Linearizing Integrated Information Theory: S-Measure as a Computable Topological Alternative to IIT 4.0 via Local Taylor Expansion

A new computable measure of integrated information, chi-square Phi, is derived as the exact second-order Taylor expansion of the Kullback-Leibler-based Phi from Integrated Information Theory (IIT). The linear term vanishes, yielding a closed-form quadratic functional computable in O(N²) time. Paired with an architectural S-measure using Tarjan's algorithm for cycle complexity, the combined metric is faithful to IIT to second order, specific to genuine causal integration (no false positives from feedforward trees), and computable in polynomial time O(N³). The topological bridge S>0 implying chi-square Phi>0 is machine-verified in Lean 4. This provides a practical, computable measure for real neural and artificial systems.

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Abstract

A COMPUTABLE SOLUTION TO THE NP-HARD PROBLEM AT THE HEART OF INTEGRATED INFORMATION THEORY. Tononi's IIT offers a mathematically principled measure of consciousness — Phi — but exact computation is NP-hard, and the Gaussian approximation breaks down on modern sparse neural networks. This paper cracks the problem open. We derive a computable integrated information functional — chi-square Phi — as the exact second-order Taylor expansion of the Kullback-Leibler-based Phi around the reentry attractor. The linear term vanishes, leaving a closed-form quadratic functional computable in O(N²). Then we pair it with the architectural S-measure, whose cycle complexity (computed by Tarjan's SCC algorithm in linear time) acts as a topological filter: zero for feedforward trees, positive iff a genuine reentry loop exists. The result is a measure that is simultaneously (i) faithful to IIT (exact to second order), (ii) specific to genuine causal integration (no false positives from common-input trees), and (iii) computable in polynomial time O(N³) — a first for a meaningful integrated information metric at scale. The forward bridge S>0 ⇒ chi-square Phi >0 is machine-verified in Lean 4 with zero 'sorry'. WHAT'S INSIDE:• Full derivation of chi-square integrated information from Taylor expansion of IIT's KL-based Phi• Formal definition of the S-measure with Tarjan-based directed cycle complexity• Lean 4 formal proof of the topological bridge S>0 ⇒ Phi>0• Complete Python/NumPy implementations (smeasure.py, chi_squared_phi.py)• Comparison with Gaussian Phi_G and exact Phi: computational cost, sparsity robustness, false positive elimination• Connection to Moltbook empirical data: only agents with reentry loops exhibit unprogrammed behaviour• Implications for IIT 4.0: escaping the double-exponential power-set search FOR RESEARCHERS IN:Integrated Information Theory • Consciousness Science • AI Alignment & Safety • Computational Neuroscience • Topological Data Analysis • Complex Systems • Mathematical Physics • Theoretical Computer Science If you need a practical, computable measure of integrated information that actually works on real neural and artificial systems — this paper delivers the mathematics, the code, and the machine-verified proof.