Spacetime Imprint and Emergent Time: Observational and Quantum Tests of Spacetime Memory

We present Spacetime Imprint and Emergent Time (SIET), a unified theoretical and observational framework in which spacetime possesses a physical memory of past dynamical events. In SIET, gravitational dynamics depend not only on the instantaneous baryonic mass distribution, but also on a spacetime memory field encoding the historical evolution of matter and interactions. Time is interpreted as an emergent parameter associated with the local rate of memory change, providing a natural bridge between classical and quantum regimes. We develop the full mathematical structure of the theory, define its weak-field and cosmological limits, and formulate explicit, falsifiable observational predictions. Using publicly available SPARC galaxy rotation curve data, we introduce and test a gravitational hysteresis effect, demonstrating statistically significant correlations between dynamical anomalies and proxies of interaction history. The framework further predicts measurable consequences in gravitational lensing, gravitational-wave memory, atomic clocks, quantum decoherence, and interferometric experiments. All symbols are consistently defined and rendered to ensure cross-platform reproducibility. SIET provides a testable alternative to dark matter interpretations by replacing unseen substances with historical degrees of freedom of spacetime itself.