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 model built on the inertia directions, matched to the global units, is shown to carry no thermodynamics (Theorem 6.1): inertia is local and thermodynamically inert, while Frobenius is global and carries the linking matrix.