quotients and displaces equilibrium ranges throu...

Sato-Tate families in localized Bost-Connes systems: open phase, no congruence obstruction, and the symmetric power expansion

For a non-CM elliptic curve over Q and a Sato-Tate interval of angles, the associated prime family has positive density, critical exponent one, and an open symmetry-broken phase, with zeta_S(beta) = (delta + o(1)) log(1/(beta-1)) unconditionally, sharpening to O(1) under any effe...

Circle actions on function field Bost-Connes systems: the compact group advantage is illusory, and the Weil zeta function is thermodynamic

The circle action coming from the degree grading on a function field Bost-Connes system is analyzed: the apparent advantage of a compact acting group is shown to be illusory for classification purposes, while the Weil zeta function of the underlying curve is exhibited as a thermo...

The statistical mechanics of sieve-theoretic primes: the critical spectrum, the log-power dichotomy, and two families that are never unique

For six sieve-theoretic prime families the critical exponents equal the counting exponents, and the open/closed character of the symmetry-broken phase is the power of the logarithm, hence the number of simultaneous prime conditions. Confinement to a single residue class obstructs...

Localized GL_2 Bost-Connes systems: the equilibrium range and the automorphic obstruction range are disjoint, by exactly one

For GL_2 the semigroup crossed product is unavailable (the unit group SL_2(Z) is infinite, by the unit-group exclusion of Paper X) and one must pass to the Connes-Marcolli Hecke quotient, whose local factor is the Solomon zeta of M_2(Z_p), computed by the sublattice count sigma(p...

Isogeny, zeta-equivalence, and the Picard torsor in function field Bost-Connes systems

For function field Bost-Connes systems the deep stratum of the boundary is a torsor under the Picard group, and under the full gauge action of Paper XIV the closure of the translation monoid in its homeomorphism group is a compact group acting simply transitively, recovering the...

Isogeny, zeta-equivalence, and the Picard torsor in function field Bost-Connes systems

For function field Bost-Connes systems the deep stratum of the boundary is a torsor under the Picard group, and under the full gauge action of Paper XIV the closure of the translation monoid in its homeomorphism group is a compact group acting simply transitively, recovering the...

The Galois-twisted arithmetic groupoid: its KMS classification, and why the Tannakian obstruction does not compute it

The Galois-twisted groupoid model, in which the Frobenius elements act through a homomorphism of the abelian ideal group (hence with commuting lifts), is classified: its KMS_beta simplex is Prob(E_beta \ G) for the abelian essential-value group E_beta, which agrees with the Tanna...

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

Let Q47(x) = x^47 - (x-1)^47, an irreducible polynomial of degree 46, and let S be the set of primes it represents. We analyse the localized Bost-Connes system A_{Q,S} in the framework of the author's paper "KMS state uniqueness and phase transitions in localized Bost-Connes syst...

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

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...

Conserved dynamics make the solute hyperuniform, not the domains

Analysis code, derived data and manuscript source for a study of spatial order in Cahn–Hilliard coarsening. In a phase-separating system with a locally conserved order parameter, the concentration field is hyperuniform, with Sc(k) ~ k⁴, while the point pattern of domain centroids...

A Statistical Thermodynamic Lattice Model for Cu–In Coupled Substitution in Sphalerite

Version 2.0. This version corrects the equilibration argument of Section 4.7. Replica-exchange sampling reaches a state 3.1 eV over the box below anything the version 1 annealing protocol attained, showing that the two-trajectory test reported there was not a sufficient test of e...