Experiments

Integration Measures

Every claim in this book is either tested, testable, or honestly labeled as speculative. This appendix catalogs the experimental program as a research archive: 43 canonical study records spanning LLM, MARL, Lenia CA, protocell agency, the measurement program, and a cross-substrate convergence test. The point is not to present a trophy wall of positive results. The point is to show the dependency structure: which claims survived, which failed, which were retracted, and which architectural walls forced the next experiment.

The measures the experiments report, defined once. Part I uses these by name and points here for the definitions; the caution at the end applies wherever they appear.

The first is transfer entropy — directed causal influence between components. From process $X$ to process $Y$ , it measures the information $X$ provides about the future of $Y$ beyond what $Y$ ’s own past provides:

\text{TE}_{X \to Y} = \MI(X_t; Y_{t+1} | Y_{1:t})

The deepest measure is integrated information ( $\Phi$ ). Following IIT, for a system in state $\state$ it is the extent to which the system’s causal structure exceeds the sum of its parts:

\Phi(\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]

where the minimum is over all bipartitions of the system, and $D$ is an appropriate divergence (typically Earth Mover’s distance in IIT 4.0).

In practice, computing $\Phi$ exactly is intractable. Three proxies make it operational:

Transfer entropy density—average transfer entropy across all directed pairs: $\bar{\text{TE}} = \frac{1}{n(n-1)} \sum_{i \neq j} \text{TE}_{i \to j}$
Partition prediction loss—the cost of factoring the model: $\Delta_P = \mathcal{L}_{\text{pred}}[\text{partitioned model}] - \mathcal{L}_{\text{pred}}[\text{full model}]$
Synergy—the information that components provide jointly beyond their individual contributions: $\text{Syn}(X_1, \ldots, X_k \to Y) = \MI(X_1, \ldots, X_k; Y) - \sum_i \MI(X_i; Y | X_{-i})$

A complementary measure captures the system’s representational breadth rather than its causal coupling. The effective rank of a system with state covariance matrix $C$ measures how many dimensions it actually uses:

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

where $\lambda_i$ are the eigenvalues of $C$ . This is bounded by $1 \leq \effrank \leq \rank(C)$ , with $\effrank = 1$ when all variance is in one dimension and $\effrank = \rank(C)$ when variance is uniformly distributed across all active dimensions.

A fifth measure captures something the others miss: the topology of mode coupling over time. Given state covariance $C(t)$ at each timestep, eigendecompose and align frames across adjacent timesteps via Procrustes: $R(t, t{+}1) = \arg\min_R \|V(t{+}1) - V(t)R\|_F$ . Accumulate the rotation around a cycle — a drought-recovery loop, say — to obtain the holonomy $H_\gamma = \prod R(t, t{+}1)$ . The holonomy index:

\mathcal{H}_\gamma = \|H_\gamma - I\|_F

measures how much the eigenmodes twist through the cycle. $\mathcal{H} = 0$ : modes return to their starting configuration — flat eigenskeleton, decomposable computation. $\mathcal{H} > 0$ : modes coupled through the cycle — curved eigenskeleton, irreducibly integrated. This is computable from covariance matrices already tracked in the experiments and captures a structural feature distinct from both $\intinfo$ (partition cost at a single timepoint) and $\effrank$ (eigenvalue concentration without topology). $\intinfo$ asks: does breaking the system lose information? $\effrank$ asks: how many modes are active? $\mathcal{H}$ asks: do the modes talk to each other when the system moves?

One caution about all five. Effective rank measures how many representational dimensions are active. Partition loss measures how much predictive performance is lost when a system is divided. Transfer entropy measures directed statistical dependence. The holonomy index measures how representational modes reorganize along a trajectory. None is automatically identical to another, and none by itself measures consciousness.