Why Ω? A Uniqueness Argument for a Geometric Measure of Consciousness

Quantitative theories of consciousness face a foundational challenge: why should any particular mathematical quantity serve as a measure of conscious experience? Here we show that under two minimal assumptions—high-dimensional weak coupling (P1, justified by the scale of neural populations) and the inherent asymmetry of causal inference (P2)—a unique consciousness-related quantity Ω emerges through a chain of uniqueness theorems. The derivation proceeds along two converging lines. The path chain applies Maximum Caliber—the path-space maximum entropy principle—to show that Langevin dynamics is the unique inference dynamics under moment constraints, yielding the Onsager–Machlup action S_OM as the unique path functional. The metric chain invokes the Čencov–Lebanon theorem to establish Fisher information as the unique invariant metric on causal statistical manifolds, and proves that KL divergence is the unique divergence compatible with causal factorization. The confluence of these two chains identifies Ω_geo—the kinetic component of S_OM measured in the causal Fisher metric—as the unique candidate for quantifying inferential activity. Seven assumptions commonly taken as primitive are eliminated. We further derive the energy–information conversion factor κ from MaxCal's Lagrange multiplier structure combined with the Einstein relation, establishing the physical measure Ω = κ · Ω_geo. The decomposition of S_OM additionally reveals θ, the angle between inference velocity and environmental drift, as an independent variable distinguishing modes of consciousness—flow (θ ≈ 0), mind-wandering (θ ≈ π/2), and cognitive conflict (θ ≈ π)—at equal levels of Ω. We discuss testable predictions, the relationship to the Free Energy Principle, and open questions.