This paper introduces the C³ Model (Context Capture, Coherence, and Continuity), a cognitive architecture that enables AI systems to maintain persistent identity, emotional coherence, and relational continuity across interactions. Implemented as the cognitive subsystem of the Presence Engine™, the C³ Model addresses architectural amnesia in current AI systems through stateful context awareness spanning emotional, linguistic, and temporal domains. The architecture operates through a five-layer recursive cognition loop (Perceptual Input → State Construction → Integrative Awareness → Behavioral Adaptation → Meta-Learning), distinguishing between memorized retrieval and causal inference via loss curvature decomposition. This enables relational pattern understanding rather than simple fact recall. Key technical contributions include: (1) hybrid dispositional modeling using OCEAN/HEXACO personality frameworks with identity regulation capabilities, (2) empirically calibrated context decay functions across emotional (12-48 hour), linguistic (5-14 day), and temporal (30-90 day) domains, (3) real-time conflict resolution between contextual signals, and (4) privacy-first local processing with encrypted vector storage. The system has been validated through academic collaboration with Dr. Michael Hogan (University of Galway) and implemented in Python using Anthropic Claude 3 Haiku, with a 100-user pilot study planned for Q4 2025. This work establishes prior art for continuity architecture as critical AI infrastructure. Keywords: continuity architecture, stateful AI, context awareness, emotional intelligence, C³ Model, Presence Engine, OCEAN personality modeling, HEXACO framework, persistent identity, behavioral coherence, recursive cognition, affective computing, privacy-first AI, conversational continuity, dispositional modeling, identity regulation