The Geometry of Project Dynamics: A Frenet–Serret Trajectory Framework for Monitoring Performance and Systemic Complexity

Project monitoring in complex environments is still largely dominated by stage-based Key Performance Indicators (KPIs) and static complexity metrics, which limit managers’ ability to interpret structural changes unfolding between reporting milestones. This study addresses this limitation by conceptualizing projects not merely as sequences of states, but as trajectories evolving through a joint performance–complexity space. The paper introduces a geometry-based and model-agnostic framework that represents project evolution as a continuous trajectory constructed from discrete KPI and Systemic Complexity Index (SCI) observations. Using the Frenet–Serret formalism, curvature (κ) and torsion (τ) are derived as intrinsic descriptors of project motion. These indicators are obtained by interpolating stage-based monitoring data into a differentiable trajectory and are evaluated at managerial decision points. The framework is illustrated using real project monitoring data. The results show that joint κ–τ patterns provide a compact and interpretable description of project regime transitions, structural instability, and stabilization phases. In the empirical case, peaks in curvature and torsion coincide with periods of strong trade-off conflict and systemic decoupling, while their subsequent reduction reflects restored alignment and governability. Rather than proposing a new performance metric or predictive control tool, this study reframes project monitoring as the observation of project motion through a joint performance–complexity space. The SCI–Frenet framework offers a new conceptual and methodological lens that complements existing project control approaches by supporting managerial sensemaking under complexity. This manuscript is a preprint and has been submitted for peer review. The purpose of this preprint is to establish the date of public disclosure of the proposed framework and to document the originality of applying Frenet–Serret geometry to the field of project management.