Ruqing Chen

The abelian scaling bottleneck, and the angular Hurwitz system: where noncommutative arithmetic thermodynamics can and cannot live

Two structural results on the possibility of a genuinely noncommutative arithmetic thermodynamics. First, a universal no-go, the abelian scaling bottleneck: for a right LCM semigroup with scalar one-parameter dynamics driven by a multiplicative weight N, every KMS state is tracia...

The ray-class repair for affine systems: the class group as the index of definable scalings

For a non-principal prime p of a number field the affine Bost-Connes-type monoid is not well defined over the ring of integers. This paper proves that the obstruction is itself the invariant: with h the order of [p] in the class group, the globally definable scalings form exactly...

K-theory of the affine boundary quotient: the return of the norm and the trace

For the affine control system attached to a principal prime p=(pi) of a number field K of degree n, with q = N(p), the boundary quotient is stably isomorphic to C_0(K_p) crossed by (O_K[1/pi] semidirect Z), and an iterated Pimsner-Voiculescu computation over the exterior-power to...

The adaptive coupling upper bound for the Wasserstein separation: microscopic flips against the macroscopic symmetry cost

For the extreme broken-phase KMS states of a localized Bost-Connes system, separated by the order parameter chi(g) prod_j epsilon_j(x_j), the only known upper bound on the Connes-Rieffel distance W_1 was the trivial group-flip cost w_0. This paper constructs an x-adaptive couplin...

The ray-class repair for affine systems: the class group as the index of definable scalings

For a non-principal prime p of a number field the affine Bost-Connes-type monoid is not well defined over the ring of integers. This paper proves that the obstruction is itself the invariant: with h the order of [p] in the class group, the globally definable scalings form exactly...

The Wasserstein separation of the broken phases: a finite-dimensional reduction and the quadratic chaos bound

For a localized Bost-Connes system in its symmetry-broken phase, the Connes-Rieffel distance W_1 between the two extreme KMS states was previously squeezed between w_0 Pi_infinity(beta) and w_0. This paper sharpens both sides. A finite-dimensional reduction sandwiches W_1 between...

K-theory of the affine boundary quotient: the return of the norm and the trace

For the affine control system attached to a principal prime p=(pi) of a number field K of degree n, with q = N(p), the boundary quotient is stably isomorphic to C_0(K_p) crossed by (O_K[1/pi] semidirect Z), and an iterated Pimsner-Voiculescu computation over the exterior-power to...

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...

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...

A non-commutative Ore LCM semigroup with arithmetic thermodynamics: the Hurwitz quaternions

The Hurwitz quaternion order provides a genuinely non-commutative right Ore, right LCM monoid whose Nica-Toeplitz system has arithmetic thermodynamics, with partition function zeta(beta) zeta(beta-1) and critical exponent 2. Conditions (C1)-(C4) for a non-injective arithmetic Bos...

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...

The equal-norm case: norm-separation is not a hypothesis but a characterization, and the gauge-equivariant repair

When distinct primes of S share the same norm, the naive reconstruction arguments of the series degenerate. This paper shows that norm-separation is exactly the condition under which the dual gauge action separates the prime coordinates, characterizes the equal-norm degeneracy, a...

The limits of abelian reconstruction: twin curves exist, genus one suffices, and the obstruction is a torsor coordinate

Twin curves, non-isomorphic curves whose localized Bost-Connes systems are isomorphic as abelian systems, exist already in genus one: the point-count and Picard-class profiles are proved equal for isogenous elliptic curves (with the degree-n class equal to nu_n times a fixed clas...

Equivariant rigidity of localized Bost-Connes C*-dynamical systems: the fixed-point algebra is the Cartan subalgebra

For localized Bost-Connes C*-dynamical systems the fixed-point algebra of the gauge action is identified with the canonical Cartan subalgebra, and equivariant isomorphisms of the systems are shown to be governed by this identification: an equivariant rigidity theorem reducing dyn...

Localized Bost-Connes systems III: invariants, entropy, and the measure class

This paper isolates which invariants of a localized Bost-Connes system depend on the inverse temperature and which do not. The measurable cohomology of the tail relation with circle coefficients is computed through its kernel, the beta-dependence is located exactly (Corollary 2.4...

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...

Localized Bost-Connes systems in arithmetic topology: local class field theory, linking Kakutani dichotomies, and topological phase transitions

The arithmetic-topology dictionary (primes as knots, fields as three-manifolds) is transported to localized Bost-Connes systems. The Gauss linking matrix formula (Theorem 3.2) expresses the obstruction data through linking numbers via local class field theory, and the meridian mo...

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...

Localized Bost-Connes systems III: invariants, entropy, and the measure class

This paper isolates which invariants of a localized Bost-Connes system depend on the inverse temperature and which do not. The measurable cohomology of the tail relation with circle coefficients is computed through its kernel, the beta-dependence is located exactly (Corollary 2.4...

Towards non-abelian localized Bost-Connes systems: the matrix Kakutani dichotomy and a Tannakian obstruction

For a compact group G with Frobenius classes indexed by a set S of primes, the obstruction series Xi_beta(rho,S) = sum over p of N(p)^{-beta} ||I - rho(Frob_p)||_HS^2 defines a Tannakian category T_beta = Rep(G/N_beta) (the matrix Kakutani dichotomy, for commuting lifts), with th...