What question did this study set out to answer?

The aim is to establish the uniqueness and necessity of the MCCO framework for optimizing decision-making in complex multi-chain scenarios.

May 15, 2026Open Access

Real Fog Landscape: Path-Functional Optimisation as the Mathematical Stage for MCCO

Key Points

The aim is to establish the uniqueness and necessity of the MCCO framework for optimizing decision-making in complex multi-chain scenarios.
Developed a path-functional optimisation framework on empirical path measures.
Established two central theorems to prove well-posedness and optimal mixed control.
Highlighted the inadequacy of traditional bridge-reduced approaches.
MCCO framework showed a 3.56× speedup over classical reduction methods.
Theorem 1 proved the value function as a unique solution to the coupled HJB / quasi-variational inequality system.
Theorem 2 verified optimal mixed control while ensuring robustness to cost changes.

Abstract

FOG-4 is the closing paper of the dynamic fog landscape programme. Where FOG-1 through FOG-3 describe the multi-chain fog landscape under increasingly refined formalisms (static density, time-resolved density, multi-scale homogenised density), FOG-4 makes the programme-level statement: on the real fog landscape, the multi-chain collapse-centred optimisation (MCCO) framework is the unique structurally adequate tool—not one alternative among many. The argument is in three parts. First, FOG-1 through FOG-3 each "cross a bridge" from the multi-chain reality to a reduced description, and each bridge crossing loses information essential for dynamic decision systems: chain identity, full trajectories, and inter-chain coordination structure. Second, tools that do not cross the bridge must process this chain structure intrinsically. Third, MCCO operates directly on the path-empirical measure without crossing the bridge, and its operational 3. 56×3. 56 3. 56× speedup over classical reductions is empirical evidence that chain structure carries decision-relevant information unavailable to bridge-reduced approaches. The mathematical scaffold is path-functional optimisation on Ω (0, T;DK) (0, T; DK) Ω (0, T;DK). The basic object is the empirical path measure μN=N−1∑cδρc (⋅) N = N^-1c ₂ () μN=N−1∑cδρc (⋅) ; the mean-field limit gives a path measure governed by McKean-Vlasov dynamics. Control is mixed: continuous control of the Lindblad generator (within-period dynamics) and impulse control of deployment events (the collapse-chain-triggered stopping times of FOG-3, now operator-choosable). Cost combines running and terminal. Two central theorems with full proofs anchor the framework: Theorem 1 (HJB-QVI well-posedness, central #1) establishes the value function as the unique viscosity solution of a coupled HJB / quasi-variational inequality system; Theorem 2 (verification, central #2) constructs the optimal mixed control and establishes Lipschitz robustness to cost perturbations. Sections 8-10 instantiate the framework on MCCO and prove the classical-bridge inadequacy result, elevating MCCO from "one application" to "structural necessity. " Working Paper Version 2 (MCCO-aligned reframe), 31 pages, bilingual English/Traditional Chinese. See also FOG-1 (10. 5281/zenodo. 19852881), FOG-2 (10. 5281/zenodo. 20046651), FOG-3 (10. 5281/zenodo. 20111986), MCCO standalone (10. 5281/zenodo. 20104942).

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