Coordinate Heart System (CHS) Overview

• Coordinate Heart System (CHS) is a geometric framework that maps emotions as coordinates on a unit circle, enabling precise vector-based emotion modeling.
• It incorporates algorithms for linear interpolation, conflict resolution, and stability tracking, improving upon traditional categorical models.
• CHS guarantees full geometric coverage through convex combinations of eight base emotions, supporting real-time monitoring and effective AI applications.

The Coordinate Heart System (CHS) is a geometric, mathematically rigorous framework for emotion representation tailored to AI applications. By embedding eight foundational emotions as coordinates on the unit circle—seven peripheral and one central baseline—the CHS enables precise vectorial computation, interpolation, and combination of emotional states. The system resolves limitations in prior categorical models by providing complete geometric coverage, introducing algorithms for emotion mixing and conflict resolution, and incorporating a real-time, multidimensional stability parameter to model psychological well-being. This geometric construct, which leverages LLMs for interpretation of natural language emotional content, establishes a new mathematical basis for AI-driven emotion modeling (Al-Desi, 19 Jul 2025).

1. Geometric Construction: Eight-Coordinate Unit Circle Embedding

The CHS employs seven “peripheral” core emotions positioned on the unit circle, together with a “baseline” emotion (Love) at the origin. This arrangement facilitates direct mathematical operations on emotion states via planar coordinates.

For any peripheral emotion at angle $\theta$ on the unit circle, the coordinate assignment is:

$x = \cos(\theta),\quad y = \sin(\theta)$

The angular assignments and resulting explicit coordinates are as follows:

Emotion	Coordinates	Angle $\theta$
Love	(0.0, 0.0)	(origin)
Guilt	(1.0, 0.0)	$0^\circ$
Joy	(0.0, –1.0)	$270^\circ$
Fear	(0.5, –0.866)	$300^\circ$
Sadness	(0.866, –0.5)	$330^\circ$
Disgust	(–0.5, –0.866)	$240^\circ$
Pride	(–1.0, 0.0)	$180^\circ$
Anger	(0.0, 1.0)	$90^\circ$

This embedding enables each complex or blended emotional state to be represented as a convex combination of these anchor coordinates, making the system robust to linear operations and geometric interpolation (Al-Desi, 19 Jul 2025).

2. Theoretical Justification and Coverage Guarantees

The CHS addresses the critical deficiency of insufficient geometric coverage in prior five-emotion models. The Five-Emotion Gap Lemma demonstrates that when five peripheral points are placed at $x = \cos(\theta),\quad y = \sin(\theta)$0 intervals, the resulting angular gap leaves “blind spots”: any angle more than $x = \cos(\theta),\quad y = \sin(\theta)$1 from the nearest base emotion cannot be represented exactly by convex combinations of the anchors.

The Eight-Emotion Coverage Theorem formally establishes that the eight-point base set—including Love at the origin and the other seven on the unit circle—covers the entire unit disk. Any point $x = \cos(\theta),\quad y = \sin(\theta)$2 with $x = \cos(\theta),\quad y = \sin(\theta)$3 can be expressed as a convex combination of the eight base coordinates:

• Along a ray from the origin (Love) to a base emotion $x = \cos(\theta),\quad y = \sin(\theta)$4, all intermediate points are $x = \cos(\theta),\quad y = \sin(\theta)$5, with $x = \cos(\theta),\quad y = \sin(\theta)$6.
• For directions between adjacent base emotions, any target can be written as $x = \cos(\theta),\quad y = \sin(\theta)$7, where $x = \cos(\theta),\quad y = \sin(\theta)$8 and $x = \cos(\theta),\quad y = \sin(\theta)$9.

This explicit construction provides a mathematically guaranteed, blind-spot-free representational space for emotional states in the unit disk (Al-Desi, 19 Jul 2025).

3. Vector Operations: Mixing, Conflict Resolution, and Distances

CHS formalizes vector space operations on emotions:

• Two-Emotion Linear Interpolation: Given emotions $\theta$0 and $\theta$1 with intensities $\theta$2:

$\theta$3

• Multi-Emotion Mixing: Emotions $\theta$4 and intensities $\theta$5 are sequentially mixed using pairwise interpolation, yielding a single composite point in $\theta$6 time.
• Conflict Resolution: Predefined opposing pairs (Joy$\theta$7Anger, Guilt$\theta$8Pride) are resolved by subtracting the minimum intensity from both, discarding the lesser, and incrementing a penalty term $\theta$9:

$0^\circ$0

• Distance Measures: Both standard Euclidean and angular distances are defined within the emotion plane:

$0^\circ$1

$0^\circ$2

These operations enable quantification of emotional blends, degree of conflict, and transitions in affective state (Al-Desi, 19 Jul 2025).

4. Stability Modeling and Temporal Dynamics

The CHS introduces a normalized stability parameter $0^\circ$3, integrating emotional, conflict, and contextual drains:

$0^\circ$4

• $0^\circ$5
• $0^\circ$6
• $0^\circ$7 is derived from contextual analysis by NLP (e.g., burnout markers)

$0^\circ$8 is typically set near $0^\circ$9 (configurable). If $270^\circ$0, the system enters crisis mode, signaling psychological destabilization or overload.

Temporal stability is tracked by a hybrid mechanism: For successive states $270^\circ$1, $270^\circ$2:

$270^\circ$3

$270^\circ$4

This mechanism maintains nuance in psychological well-being assessment over time, responsive to both rapid shifts and slow drifts in affective state (Al-Desi, 19 Jul 2025).

5. Computational Framework and Algorithmic Pipeline

The operational CHS pipeline consists of the following stages:

1. Natural Language Processing: An LLM (e.g., Gemini 1.5 Flash) processes raw user text, extracting an emotion intensity dictionary $270^\circ$5 and a contextual drain value.
2. Intensity Mapping: Linguistic cues are mapped into normalized intensities $270^\circ$6.
3. Conflict Resolution and Mixing: The conflict algorithm is applied, and final (x, y) coordinates are computed via sequential mixing.
4. Stability Computation: $270^\circ$7, $270^\circ$8, and $270^\circ$9 are used to calculate $300^\circ$0.
5. Hybrid Temporal Tracking: Smoothed coordinates and $300^\circ$1 are updated in real time.
6. Real-Time Interpolation: All core steps are implemented in $300^\circ$2 or $300^\circ$3 time, supporting responsive, interactive applications.

This computational design enables real-time emotion recognition, blending, and monitoring, adaptive to linguistically complex and psychologically conflicted states (Al-Desi, 19 Jul 2025).

6. Empirical Validation and Applications

CHS’s efficacy is demonstrated via case-based experimental validation:

• Conflict Resolution: Given the input “I’m thrilled but guilty...”—Joy $300^\circ$4, Guilt $300^\circ$5—the system applies conflict subtraction ($300^\circ$6), yielding resolved Joy $300^\circ$7, Guilt $300^\circ$8, $300^\circ$9. The stability parameter $330^\circ$0 drops sharply, signaling psychological tension.
• Contextual Drain Modeling: For input “I feel nothing, I’m exhausted,” all base emotion intensities are zero but contextual drain $330^\circ$1 produces $330^\circ$2, reflecting burnout.
• Integrated Stress: Intensities for Fear $330^\circ$3, Sadness $330^\circ$4, Joy $330^\circ$5, Pride $330^\circ$6, with $330^\circ$7, yield final $330^\circ$8 and coordinates $330^\circ$9, correctly flagging a crisis state.

These results establish that CHS fills the representational gaps of prior categorical models, quantifies emotional conflict, integrates affectively "silent" distress, and enables real-time multidimensional monitoring of psychological stability (Al-Desi, 19 Jul 2025).

7. Relationship to Anatomical Coordinate Systems in Cardiac Modeling

While CHS addresses geometric mapping of emotion space, there exists a parallel in anatomical coordinate frameworks used in cardiology, notably in the Cobiveco biventricular system (Schuler et al., 2021). Cobiveco provides continuous, bijective, normalized coordinates for localizing cardiac structures, using Laplacian and trajectory-based normalization for geometric consistency across patient-specific heart meshes. Both systems exemplify the use of convex geometric representations and coordinate interpolation to eliminate domain-specific “blind spots,” guarantee full coverage, and support transferable, physically-meaningful computations.

In summary, the Coordinate Heart System constitutes a novel, mathematically grounded approach to emotion modeling in artificial intelligence, providing a rigorous geometric basis for parsing, combining, and tracking emotional states in both static and temporally-varying contexts.