Operator Theory of Distribution Dynamics (OTDD v2.0)

We introduce a mathematical framework in which probability distributions are fixed points of a nonlinear composite operator T_θ acting on the space of probability measures P (X). The operator combines dissipation, integral convolution, multiplicative modulation, thresholding, and normalisation. We prove existence of fixed points (Schauder), spectral structure (Krein–Rutman), and exponential convergence. For a scale-invariant Gamma kernel, we derive a closed-form dominant eigenvalue: λ₁ (a; κ, β) = β⁻ᵃ · Γ (κ+a) / Γ (κ) and explicit spectral gap ratio r = κ/ (κ+a). The tail class of the fixed point is determined entirely by the growth order of η (x): power-law, exponential, Gaussian, lognormal, and stretched-exponential distributions all emerge from a single operator family. The critical boundary q = 3/2 ↔ κ = 4 connects to the GPM–GARCH CLT breakdown condition. Operator non-commutativity E_η, W generates a canonical Dirichlet energy whose monotone dissipation provides a second, independent convergence mechanism. Includes: full paper (Sections 1–9), appendices (A1: Krein–Rutman proof, 3: dissipation proof, B: extended tail classification, C: (α, γ) universality plane), four figures, and Python simulation code.