Log-Density Tensor-Train Control with Multi-Well Metastability: Phase-Space Frames, Fokker–Planck Transport, Dual Riccati Architecture, CVaR Stability, and Multi-Well Switching Stability

We present a unified mathematical framework that integrates two complementary componentsof distributional control under heavy-tailed uncertainty: (I) the log-density tensor-train(TT) architecture for single-mode belief propagation, and (II) multi-well log-density extensionscapturing metastable overshoot and controlled switching. The combined framework proceedsthrough five interlocking layers: phase-space frame construction via Gabor–Hermite andSchr¨odinger wavelet bases; TT decomposition of the joint log-density; Fokker–Planck transportof TT cores; dual Riccati control (PSD/NSD duality) with CVaR-augmented objectives; andmulti-well log-density structure governing transitions between metastable operating regimes.The central new contribution is Theorem 12.1, which establishes exponential α-divergencestability for systems undergoing multi-well switching, bounding both the inter-well transitiontransients and the post-switch residual tail risk. The result unifies the single-well convergencetheory (Theorem 10.2) with the multi-well overshoot characterisation (Theorem 11.1) into asingle Lyapunov-based stability certificate valid across all wells and all transition events.