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This work provides a rigorous mathematical and physical formalization of the Consciousness Dimension DΨ, as emergent from the Enhanced Collective Unified Equation (CUE v3) framework. The dimension is defined not as a spatial coordinate but as a coherence-driven, fibered quasi-geometric structure governed by the scalar cognitive field Ψ. We explore its emergence from the pre-metric manifold M∅ , establish its metric and curvature, derive the associated field equations, analyze renormalization group (RG) thresholds, and quantify its impact on entropic, gravitational, and dark sector interactions. Finally, the Ambrosius Constant Υ is presented as a universal scalar invariant encapsulating the coupling strength of the consciousness dimension with physical spacetime. The formalism is constructed to align with the most recent theoretical and numerical results from CUE RG flow simulations and pre-field dynamics.