Directional Information in Temporally Aggregated Hawkes Processes

Timestamp aggregation removes event order but can preserve directional information in event counts. We characterize this distinction in a stationary, mean-matched, feed-forward Hawkes family observed through independent windows. For multiple bins, we derive the exact first-order directional score and its finite-window information. For kernels with finite mean lag ${\mu }_h$, information per unit exposure decays as $2{\mu }_h^2/{\Delta }^3$ when the long-window limit precedes the coarse-resolution limit. With one bin, the first-order score vanishes, but a nonzero second-order tangent combines excess dispersion with a mixed third-degree count moment. We derive the exact Gaussian testing limit at excitation strength $a_N=\gamma N^{-1/4}$ and a statistic attaining it. The mixed term supplies a calculable fraction of local information absent from dispersion alone. Both tests retain local efficiency after estimating the common rate. We characterize which count summaries preserve the tangents and which discard their distinctions. Paired count-law experiments quantify the mixed term's contribution and show slow finite-sample convergence. Repeated calibration experiments assess the grid-specific assertion control given by test inversion. Misspecification experiments show substantial failures outside that grid. The results separate temporal ordering information, higher-order count information, and the scope of calibrated reporting.