FOG-3 extends the Dynamic Fog Landscape framework (FOG-2) to multi-scale operational systems with two coupled time clocks: a fast clock τ governing intra-period dynamics and a slow clock t governing cross-period redeployment, coupled by a small parameter ε = Δτ/Δt. The framework's defining design choice is that cross-period deployment events are triggered by the collapse chain itself, not by a deterministic schedule or an exogenous Poisson process: deployment is the structural consequence of observation–collapse events accumulating to threshold. This preserves the strong-Copenhagen modelling commitment of the broader programme through the multi-scale extension. In the limit ε → 0, two-scale convergence theory reduces the coupled (τ, t) system to an effective slow-clock Fokker–Planck equation whose drift and diffusion coefficients are averages over the fast-clock invariant measure. Two central theorems anchor the framework, both with full proofs: Theorem 1 (homogenisation, central #1): convergence in Wasserstein-2 metric with explicit rate O (ε^1/2) + O (N^−1/2). Theorem 2 (multi-scale sensitivity bound, central #2): extends FOG-2's single-scale sensitivity bound via the effective contractivity constant αᵉff = αˢlow − c λᶠast, identifying three multi-scale regimes (contractive, marginal, expansive). The fast-clock contraction term reveals that strong fast-clock mixing reduces multi-scale expansiveness even when the slow-clock generator is itself expansive. A hierarchical tracking lemma generalises FOG-2's component lifecycle framework to nested time scales. To our knowledge, the configuration of collapse-chain-triggered stopping-time deployment with Lindblad–Fokker–Planck homogenisation in this combined form is novel. Working Paper Version 1, 28 pages, bilingual (English / Traditional Chinese). Empirical validation is the subject of a separate companion paper. See also FOG-1 (10. 5281/zenodo. 19852881) and FOG-2 (10. 5281/zenodo. 20046651).
Huiying Rao (Sun,) studied this question.