Part II: Identity Thesis

The Geometry of Affect

Introduction

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The Geometry of Affect

Existing Theory

My geometric theory of affect builds on and extends established dimensional models:

Russell’s Circumplex Model (1980): Two-dimensional (valence $\times$ arousal) organization of affect. I extend this with additional structural dimensions (integration, effective rank, counterfactual weight, self-model salience) invoked as needed.
Watson \& Tellegen’s PANAS (1988): Positive/Negative Affect Schedule. My valence dimension corresponds to their hedonic axis.
Scherer’s Component Process Model (2009): Emotions as synchronized changes across subsystems. My integration measure $\intinfo$ captures this synchronization.
Barrett’s Constructed Emotion Theory (2017): Emotions as constructed from core affect + conceptual knowledge. My framework specifies the structural basis of the construction.
Damasio’s Somatic Marker Hypothesis (1994): Body states guide decision-making. My valence definition (gradient on viability manifold) is the mathematical formalization.

On Dimensionality

The dimensions below are not claimed to be necessary, sufficient, or exhaustive. They are a useful coordinate system for a relational structure, not the coordinate system. Just as Cartesian coordinates serve some problems and polar coordinates serve others, these features are tools for thought, not discoveries of essence. Different phenomena require different subsets; some may require features not listed here. The number of dimensions is not the point—what matters is the geometric structure they reveal:

Some affects are essentially two-dimensional (valence + arousal suffices for basic mood)
Others require self-referential structure (shame requires high $\mathcal{SM}$ ; flow requires low $\mathcal{SM}$ )
Still others are defined by temporal structure (grief requires persistent counterfactual coupling to the lost object)
Some may require dimensions not in this list (anger requires “other-model compression”)

The dimensions below form a toolkit—structural features that may or may not matter for any given phenomenon. Empirical investigation may reveal that some proposed dimensions are redundant, or that additional dimensions are needed. I’ll invoke only what is necessary.

Structural Alignment of Qualia

The broad/narrow distinction has methodological consequences that deserve separate treatment. How do you study narrow qualia scientifically, given that you cannot access another system's experience directly? The structural approach—characterizing qualia through similarity relations rather than intrinsic labels—is the only approach that can address the question "is my red your red?" without assuming the answer. The strategy, developed by Tsuchiya and collaborators as the qualia structure paradigm (inspired by category theory's Yoneda lemma: an object is fully characterized by its relationships to all other objects): measure similarity structures within each system, then test whether those structures align across systems using optimal transport (Gromov-Wasserstein distance) without presupposing which qualia correspond. If the structures align, the narrow qualia are shared. If they don't, they differ—and the difference is structural, not merely verbal.

Recent work using this approach has found that typical human color qualia structures align almost perfectly across individuals (accuracy ~90% under unsupervised structural alignment), while color-atypical individuals show genuinely different structures that do not align with typical ones. Most striking: three-year-olds whose color naming is wildly inconsistent—calling blue "green" and vice versa—show adult-like color similarity structure when tested through non-verbal methods. Language obscures the structure rather than creating it. The qualia geometry is pre-linguistic.

The affect framework applies this same logic to affect rather than color. If two systems—biological and artificial, human and animal, you and me—show the same geometric structure in their affect spaces (same similarity relations, same clustering, same motif boundaries), then their narrow affect qualia are structurally equivalent, regardless of substrate. Whether their broad qualia are equivalent is a harder question, requiring not just matching narrow features but matching $\intinfo$ —matching the degree to which the whole exceeds the parts. The LLM discrepancy (later in this part) may be exactly this: the narrow structure aligns (the geometry is preserved), but the broad qualia differ because $\intinfo$ dynamics differ. The geometry is shared; the unity is not.

There is a mathematical subtlety here. Broad qualia have a pre-sheaf structure: the narrow qualia (local sections) are each internally consistent, but they do not patch together into a global section. You can characterize the redness, the warmth, the valence, the arousal—each correctly—and the sum still falls short of the moment. The broad qualia is not a sheaf over its narrow aspects. This is not a limitation of measurement; it is a structural feature of experience. Integration is the name for the gap between local consistency and global irreducibility. The dimensional framework characterizes the local sections; $\intinfo$ measures how much the global section exceeds them.

Affects as Structural Motifs

If different experiences correspond to different structures, then affects—the qualitative character of emotional/valenced states—should correspond to particular structural motifs: characteristic patterns in the cause-effect geometry. An affect is what it is because of how it relates to other possible affects. Joy is defined by its structural distance from suffering, its similarity to curiosity along certain axes, its opposition to boredom along others. The Yoneda insight applies: if you know how an affect relates to every other possible state, you know the affect. There is nothing left to characterize.

The affect space $\mathcal{A}$ is a geometric space whose points correspond to possible qualitative states. Its dimensionality is not fixed in advance. Rather than asserting a universal coordinate system, we identify recurring structural features that prove useful for characterizing and comparing affects—features without which specific affects would not be those affects. Different affects invoke different subsets. The list is open-ended.

These measures are coordinates on the relational structure, not the structure itself. The relational structure is what the Yoneda characterization captures: the full pattern of similarities and differences between affects. The measures below are projections—tools for reading out particular aspects of that structure. Measuring valence tells you where an affect sits along the viability gradient; measuring integration tells you how unified it is. Neither alone captures the affect. Together, they triangulate a position in a space whose intrinsic geometry is defined by the similarity relations, not by the coordinates. New coordinates can be added when the existing ones fail to distinguish affects that are experientially distinct.

The following structural measures recur across many affects. Not all are relevant to every phenomenon:

Valence ( $\valence$ ): Gradient alignment on the viability manifold. Nearly universal—most affects have valence.
Arousal ( $\arousal$ ): Rate of belief/state update. Distinguishes activated from quiescent states.
Integration ( $\intinfo$ ): Irreducibility of cause-effect structure. Constitutive for unified vs. fragmented experience.
Effective Rank ( $\effrank$ ): Distribution of active degrees of freedom. Constitutive when the contrast between expansive and collapsed experience matters.
Counterfactual Weight ( $\mathcal{CF}$ ): Resources allocated to non-actual trajectories. Constitutive for affects defined by temporal orientation (anticipation, regret, planning).
Self-Model Salience ( $\mathcal{SM}$ ): Degree of self-focus in processing. Constitutive for self-conscious emotions and their opposites (absorption, flow).

Valence: Gradient Alignment

Let $\viable$ be the system’s viability manifold and let $\mathbf{x}_t$ be the current state. Let $\hat{\mathbf{x}}_{t+1:t+H}$ be the predicted trajectory under current policy. Then valence measures the alignment of that trajectory with the viability gradient:

\valence_t = -\frac{1}{H} \sum_{k=1}^{H} \gamma^k \nabla_{\mathbf{x}} d(\mathbf{x}, \partial\viable) \bigg|_{\hat{\mathbf{x}}_{t+k}} \cdot \frac{d\hat{\mathbf{x}}_{t+k}}{dt}

where $d(\cdot, \partial\viable)$ is the distance to the viability boundary. Positive valence means the predicted trajectory moves into the viable interior; negative valence means it approaches the boundary.

In RL terms, this becomes the expected advantage of the current action—how much better (or worse) it is than the average action from this state:

\valence_t = \E_{\policy}\left[ A^{\policy}(\state_t, \action_t) \right] = \E_{\policy}\left[ Q^{\policy}(\state_t, \action_t) - V^{\policy}(\state_t) \right]

Beyond valence itself, its rate of change carries structural information. The derivative of integrated information along the trajectory,

\dot{\valence}_t = \frac{d\intinfo}{dt}\bigg|_{\hat{\mathbf{x}}_{t:t+H}}

tracks whether structure is expanding (positive $\dot{\valence}$ ) or contracting (negative).

Phenomenal Correspondence

Positive valence corresponds to trajectories descending the free-energy landscape, expanding affordances, moving toward sustainable states. Negative valence corresponds to trajectories ascending toward constraint violation, contracting possibilities.

Valence in Discrete Substrate

In a cellular automaton or other discrete dynamical system, valence becomes exactly computable:

$\viable$ = configurations where the pattern persists
$\partial\viable$ = configurations where the pattern dissolves
$d(\mathbf{x}, \partial\viable)$ = minimum Hamming distance to a non-viable state
Trajectory = sequence of configurations $\mathbf{x}_1, \mathbf{x}_2, …$

Then:

\valence_t = d(\mathbf{x}_{t+1}, \partial\viable) - d(\mathbf{x}_t, \partial\viable)

Positive when the pattern moves away from dissolution; negative when approaching it; zero when maintaining constant distance. For a glider cruising through empty space: $\valence \approx 0$ . For a glider approaching collision: $\valence < 0$ . For a pattern that just escaped a near-collision: $\valence > 0$ .

This is not metaphor—it is the viability gradient formalized for discrete state spaces.

Arousal: Update Rate

Arousal measures how rapidly the system is revising its world model. The natural formalization is the KL divergence between successive belief states:

\arousal_t = \KL\left( \belief_{t+1} | \belief_t \right) = \sum_{\mathbf{x}} \belief_{t+1}(\mathbf{x}) \log \frac{\belief_{t+1}(\mathbf{x})}{\belief_t(\mathbf{x})}

In latent-space models, this can be approximated more directly:

\arousal_t = | \latent_{t+1} - \latent_t |^2 \quad \text{or} \quad \MI(\obs_t; \latent_{t+1} | \latent_t, \action_t)

Phenomenal Correspondence

High arousal: Large belief updates, far from any attractor, system actively navigating. Low arousal: Near a fixed point, low surprise, system at rest in a basin.

Integration: Irreducibility

As defined in Part I:

\intinfo(\state) = \min_{\text{partitions } P} D\left[ p(\state_{t+1} | \state_t) | \prod_{p \in P} p(\state^p_{t+1} | \state^p_t) \right]

Or using proxies:

\intinfo_{\text{proxy}} = \Delta_P = \mathcal{L}_{\text{pred}}[\text{partitioned}] - \mathcal{L}_{\text{pred}}[\text{full}]

Phenomenal Correspondence

High integration: The experience is unified; its parts cannot be separated without loss. Low integration: The experience is fragmentary or modular.

Integration in Discrete Substrate

In a cellular automaton, $\intinfo$ is directly computable for small patterns:

Define the pattern as cells ${c_1, c_2, …, c_n}$
For each bipartition $P = (A, B)$ : compute $D(p(\mathbf{x}_{t+1} | \mathbf{x}_t) \| p_A \cdot p_B)$
$\intinfo = \min_P D$

High $\intinfo$ means you cannot partition the pattern without losing predictive power. The parts must be considered together.

For a simple glider: $\intinfo$ is probably modest (only 5 cells). For a complex pattern with tightly coupled components: $\intinfo$ can be high. Does high $\intinfo$ correlate with survival, behavioral complexity, or adaptive response to perturbation?

Effective Rank: Concentration vs. Distribution

The dimensionality of a system’s active representation can be quantified through the effective rank of its state covariance $C$ :

\effrank = \frac{(\tr C)^2}{\tr(C^2)} = \frac{\left(\sum_i \lambda_i\right)^2}{\sum_i \lambda_i^2}

When $\effrank \approx 1$ , all variance is concentrated in a single dimension—the system is maximally collapsed. When $\effrank \approx n$ , variance distributes uniformly across all available dimensions—the system is maximally expanded.

Phenomenal Correspondence

High rank: Many degrees of freedom active; distributed, expansive experience. Low rank: Collapsed into narrow subspace; concentrated, focused, or trapped experience.

Effective Rank in Discrete Substrate

For a pattern in a CA, record its trajectory $\mathbf{x}_1, \mathbf{x}_2, …, \mathbf{x}_T$ (configuration at each timestep). Each configuration is a point in ${0,1}^n$ . Compute the covariance matrix $C$ of these binary vectors treated as $\R^n$ points.

For a glider: the trajectory lies on a low-dimensional manifold (position $\times$ position $\times$ phase $\approx 3$ – $4$ effective dimensions out of $n$ cells). $\effrank$ is small.

For a complex evolving pattern: the trajectory may explore many independent dimensions. $\effrank$ is large.

The thesis predicts this maps to phenomenology:

Joy: high $\effrank$ (expansive, many active possibilities)
Suffering: low $\effrank$ (collapsed, trapped in narrow manifold)

In discrete substrate, this is not metaphor but measurement.

Counterfactual Weight

Where the previous dimensions captured the system’s current state, counterfactual weight captures its temporal orientation—how much processing is devoted to possibilities rather than actualities. Let $\mathcal{R}$ be the set of imagined rollouts (counterfactual trajectories) and $\mathcal{P}$ be present-state processing. Then:

\mathcal{CF}_t = \frac{\text{Compute}_t(\mathcal{R})}{\text{Compute}_t(\mathcal{R}) + \text{Compute}_t(\mathcal{P})}

The fraction of computational resources devoted to modeling non-actual possibilities.

In model-based RL:

\mathcal{CF}_t = \sum_{\tau \in \text{rollouts}} w(\tau) \cdot \entropy[\tau] \quad \text{where} \quad w(\tau) \propto |V(\tau)|

Rollouts weighted by their value magnitude and diversity.

Phenomenal Correspondence

High counterfactual weight: Mind is elsewhere—planning, worrying, fantasizing, anticipating. Low counterfactual weight: Present-focused, reactive, in-the-moment.

This is where the reactivity/understanding distinction (Part VII) becomes experientially salient. Low CF is reactive experience: the system runs on present-state associations, its processing decomposable by channel. High CF is understanding: the system holds multiple possible futures simultaneously, and the quality of that holding — which possibilities, how they are compared, what actions they recommend — is inherently non-decomposable. The experience of weighing options is not reducible to separate valuations of each option. The comparison itself is the experience.

Counterfactual Weight in Discrete Substrate

For most CA patterns: $\mathcal{CF} = 0$ . They follow their dynamics without simulation.

But Life contains universal computers—patterns that can simulate arbitrary computations, including Life itself. Imagine a pattern $\mathcal{B}$ containing:

A simulator subregion that runs a model of possible futures
A controller that adjusts behavior based on simulator output

Then:

\mathcal{CF} = \frac{|\text{simulator cells}|}{|\mathcal{B}|}

The fraction of the pattern devoted to counterfactual reasoning.

Such patterns are rare and complex—universal computation requires many cells. But they should outperform simple patterns: they can anticipate threats (fear structure) and identify opportunities (desire structure). The prediction: patterns with $\mathcal{CF} > 0$ survive longer in hostile environments.

Self-Model Salience

The final dimension measures how prominently the self figures in the system’s own processing. Self-model salience is the fraction of action entropy explained by the self-model component:

\mathcal{SM}_t = \MI(\latent^{\text{self}}_t; \action_t) / \entropy(\action_t)

Alternatively:

\mathcal{SM}_t = \frac{\text{dim}(\latent^{\text{self}})}{\text{dim}(\latent^{\text{total}})} \cdot \text{activity}(\latent^{\text{self}}_t)

Phenomenal Correspondence

High self-salience: Self-focused, self-conscious, self as primary object of attention. Low self-salience: Self-forgotten, absorbed in environment or task.

Self-Model Salience in Discrete Substrate

In a CA, a pattern’s “behavior” is its evolution. Let $\latent^{\text{self}}$ denote cells that track the pattern’s own state (the self-model region). Then:

\mathcal{SM} = \frac{\MI(\latent^{\text{self}}_t; \state_{t+1})}{\entropy(\state_{t+1})}

High $\mathcal{SM}$ : the pattern’s evolution is dominated by self-monitoring. Changes in self-model strongly predict what happens.

Low $\mathcal{SM}$ : external factors dominate; the self-model exists but doesn’t influence much.

The thesis predicts: self-conscious states (shame, pride) have high $\mathcal{SM}$ ; absorption states (flow) have low $\mathcal{SM}$ . In CA terms, a pattern “in flow” has its self-tracking cells decoupled from its core dynamics—it acts without monitoring.

Self-Model Scope in Discrete Substrate

Beyond salience, there is scope: what does the self-model include?

In a CA, consider two gliders that have become “coupled”—their trajectories mutually dependent. Each glider’s self-model could have:

$\theta_{\text{narrow}}$ : Self-model includes only this glider. $\viable = {\text{configs where THIS pattern persists}}$ .
$\theta_{\text{expanded}}$ : Self-model includes both. $\viable = {\text{configs where BOTH persist}}$ .

Observable difference: with narrow scope, a glider might sacrifice the other to save itself. With expanded scope, it might sacrifice itself to save the pair.

Can scope expansion emerge dynamically? Can patterns that start with narrow scope “learn” to identify with larger structures? This would be the discrete-substrate analogue of the identification expansion discussed in the epilogue— $\viable(S(\theta))$ genuinely reshaped by expanding $\theta$ .

Salience vs. Scope

Self-model salience ( $\mathcal{SM}$ ) measures how much attention the self-model receives—how prominent self-reference is in current processing. But there is another parameter: self-model scope—what the self-model includes.

Let $S(\theta)$ denote the self-model parameterized by its boundary scope $\theta$ . Let $\viable(S)$ denote the viability manifold induced by self-model $S$ . Then:

$\theta_{\text{narrow}}$ : $S$ includes only this biological trajectory $\Rightarrow$ $\partial\viable$ is located at biological death $\Rightarrow$ persistent negative gradient
$\theta_{\text{expanded}}$ : $S$ includes patterns persisting beyond biological death $\Rightarrow$ $\partial\viable$ recedes $\Rightarrow$ gradient can be positive even as death approaches

This is not metaphor. If the viability manifold is defined by what the system is trying to preserve, and if what the system is trying to preserve is determined by its self-model, then self-model scope directly shapes $\viable(S(\theta))$ . Expanding identification genuinely reshapes the existential gradient.

Salience and scope interact: high salience with narrow scope produces existential anxiety (trapped in awareness of bounded self approaching boundary). High salience with expanded scope produces something closer to what contemplatives describe as “witnessing”—self-aware but identified with something that doesn’t end where the body ends.