SAIM Framework: Kibble-Zurek Mechanism & Quantum Memory Hysteresis Analysis

This report presents a comprehensive dashboard analysis of the Non-Markovian Kibble-Zurek Mechanism (KZM) in quantum systems. We explore the interplay between the freeze-out time (tau_q), the non-local memory strength (K-Kernel), and the Euler topological phase (Theta). The simulations demonstrate three distinct phenomena: (1) Control (Markovian limit), where a sharp defect peak appears at the critical point; (2) Hysteresis (memory-induced time-lag), where the defect peak shifts to the right and broadens due to critical slowing down; and (3) Phase-Controlled Splitting, where tuning the Euler phase near Pi acts as a quantum protection mechanism, splitting the defect peak and minimizing errors. This provides a mathematically rigorous strategy for enhancing coherence in near-term quantum processors.