A mathematical model of temporal dynamics in adaptive economic systems based on sheaf theory

Coordinating distributed agents in digital platforms and ecosystems remains an open theoretical problem: existing models — from general equilibrium to system dynamics — provide no rigorous criterion for determining under what conditions locally consistent agent strategies cohere into a unified global trajectory and when they do not. Traditional platform theory, in turn, focuses on the structure of market interactions but does not offer tools for diagnosing the temporal gaps that accumulate as agents adapt to a changing environment at different rates. This paper addresses both deficiencies by developing a mathematical model of temporal consistency and inconsistency based on the apparatus of sheaf theory. An adaptive economic system is modeled as a topological space in which open sets correspond to local markets or agent groups, and a sheaf assigns feasible strategic configurations to each point of the system. The first cohomology group serves as a quantitative measure of coordination gaps, while the rate of their resolution is linked to the spectral properties of the sheaf Laplacian. A key methodological contribution is the use of the orthogonal Procrustes method to calibrate restriction maps directly from observed agent data, translating the theoretical construct into a practically applicable diagnostic tool. The model is validated on operational data from a real digital platform covering the second half of 2025: 26 weekly observations across three agents — a supplier, a platform operator, and a logistics partner. Diagnostics reveal four distinct phases of the misalignment life cycle; angular misalignment measures between agents (ranging from 10.9° to 42.6°) are interpreted as cognitive gaps — a quantitative indication of how differently participants construct a shared strategic space even under formally aligned objectives. Spectral analysis of the sheaf Laplacian demonstrates that, given the current interaction architecture, global agent coordination is topologically unattainable (H0 (G; F) = 0), with a characteristic recovery horizon of approximately 320 weeks; a threefold increase in misalignment energy under an exogenous shock confirms the diagnostic sensitivity of the model. The scientific contribution lies in a mathematical model that combines sheaf-cohomological diagnostics of temporal gaps with empirical calibration of restriction maps using the Procrustes method. The findings provide a foundation for developing monitoring and adaptive governance systems for digital platforms and multi-agent economic ecosystems.