Part II: Identity Thesis

Effective Rank: Concentration vs. Distribution

Introduction
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Effective Rank: Concentration vs. Distribution

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

reff=(tr⁡C)2tr⁡(C2)=(∑iλi)2∑iλi2\effrank = \frac{(\tr C)^2}{\tr(C^2)} = \frac{\left(\sum_i \lambda_i\right)^2}{\sum_i \lambda_i^2}

When reff≈1\effrank \approx 1, all variance is concentrated in a single dimension—the system is maximally collapsed. When reff≈n\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 x1,x2,…,xT\mathbf{x}_1, \mathbf{x}_2, …, \mathbf{x}_T (configuration at each timestep). Each configuration is a point in 0,1n{0,1}^n. Compute the covariance matrix CC of these binary vectors treated as Rn\R^n points.

For a glider: the trajectory lies on a low-dimensional manifold (position ×\times position ×\times phase ≈3\approx 3–44 effective dimensions out of nn cells). reff\effrank is small.

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

The thesis predicts this maps to phenomenology:

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

In discrete substrate, this is not metaphor but measurement.