The Universal Trajectory Architecture: Continuous Flows, Non-Archimedean Branching, Cauchy–Poisson Spectral Windows, and the Tripartite Index on the Adèle Class Space

We present the complete mathematical formulation of the Universal Trajectory Architecture (UTA), an operator-algebraic, spectral-geometric, and topological framework that embeds continuous geometric flows, non-Archimedean arithmetic branching, and non-equilibrium state transitions into a single \theta-twisted algebra \mathcal{A}(\theta) over the global Adèle Class Space: \mathbb{X}_{\mathbb{Q}} \equiv \mathbb{A}_{\mathbb{Q}}/\mathbb{Q}^\times

Physical trajectories are demonstrated to be neither purely smooth classical continuum curves nor disconnected stochastic jumps, but complementary spectral projections of the global infinitesimal scaling generator: \mathbf{H} \equiv x \frac{d}{dx} + \frac{1}{2} = -i \frac{d}{du}, \qquad u \equiv \ln|x|