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.