Integrated Information as Irreducible Dissipation: A Low-Order, Thermodynamically Consistent Derivation

The canonical measure Φ of Integrated Information Theory (IIT) requires exhaustive cause-effect enumeration and suffers from partition ambiguities. We reformulate IIT under three response laws: (i) finite energetic capacity, (ii) a uniform non-zero delay, and (iii) low-order bounded-input-bounded-output stability. Approximating the delay τ₀ with a Padé-(1,1) expansion keeps every transfer function rational and at most second order, so any causal network reduces to two stable poles (−α₁,−α₂) with a weight ratio θ₂. From these parameters we obtain the closed-form scalar:λ S_c^2/4θ_2^2(α_2⁻¹-α_1⁻¹),λ=Δμ/ T,which fulfils the five phenomenological axioms of IIT, scales O(N) for sparse graphs, and equals the minimum decrement in a quadratic entropy-production functional when couplings are intact. Thus Φ embeds IIT within Prigogine's minimum-dissipation principle and provides a tractable consciousness metric for large-scale neural data.