The Critical Circle: a Topological Cartography of the Critical Line via the Canonical Logarithm

Starting from a single operation — the polar decomposition of the canonical logarithm applied to the Euler product — we derive the topological structure of the critical line of the Riemann zeta function in the b-plane of the specialization map Φ: b ↦ p−s introduced in Paper 1. The unconditional content — tori at every depth, phase coordinates, incommensurability, density, and the strict complexity of zeros — follows from one theorem whose only inputs are the canonical logarithm, the Fundamental Theorem of Arithmetic, and the Kronecker–Weyl equidistribution theorem. The conditional content — the critical circles, the Bohr torus at σ = 1/2, the supercritical/subcritical partition at |b| = 1/2, the spectral compression, and the phase-suppression mechanism — requires one additional datum: the involution ρ(s) = 1 − s from Riemann's functional equation, which depends on the archimedean factor Λ∞(s) = π−s/2 Γ(s/2). This factor is identified as the unique obstruction to closing the chain from the elementary traversals of Paper 1 to the functional equation. Five independent appearances of 1/2 are documented and classified by epistemic status.