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

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.