IIT 4.0's Phi: Wasserstein Metric and the Integration-Information Tradeoff — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Integrated Information Theory (IIT) formalizes consciousness as a quantity Φ (phi) — the irreducible cause-effect power of a system's state — but current formalizations face a fundamental tension between information loss and integration, with algorithmic information theory offering a lossless alternative. | MATH: IIT 4.0 defines Φ via the *Earth mover's distance* (Wasserstein metric) between the cause-effect repertoire of a mechanism and its partitioned counterpart: Φ = min_partition W( p(mechanism_state) || p(partitioned_state) ). Tononi's original (2008) formulation used effective information: EI(X→Y) = H(Y) − H(Y|X) summed over minimal partitions. The algorithmic alternative (arXiv:1405.0126) replaces Shannon entropy with Kolmogorov complexity: Φ_AIT = K( system_state ) − Σ K( subsystem_states ), requiring no lossy compression. | CONNECTION: The Wasserstein metric is a *transportation* distance — its optimal solution on discrete state spaces corresponds to a linear programm
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026wasserstein,
title = {{IIT 4.0's Phi: Wasserstein Metric and the Integration-Information Tradeoff — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23526314},
url = {https://zenodo.org/records/23526314},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - IIT 4.0's Phi: Wasserstein Metric and the Integration-Information Tradeoff — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/21
PB - Zenodo
DO - 10.5281/zenodo.23526314
UR - https://zenodo.org/records/23526314
AB - FINDING: Integrated Information Theory (IIT) formalizes consciousness as a quantity Φ (phi) — the irreducible cause-effect power of a system's state — but current formalizations face a fundamental tension between information loss and integration, with algorithmic information theory offering a lossless alternative. | MATH: IIT 4.0 defines Φ via the *Earth mover's distance* (Wasserstein metric) between the cause-effect repertoire of a mechanism and its partitioned counterpart: Φ = min_partition W( p(mechanism_state) || p(partitioned_state) ). Tononi's original (2008) formulation used effective information: EI(X→Y) = H(Y) − H(Y|X) summed over minimal partitions. The algorithmic alternative (arXiv:1405.0126) replaces Shannon entropy with Kolmogorov complexity: Φ_AIT = K( system_state ) − Σ K( subsystem_states ), requiring no lossy compression. | CONNECTION: The Wasserstein metric is a *transportation* distance — its optimal solution on discrete state spaces corresponds to a linear programm Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -