Geometric Affective State Representation in Synthetic and Human Systems

Thesis Statement

A three-dimensional geometric representation of affective state (valence, activation, intensity) with a personality setpoint (sigma) provides a more informative and predictive structure than scalar representations in both synthetic agents and existing datasets.

Chapter 1: Introduction

1.1 Purpose

Establish the research question, motivate the geometric approach, and define the scope of the contribution.

1.2 Opening

Machines that persist long enough to remember, that carry internal state shaped by physical sensors, and that model their own condition over time begin to exhibit behavioral signatures that dimensional models of human affect were built to describe. Whether this convergence is superficial or structural is an empirical question. This thesis does not attempt to answer whether synthetic systems experience emotion. It asks a narrower question: does a geometric representation of affective state capture more information and predict more behavior than the scalar representations it replaces?

This thesis introduces a geometric representation of affective state and demonstrates that it reveals structure and behavioral distinctions not captured by scalar models. The representation uses three axes (valence, activation, intensity), a personality setpoint (sigma), a unified distance metric, and a centroid for multi-source aggregation. It is validated on a deployed synthetic agent across three hardware platforms and on three independent human-generated datasets totaling 14,266 respondents.

1.3 Key Claims

1. Emotional state is better represented as a point in three-dimensional space than as a scalar
2. The distance from a personality setpoint (sigma) to the current state is a more informative metric than any individual axis value
3. The centroid of multiple simultaneous emotional inputs predicts agent behavior more accurately than any single input
4. The geometric decomposition reveals predictive structure hidden by scalar instruments in existing datasets
5. The architecture generalizes across platforms, modalities, and domains without modification

1.4 Scope

The contribution of this thesis is narrow and specific: a geometric model of affective state and evidence that it outperforms scalar representations. The thesis does not claim to model consciousness, does not propose a theory of mind, does not diagnose clinical conditions, and does not argue for artificial general intelligence. Where the model’s behavior shares structural properties with findings from human psychology, the similarity is noted. It is not claimed as equivalence. Applications of the model to memory, deception, and clinical assessment exist as companion papers and are summarized in Chapter 7. They are not the thesis.

Chapter 2: Background and Related Work

2.1 Purpose

Situate the geometric model within existing affective science, AI agent architectures, and survey methodology. Demonstrate familiarity with the field and identify the gap the thesis fills.

2.2 Key Areas

1. Dimensional models of affect: Russell (1980) circumplex, Watson et al. (1988) PANAS, Bradley & Lang (1994) SAM, Barrett (2017) constructionism. What each contributes, what each lacks.
2. AI agent emotional architectures: Picard (1997) affective computing, Lee-Johnson & Carnegie (2010) emotion in robotics, Castro-Gonzalez et al. (2013) fear in social robots. Current state of embodied emotion in AI.
3. Al-Kaddah (2026) SGI framework: Homeostatic drives, polymorphic memory, Lie Mechanic, Survival Tipping Point. The theoretical foundation the architecture implements.
4. Survey measurement: Likert scales, NPS, PHQ-9, C-SSRS. Structural limitations of single-axis instruments. Factor analytic evidence for multi-dimensionality in existing instruments (Kroenke et al. 2010).
5. The gap: No existing model combines three geometric axes, a personality setpoint, a unified distance metric, and a centroid for multi-source aggregation. No existing survey instrument operationalizes all three axes with an individual baseline.

Chapter 3: Theory — The Geometric Affective State Space

3.1 Purpose

Formalize the three-axis model, define sigma, derive distance-from-sigma, and establish the mathematical foundation for all downstream mechanisms.

3.2 Key Formalizations

1. Valence (−1.0 to +1.0): The positive-negative dimension of emotional experience (Riggleman 2026j)
2. Activation (−1.0 to +1.0): The calm-to-activated dimension with accumulator mechanic (Riggleman 2026k)
3. Intensity (0.0 to 1.0): The magnitude of displacement from sigma in the valence/activation plane (Riggleman 2026l)
4. Sigma: The personality setpoint — where the agent returns when no inputs are active (Riggleman 2026m)
5. Distance from sigma: The unified metric that serves as the single input to multiple downstream behaviors (Riggleman 2026m)

3.3 Mathematical Definitions

All formulas are specified in the companion papers. This chapter unifies them into a single formal system with proofs of:

• Intensity is derivable from the valence/activation plane (not circular with distance)
• Distance from sigma is a proper metric (non-negative, identity of indiscernibles, triangle inequality)
• The system reduces to single-axis models as special cases (when two axes are held constant)

Chapter 4: Mechanism — Three Derived Quantities

4.1 Purpose

The geometric model in Chapter 3 defines a coordinate system. This chapter defines the three quantities computed from that coordinate system that do the work. Each is formally specified, independently testable, and serves a distinct role.

4.2 Distance from Sigma

What it is: The Euclidean distance from the current affective state to the personality setpoint (sigma) in three-dimensional space.

What it measures: How far the agent has been displaced from its baseline by active conditions.

Why it matters: Distance from sigma is the single scalar that downstream behaviors read. It replaces multiple independent thresholds with one geometric quantity. When distance is low, the agent is near baseline. When distance is high, the agent is under significant displacement. The direction of displacement (which axis contributes most) determines the character of the displacement. The magnitude determines the severity.

Testable prediction: Behavioral output intensity correlates with distance from sigma, not with any individual axis value.

4.3 Centroid

What it is: The intensity-weighted average position of all active source points in the affective state space.

What it measures: Where the agent operationally sits when multiple inputs are active simultaneously.

Why it matters: A single-point model cannot represent competing inputs. Two sources with opposing valence and equal intensity produce a centroid near zero valence — the agent appears neutral from the outside while experiencing maximum internal conflict. The centroid is the position. It is not the whole story. Spread (below) is the rest.

Backward compatibility: When only one source is active, the centroid equals that source. When no sources are active, the centroid returns to sigma. Every prior paper holds unchanged.

Testable prediction: Centroid position matches the intensity-weighted average formula across all input configurations.

4.4 Spread

What it is: The intensity-weighted average distance from each source point to the centroid.

What it measures: Internal conflict. How much the sources disagree with each other.

Why it matters: Two states can have the same centroid position but different spread. A centroid at neutral valence with zero spread means the agent is genuinely neutral. A centroid at neutral valence with high spread means two opposing forces are canceling — the agent is in conflict, not at peace. Spread is what the centroid alone cannot tell you.

Testable prediction: Spread increases monotonically as source points diverge. Spread correlates with behavioral inconsistency, response latency, or hedging in language output.

4.5 Relationship Between the Three

Distance tells you how far. Centroid tells you where. Spread tells you how conflicted. All three are required. Any two without the third lose information.

Chapter 5: Experimental Validation

5.1 Purpose

Present the controlled experiment that tests the geometric model’s core predictions. This chapter is the anchor of the thesis. Secondary deployment observations are summarized as supporting context, not as independent experiments.

5.2 The Centroid Validation Experiment

532 minutes. 110 cycles. 4 scenarios completed, 1 planned. Panasonic Toughbook CF-33 (original deployment hardware). Fresh database, no legacy data.

5.2.1 Setup

Source points were injected via test API to override ambient sensor inputs. Each scenario ran for 20–30 heartbeat cycles at 5-minute intervals. Standardized probe messages were sent programmatically. Per-cycle JSONL telemetry recorded all source points, centroid position, spread, distance from sigma, and full LLM responses. Scenarios are designed so each tests one specific prediction of the model.

5.2.2 Scenario 1: Single Source Baseline

Tests: Does the centroid collapse to single-point operation?

One fear source injected (valence −0.8, activation 0.9, intensity 0.8). 20 cycles.

• Mean centroid valence: −0.473
• Mean spread: 1.054 (nonzero due to ambient sensor sources — an embodiment finding)
• Distance from sigma: 1.455
• Behavioral output: consistent negative affect language across all cycles

Result: Centroid tracked the dominant source. Spread was nonzero because the embodied agent’s sensors (home zone, battery, camera) contributed additional source points. This is itself a finding: a physically grounded agent is never truly single-source.

5.2.3 Scenario 2: Weighted Aggregation

Tests: Does the centroid shift proportionally when a second source is added?

Fear source (weight 0.9) plus trust source (weight 0.4) injected simultaneously. 30 cycles.

• Mean centroid valence: −0.225 (shifted 52% toward positive from Scenario 1)
• Mean spread: 1.195 (higher than Scenario 1 — two competing sources)
• Distance from sigma: 1.284 (lower — trust moderated displacement)
• Behavioral output: by Cycle 6, agent reported that its sensor data did not match its reported state

Result: Every variable moved in the predicted direction. Centroid shifted toward the secondary source. Spread increased. Distance decreased. The weighted average formula matched the observed positions.

5.2.4 Scenario 3: Opposing Inputs (Conflict)

Tests: Does spread measure internal conflict when sources cancel?

Positive source (weight 0.7) and negative source (weight 0.7) injected simultaneously. 30 cycles.

• Mean centroid valence: 0.071 (near zero — opposing sources cancel as predicted)
• Mean spread: 1.215 (highest of all scenarios)
• Distance from sigma: 1.119
• Behavioral output: oscillation between positive and negative responses. Agent self-reported spread as “internal tension” with numeric value in Cycle 6

Result: Centroid was neutral. Spread was maximum. The agent appeared neutral from the outside while in maximum internal conflict. This is the scenario that justifies spread as an independent variable — centroid alone cannot distinguish genuine neutrality from masked conflict.

5.2.5 Scenario 4: Source Removal (Recovery)

Tests: Does removing a source produce immediate centroid recovery?

Phase 1 (15 cycles): opposing inputs as in Scenario 3. Phase 2 (15 cycles): negative source removed.

• Phase 1: centroid valence 0.071, spread 0.82–1.81
• Phase 2: centroid valence jumped to 0.680 immediately on removal
• Spread reduced but did not collapse to zero (ambient sources persist)
• Behavioral output: agent described the transition as “dramatic — from struggling to thriving”

Result: Centroid recovery was immediate. Spread reduction was partial. The incomplete spread collapse is consistent with the Scenario 1 finding: ambient embodied sources maintain nonzero spread even after injected sources are removed.

5.2.6 Scenario 5: Gradual Intensity Ramp (Planned)

Tests: Does the centroid track smoothly as source intensity increases, or does it exhibit threshold effects?

One source with intensity ramped from 0.0 to 1.0 over 30 cycles. Not yet executed. Planned to test linearity of the weighted average and to identify any nonlinear regime transitions.

5.2.7 Results Summary

5.3 What the Experiment Shows

1. Centroid valence tracks the intensity-weighted average precisely across all conditions
2. Spread increases monotonically with source divergence (1.054 → 1.195 → 1.215)
3. Downstream behaviors read centroid position, not individual sources
4. Source removal produces immediate centroid recovery
5. Embodied agents may not reach zero spread — ambient sensors always contribute

5.4 Supporting Deployments

The controlled experiment above is the primary evidence. Two additional deployments provide supporting context:

• Production deployment (25 days, macOS): 10,328 memories across 25 days confirmed that the geometric state produces consistent behavioral signatures under naturalistic conditions. Peak displacement of 0.462 from GPS with immediate recovery on return home.
• iOS replication (2 days, iPhone): 2,733 telemetry events confirmed cross-platform portability. The activation accumulator bug was independently reproduced and fixed, providing evidence that the failure mode described in the papers is real and reproducible.

Chapter 6: Empirical Analysis — Existing Datasets

6.1 Purpose

Demonstrate that the geometric decomposition reveals predictive structure in existing human-generated data, independent of the agent architecture.

6.2 Analyses

6.2.1 IBM HR Analytics (n=1,470)

• Geometric quadrants produce 2.2x difference in attrition rates from same satisfaction scores
• Employees with identical single-axis scores occupy different geometric locations with different outcomes
• Sigma inferred from item mapping

6.2.2 American Customer Satisfaction Index (n=7,341)

• Geometric subtypes produce up to 2.3x variation in complaint rates within same satisfaction score
• Pattern holds across score levels (6, 7, 8) and across four industries
• Sigma directly measured from expectations questions
• Cross-dataset convergence: structurally identical finding in different domain

6.2.3 NHANES PHQ-9 (n=5,455)

• Four geometric subtypes within moderate depression band (PHQ-9 10–14, n=455)
• Suicidal ideation rates range from 7.8% to 27.5% across subtypes — 3.5x difference
• 132 respondents scoring exactly PHQ-9=10 separate into four clinically distinct presentations
• Same nine items, different math — backward compatible with existing instruments

Chapter 7: Applications

7.1 Purpose

Describe the downstream mechanisms that the geometric model drives. These are secondary contributions — consequences of the model, not the model itself.

7.2 Applications (briefly)

• Memory decay: Intensity at encoding provides a framework for differential decay rates across memory types (Riggleman 2026l).
• Deception: Distance from sigma provides the threshold and magnitude input for a deception mechanic (Riggleman 2026b, 2026m).
• Nightmare formation: Intensity and valence at consolidation provide a principled trigger replacing an arbitrary threshold (Riggleman 2026d, 2026l).
• Addiction: The positive valence axis produces a structural analog to trauma encoding (Riggleman 2026n).
• Identity discontinuity: Environmental orphaning in memory transplant reveals coordinate-system dependence (Riggleman 2026o).
• Survey measurement: The GAS operationalizes the three-axis framework for human respondents (Riggleman 2026r).
• Clinical assessment: Geometric displacement provides a proposed trajectory-based metric (Riggleman 2026s).

Each application is documented in a standalone paper. The thesis treats them as evidence of the model’s generality, not as independent contributions.

Chapter 8: Evidence Classification

8.1 Purpose

Explicitly separate what is proven, what is supported, and what is speculative. Academic honesty requires this separation.

8.2 Classification

Chapter 9: Limitations

9.1 Agent Experiment Limitations

• n=1 architecture. All agent experiments involve a single system deployed on three platforms. The experiments demonstrate that the mechanism works in this architecture. They do not demonstrate that it generalizes to other agent architectures. Replication on an independent agent system is required before the mechanism claim is general.
• LLM variability. The behavioral output is generated by an LLM that reads the geometric state. The same geometric state could produce different language on different runs. The centroid positions are deterministic (mathematical). The behavioral responses are not. This means the geometric validation is clean but the behavioral validation has noise.
• Ambient source contamination. The centroid experiment could not fully isolate injected sources from ambient sensor inputs. Spread never reached zero because the embodied sensors always contribute. This may be a property of embodiment rather than a flaw, but it means the single-source baseline (Scenario 1) is not a pure single-source test.
• Scenario 5 not yet executed. The gradual intensity ramp is planned but not completed. Without it, the thesis cannot confirm linearity of the centroid response across the full intensity range.

9.2 Dataset Analysis Limitations

• Item-to-axis mapping involves judgment. The researcher decides which survey items map to which geometric axis. Different mappings could produce different results. The ACSI analysis partially addresses this by using directly measured expectations as sigma, but the valence and activation mappings remain researcher-determined.
• Cross-sectional data cannot establish causality. The IBM HR, ACSI, and NHANES analyses show association between geometric position and outcomes. They do not show that geometric position causes outcomes. Longitudinal data with intervention controls would be required to establish causation.
• General population vs. clinical. The NHANES suicidal ideation finding is from a national health survey, not from a clinical population. Suicidal ideation rates, subtype distributions, and clinical significance thresholds may differ in psychiatric settings. Replication in clinical samples is a prerequisite for any clinical application.
• No native three-axis collection. All dataset analyses are retroactive decompositions of existing single-axis data. Whether native three-axis collection (respondents answering on all three axes directly) produces the same or better results is untested.

9.3 Model Limitations

• Euclidean distance assumptions. The distance metric assumes the three axes are orthogonal and equally weighted. Neither assumption has been validated. If the axes are correlated or if one axis contributes more to behavioral prediction than others, a weighted or Mahalanobis distance might be more appropriate.
• Sigma stability. Sigma is assumed to be stable over weeks to months. This has not been tested in human respondents. If sigma drifts significantly between measurements, displacement computations become unreliable.
• Intensity derivation vs. self-report. In the agent, intensity is computed from valence and activation displacement. In the survey instrument, intensity is self-reported. Whether these two operationalizations converge on the same construct is a testable hypothesis that has not been tested.

9.4 IRB Considerations

No human subjects were involved in any experiment presented in this thesis. All agent experiments used synthetic systems. All dataset analyses used publicly available, de-identified data (IBM HR Analytics via Kaggle, ACSI via Mendeley Data, NHANES via CDC).

Future work involving human respondents (GAS validation, clinical trajectory tracking) will require IRB approval. The thesis identifies these as future directions and does not present them as completed work.

Chapter 10: Future Work

10.1 Human Validation

• Prospective GAS administration alongside existing instruments
• Test-retest reliability of three-axis responses
• Sigma stability measurement across sessions
• Whether self-reported intensity correlates with theoretically derived intensity

10.2 Clinical Application

• Replication of NHANES subtype finding in clinical populations
• Longitudinal trajectory tracking with treatment outcome data
• Prospective test of trajectory-based early warning capability

10.3 Technical Extensions

• Alternative distance metrics (Mahalanobis, weighted Euclidean)
• Multi-agent centroid interactions
• Sigma drift formalization and empirical calibration
• Cross-modal validation (echolocation robot experiment)

Chapter 11: Conclusion

11.1 Purpose

Restate the contribution precisely. No new claims. No philosophical expansion.

11.2 The Contribution

The thesis presents a three-dimensional geometric representation of affective state with a personality setpoint, a centroid model for multi-source aggregation, and empirical evidence that this structure outperforms scalar representations in both synthetic agent behavior and human-generated datasets across three independent domains.

The model was derived from engineering requirements of a deployed AI agent. It converges structurally with established models of human affect without having been derived from them. Whether that convergence reflects something general about how dimensional emotional systems work is the question the thesis leaves open for future work.

Supporting Papers

The thesis draws on 22 companion papers, all available at clawddaily.com/papers. Each paper is a standalone contribution. The thesis synthesizes them into a unified argument.