Part II: Identity Thesis

Valence: Gradient Alignment

Introduction

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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.