We construct a family of creatively equipped convex topological spaces, that provide a rigorous functional analytic framework for distributional Laplace–Stieltjes transforms. These countably multinormed, complete, Hausdorff locally convex spaces and their duals of Roumieu and Beurling type ultradistributions are designed so that shift operators corresponding to time delays act continuously, while the dual spaces accommodate generalized functions that model sudden disruptions and stochastic lead time distributions. The construction is extended to product spaces, negative time domains, and additional parameters, yielding a flexible setting for Cauchy problems with distributional data.

The framework is applied to eight central problems in modern supply chain engineering: inventory systems with stochastic lead time delays, quantitative analysis of the bullwhip effect in multi-echelon networks, propagation and recovery from singular disruptions, EV battery state-of-charge and state-of-health dynamics under delay and fractional order effects, hybrid quantum classical logistics optimization, real-time digital-twin orchestration, multi echelon network propagation on graphs, and closed-loop flows in circular economy reverse logistics. In each case we derive explicit operational calculus formulae and obtain stability or recovery certificates directly from the defining seminorm family. The resulting theory supplies well posedness, quantitative bounds, and stable interfaces between classical simulators and quantum solvers, thereby offering a mathematically grounded foundation for resilient, sustainable, and real-time decision-support systems in global supply chains.

Keywords— distributional LS transforms; delay differential equations; supply chain resilience; bullwhip effect; stochastic lead times; quantum annealing; hybrid quantum-classical optimization; digital twins; circular economy; EV battery dynamics.