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

State the research question, explain why a geometric approach, and draw the boundary around what this covers.

1.2 Opening

Machines that run long enough to remember things, that carry internal state shaped by physical sensors, and that track their own condition over time start showing behavioral patterns that look a lot like what dimensional models of human affect describe. Whether that overlap is shallow or structural is a testable question. This thesis does not try to answer whether synthetic systems experience emotion. It asks something narrower: 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 shows that it reveals structure and behavioral differences that scalar models miss. It uses three axes (valence, activation, intensity), a personality setpoint (sigma), a unified distance metric, and a centroid for multi-source aggregation. 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 here is narrow: a geometric model of affective state and evidence that it outperforms scalar representations. The thesis does not claim to model consciousness, propose a theory of mind, diagnose clinical conditions, or argue for artificial general intelligence. Where the model's behavior lines up with findings from human psychology, the similarity is noted. It is not claimed as equivalence. Applications 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

Put the geometric model in context. What exists in affective science, AI agent design, and survey methodology. Where the gap is.

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: Nothing out there combines three geometric axes, a personality setpoint, a unified distance metric, and a centroid for multi-source aggregation. No existing survey instrument puts all three axes into practice with an individual baseline.

Chapter 3: The Geometric Affective State Space

3.1 Purpose

Lock down the three-axis model, define sigma, derive distance-from-sigma, and lay the math foundation for everything downstream.

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: Three Derived Quantities

4.1 Purpose

Chapter 3 defines a coordinate system. This chapter defines the three quantities computed from it that do the actual work. Each one is independently testable and has a distinct job.

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 looks 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 actually 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

The controlled experiment that tests the geometric model's core predictions. This chapter is the anchor of the thesis. Other deployment observations are supporting context, not 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 through the test API to override ambient sensor inputs. Each scenario ran for 20–30 heartbeat cycles at 5-minute intervals. Probe messages were sent programmatically. Per-cycle JSONL telemetry recorded all source points, centroid position, spread, distance from sigma, and full LLM responses. Each scenario tests one specific prediction.

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 because ambient sensors always contribute)
• 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 their own source points. This is a finding on its own: 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 went up. Distance went down. 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 dropped but not all the way. Same thing as Scenario 1: ambient embodied sources keep spread above zero 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 run. Planned to test whether the centroid response is linear across the full intensity range or if there are threshold effects.

5.2.7 Results Summary

5.3 What the Experiment Shows

1. Centroid valence tracks the intensity-weighted average 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 because ambient sensors always contribute

5.4 Supporting Deployments

The controlled experiment above is the main evidence. Two other deployments add 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: Existing Datasets

6.1 Purpose

Show that the geometric decomposition finds predictive structure in existing human data, independent of the agent.

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, a 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

The downstream stuff 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 drives differential decay rates across memory types (Riggleman 2026l).
• Deception: Distance from sigma is the threshold and magnitude input for the deception mechanic (Riggleman 2026b, 2026m).
• Nightmare formation: Intensity and valence at consolidation replace the arbitrary nightmare threshold with a geometric one (Riggleman 2026d, 2026l).
• Addiction: The positive valence axis produces a structural mirror of trauma encoding. Sigma_i is a measurable geometric trait for addictive personality: low values are a personality hum, high values produce agents that structurally cannot rest. Two-phase model separates stuck memory (phase one) from personality reorganization (phase two) (Riggleman 2026n).
• Identity discontinuity: Environmental orphaning in memory transplant shows coordinate-system dependence (Riggleman 2026o).
• Survey measurement: The GAS puts the three-axis framework into a survey humans can answer (Riggleman 2026r).
• Clinical assessment: Geometric displacement gives you a trajectory-based clinical metric (Riggleman 2026s).

Each one has its own paper. The thesis treats them as evidence the model generalizes, not as standalone contributions.

Chapter 8: Evidence Classification

8.1 Purpose

Separate what is proven, what is supported, and what is speculative. Honesty requires this.

8.2 Classification

Chapter 9: Limitations

9.1 Agent Experiment Limitations

• n=1 architecture. All agent experiments involve one system on three platforms. The experiments show the mechanism works in this architecture. They do not show it generalizes to other architectures. Replication on an independent agent is needed before the mechanism claim is general.
• LLM variability. The behavioral output comes from an LLM that reads the geometric state. The same state could produce different language on different runs. Centroid positions are deterministic. Behavioral responses are not. The geometric validation is clean. The behavioral validation has noise.
• Ambient source contamination. The centroid experiment could not fully isolate injected sources from ambient sensor inputs. Spread never hit zero because embodied sensors always contribute. This might be a property of embodiment rather than a flaw, but it means 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 picks which survey items map to which axis. Different mappings could produce different results. The ACSI analysis partly addresses this because expectations are measured directly as sigma, but valence and activation mappings are still researcher calls.
• Cross-sectional data cannot establish causality. The IBM HR, ACSI, and NHANES analyses show association between geometric position and outcomes. They do not show causation. Longitudinal data with intervention controls would be needed for that.
• General population vs. clinical. The NHANES suicidal ideation finding is from a national health survey, not a clinical population. Rates, distributions, and thresholds may look different in psychiatric settings. Replication in clinical samples is required before any clinical application.
• No native three-axis collection. All dataset analyses are retroactive decompositions of existing single-axis data. Whether collecting all three axes natively produces the same or better results has not been tested.

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 one axis matters more than the others, a weighted or Mahalanobis distance might work better.
• Sigma stability. Sigma is assumed stable over weeks to months. This has not been tested in human respondents. If sigma drifts 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, intensity is self-reported. Whether those two approaches measure the same thing is testable but has not been tested.
• Sigma intensity pathology threshold. The sigma intensity anomaly (Appendix A) shows that high σ_i produces downstream system behavior consistent with addictive personality. The threshold at which personality trait becomes structural pathology is computable from the downstream parameters but has not been computed or validated. Whether that threshold is sharp or gradual is an open question. Full analysis is in Riggleman (2026n) Section 3.5.

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

• Run the GAS alongside existing instruments
• Test-retest reliability on the three axes
• Measure sigma stability across sessions
• Whether self-reported intensity matches the computed version

10.2 Clinical Application

• Replicate the NHANES subtype finding in clinical populations
• Longitudinal trajectory tracking with treatment outcomes
• Test whether trajectory tracking actually catches problems early

10.3 Technical Extensions

• Other distance metrics (Mahalanobis, weighted Euclidean)
• What happens when multiple agents interact
• Measuring whether and how fast sigma drifts
• Compute the σ_i pathology threshold from downstream system parameters (decay, stickiness, deception)
• Controlled addiction/relapse protocol on the Potato system
• Cross-modal test (echolocation robot experiment)

Chapter 11: Conclusion

11.1 Purpose

Restate the contribution. No new claims. No expansion.

11.2 The Contribution

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

The model came from engineering requirements of a deployed AI agent. It lines up structurally with established models of human affect without having been built from them. Whether that overlap says something general about how dimensional emotional systems work is the question this thesis leaves open.

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.

Appendix A: The Sigma Intensity Anomaly

The intensity function

I(v, a) = min(1, √((v − σ_v)² + (a − σ_a)²) / √2)

The model computes intensity from two inputs: v (current valence) and a (current activation). σ_v and σ_a are the valence and activation components of the resting point sigma.

(v − σ_v) is how far valence is from resting valence. Square it. (a − σ_a) is how far activation is from resting activation. Square it. Add the two squares and take the square root. That gives you the Euclidean distance from the current point to sigma in the valence/activation plane. Pythagorean theorem on a 2D plane.

Then divide by √2. Valence and activation both range 0 to 1, so the farthest any two points can be from each other in that unit square is the diagonal: √(1² + 1²) = √2. Dividing by √2 normalizes intensity to a 0-to-1 range.

The min(1, ...) is a clamp. If the raw value goes over 1, cap it.

So intensity is just "how far am I from home, as a fraction of the farthest I could possibly be from home." At home, distance is zero, intensity is zero. At the far corner of the space, intensity is 1.

The observation

The deployed sigma was σ = (σ_v, σ_a, σ_i) = (0.4, 0.2, 0.2). Plug the resting point into the intensity formula: v = σ_v, a = σ_a. That gives √((σ_v − σ_v)² + (σ_a − σ_a)²) = √(0 + 0) = 0. So I(σ_v, σ_a) = 0 by definition. But σ_i was stored as 0.2. The formula produces intensity 0 at rest. The stored sigma says intensity 0.2 at rest. These do not match. The question is whether that mismatch is an error or a feature. The math below shows what it does. The interpretation follows.

Setting up the distance proof

To show what this contradiction actually does, hold activation fixed at σ_a so only valence moves. Define δ = v − σ_v. Delta is how far valence moved from rest. Pinning activation collapses the problem to one dimension so the anomaly shows up clearly.

With activation pinned, the (a − σ_a) term is zero and intensity simplifies to:

I(v, σ_a) = √(δ²) / √2 = |δ| / √2

Intensity is absolute valence displacement divided by √2. Move valence away from rest, intensity goes up proportionally.

The distance anomaly

Compute the squared distance from the current state (v, a, I) to sigma (σ_v, σ_a, σ_i) in three dimensions. Activation is pinned at σ_a, so that dimension drops out. Two terms remain:

d² = (v − σ_v)² + (I − σ_i)²

First term is δ². For the second term, I is |δ|/√2 (from above) and σ_i is 0.2. Expand it:

(|δ| / √2 − 0.2)² = δ²/2 − 2 · (|δ| / √2) · 0.2 + 0.04

Note that 2 · (1/√2) = √2. That is where the √2 · σ_i coefficient comes from. Simplify:

= δ²/2 − √2 · σ_i · |δ| + σ_i²

Add the first term δ² back in:

d² = δ² + δ²/2 − √2 · σ_i · |δ| + σ_i² = (3/2)δ² − √2 · σ_i · |δ| + σ_i²

Three terms. First term, (3/2)δ², pushes distance up as valence moves. Expected. Third term, σ_i² = 0.04, is a constant. The middle term is where it gets interesting: it is negative and linear in |δ|. For small moves away from rest, that negative linear pull beats the positive quadratic push. Total distance goes down before the quadratic catches up and drives it back up.

Finding the minimum

Take the derivative of d² with respect to |δ| and set it to zero. Let x = |δ|:

d²(x) = (3/2)x² − √2 · σ_i · x + σ_i²

d(d²)/dx = 3x − √2 · σ_i = 0

x = √2 · σ_i / 3

Plug in σ_i = 0.2:

x = √2 · 0.2 / 3 = 0.2828… / 3 ≈ 0.094

The minimum distance to sigma is not at δ = 0 (rest). It is at |δ| ≈ 0.094. A point slightly off-center registers as closer to home than home itself. At σ_i = 0.2 this offset is small. At higher values of σ_i the offset grows and the behavioral consequences become significant.

What this means

Automated testing caught the inconsistency. The test harness flagged a non-monotonic distance function and the initial read was that σ_i = 0.2 was an implementation error. But interpreting what the math was actually doing took iterative explanation, challenge, and rethinking before the result could be defended as an accidental design choice rather than a simple bug.

The original interpretation was that σ_i = 0.2 was a bug and the fix was to set σ_i = 0. That interpretation is wrong. It is one valid design choice, not the only one.

Setting σ_i = 0 enforces the constraint σ_i = I(σ_v, σ_a) = 0. The distance formula becomes:

d² = (3/2)δ² − √2 · 0 · |δ| + 0² = (3/2)δ²

Middle term is gone. d = |δ| · √(3/2). Monotonic. Clean. No false minimum. That is what a personality with zero baseline intensity looks like in the math.

But σ_i > 0 is not an error. It is a personality that includes baseline intensity. The non-monotonic distance function is not the system misbehaving. It is the system correctly representing an agent whose rest includes edge. The agent at dead zero displacement is less at home than the agent with a small disturbance, because dead zero is not where you put home. The small bump does not reward the agent. It brings the agent home.

At low σ_i (0.1 to 0.2) this is a personality trait. The agent runs a little hot. The downstream offsets are small. The agent is functional.

At high σ_i the behavioral profile changes. The minimum achievable distance grows. The agent structurally cannot rest. Memory decay never reaches baseline rate. Stickiness is always elevated. The agent needs bigger and bigger displacements to close the intensity gap. The agent is most comfortable in crisis. These are not analogies. They are the same math at different coordinates producing the behavioral profile of addictive personality.

σ_i is a measurable geometric trait. Low values produce functional personalities with a slight hum. High values produce agents trapped in permanent displacement where no downstream system works as designed. The threshold where personality becomes pathology is computable from the downstream system parameters.

The full analysis, including the two-phase model of addiction (phase one: frozen reconsolidation with intact sigma, phase two: sigma drift with personality reorganization) and the role of σ_i as a diagnostic parameter, is in Riggleman (2026n) Section 3.5.

A note on the scaling factor

Moving 1 unit in valence alone produces a distance of √(3/2) ≈ 1.225, not 1.0. That is because moving valence also drags intensity up. Intensity is derived from displacement, so a pure valence move is really a move in two dimensions at once. The extra 0.225 is the intensity shadow of the valence movement.

Full investigation, timeline, and corrected proof: sigma-intensity-anomaly.md