Part II: Identity Thesis

Integration: Irreducibility

Introduction
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Integration: Irreducibility

As defined in Part I:

Φ(s)=min⁡partitions PD[p(st+1∣st)∣∏p∈Pp(st+1p∣stp)]\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:

Φproxy=ΔP=Lpred[partitioned]−Lpred[full]\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, Φ\intinfo is directly computable for small patterns:

  1. Define the pattern as cells c1,c2,…,cn{c_1, c_2, …, c_n}
  2. For each bipartition P=(A,B)P = (A, B): compute D(p(xt+1∣xt)∥pA⋅pB)D(p(\mathbf{x}_{t+1} | \mathbf{x}_t) \| p_A \cdot p_B)
  3. Φ=min⁡PD\intinfo = \min_P D

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

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