Dynamic Fog Landscapes: Time-Resolved Multi-Chain Observation, Component Lifecycle, and the Lindblad–Fokker-Planck Limit

This paper extends the static fog landscape framework of FOG-1 to time-evolving multi-chain observation systems. We develop two complementary regimes connected by a mean-field convergence theorem: a streaming framework operating at finite chain population and discrete time grid, and a Lindblad–Fokker-Planck framework operating in the continuum limit. The streaming framework fits per-time Gaussian Mixture Models to the chain ensemble and tracks component identity across consecutive fits via Wasserstein-2 matching, partitioning the cost space into four interpretable lifecycle events (drift, appearance, disappearance, merge/split). The Lindblad–Fokker-Planck framework derives an explicit partial differential equation for the time-resolved density F(x, t) from a continuous-time Lindblad master equation, with drift and diffusion coefficients given in closed form for the diagonal regime. We establish five formal results. The component tracking lemma (Lemma 1) and the dynamic Wasserstein-L² sensitivity bound (Theorem 4) are stated with full proofs; Theorem 4 extends FOG-1's static Wasserstein-L² Lipschitz bound to time-evolving ensembles via a Lindblad contractivity assumption, identifying three operational regimes (contractive, marginal, expansive). Three asymptotic theorems (mean-field limit, projected Fokker-Planck, mixing time formula) are stated in sketch form with proof tools identified; full proofs are deferred to v2. The strong-Copenhagen ontological commitment of FOG-1 is preserved and structurally strengthened: the framework asserts the landscape is observation-induced at every instant, with the trajectory itself constituted by deployment events rather than tracked from a pre-existing dynamics. This is a Working Paper Version 1 establishing the framework's theoretical infrastructure. Empirical instantiation on operational systems is the subject of a separate companion paper. The framework opens two further programme extensions: FOG-3 (multi-scale homogenisation across coupled time clocks) and FOG-4 (path-functional optimisation directly on chain trajectories).