Differential Geometric Signatures of Persistent Structural Transitions in Solar Wind Dynamic Pressure via Bézier Representations

Traditional time-domain signal analysis typically characterizes a signal directly in its original domain, potentially overlooking structural information encoded in its geometry. This study proposes a geometric-algebraic framework for characterizing transient and persistent transitions in solar wind dynamic pressure. The dynamic pressure is a fundamental parameter describing the momentum flux of the solar wind and its mechanical forcing on the terrestrial magnetosphere. The analysis begins by representing the time-domain observations as points (ti, Pi), i = 0, 1, 2, … followed by segmentation and parametric Bézier curve representation using quadratic, cubic, and quartic polynomial forms. Differential geometric and algebraic quantities derived from these representations are combined into a multidimensional signature. Rather than assigning anomalous labels directly to individual observations, empirical reference structures are constructed from normal and anomalous signal segments, while the investigated observations remain blind during reference construction. Their relative distances to the normal and anomalous reference manifolds are then evaluated using a degree-consensus strategy to assess the persistence of directional transitions across polynomial representations. The final rolling sequence exhibits a coherent transition from consecutive normal-directed states to consecutive anomalous-directed states, with the final window showing positive consensus across all investigated polynomial degrees. The resulting signature therefore characterizes the transition as a persistent geometric movement toward the anomalous reference structure rather than merely as a pointwise amplitude anomaly. The framework provides an alternative geometric-algebraic representation of solar wind dynamic pressure variability and establishes a basis for its application to related solar wind and space weather parameters.