Coordinate Heart System: A Geometric Framework for Emotion Representation

Abstract: This paper presents the Coordinate Heart System (CHS), a geometric framework for emotion representation in artificial intelligence applications. We position eight core emotions as coordinates on a unit circle, enabling mathematical computation of complex emotional states through coordinate mixing and vector operations. Our initial five-emotion model revealed significant coverage gaps in the emotion space, leading to the development of an eight-emotion system that provides complete geometric coverage with mathematical guarantees. The framework converts natural language input to emotion coordinates and supports real-time emotion interpolation through computational algorithms. The system introduces a re-calibrated stability parameter S in [0,1], which dynamically integrates emotional load, conflict resolution, and contextual drain factors. This stability model leverages advanced LLM interpretation of textual cues and incorporates hybrid temporal tracking mechanisms to provide nuanced assessment of psychological well-being states. Our key contributions include: (i) mathematical proof demonstrating why five emotions are insufficient for complete geometric coverage, (ii) an eight-coordinate system that eliminates representational blind spots, (iii) novel algorithms for emotion mixing, conflict resolution, and distance calculation in emotion space, and (iv) a comprehensive computational framework for AI emotion recognition with enhanced multi-dimensional stability modeling. Experimental validation through case studies demonstrates the system's capability to handle emotionally conflicted states, contextual distress factors, and complex psychological scenarios that traditional categorical emotion models cannot adequately represent. This work establishes a new mathematical foundation for emotion modeling in artificial intelligence systems.