Energoservis (Czechia)

Biologically reachable spectral fibres: repeat shuffling, copy ambiguity and the scarcity of cryptic pathogenic missense variants in ClinVar

A protein sequence encoder that depends only on k-mer statistics is blind to the rearrangements that preserve them: the spectral fibre of the protein. Most members of a fibre are not proteins that evolution or disease could produce. We ask which part of the fibre is biologically...

Biologically reachable spectral fibres: repeat shuffling, copy ambiguity and the scarcity of cryptic pathogenic missense variants in ClinVar

A protein sequence encoder that depends only on k-mer statistics is blind to the rearrangements that preserve them: the spectral fibre of the protein. Most members of a fibre are not proteins that evolution or disease could produce. We ask which part of the fibre is biologically...

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 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...

Galois descent for affine Bost-Connes boundaries: capitulation, the obstruction cocycle, and the emergence of the class group

Let p be a non-principal prime of a number field K, h the order of its class, H the Hilbert class field, G = Gal(H/K) = Cl_K. By the principal ideal theorem p capitulates, pO_H = (Pi), and p splits into |Cl_K|/h primes of residue degree h, the Frobenius being the class itself. Th...

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...

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...

Galois descent for affine Bost-Connes boundaries: capitulation, the obstruction cocycle, and the emergence of the class group

Let p be a non-principal prime of a number field K, h the order of its class, H the Hilbert class field, G = Gal(H/K) = Cl_K. By the principal ideal theorem p capitulates, pO_H = (Pi), and p splits into |Cl_K|/h primes of residue degree h, the Frobenius being the class itself. Th...

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...

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 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...

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...

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...

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 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...

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...

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...

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...

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...