The Axioms of Cognitive Geometry: A Formal Model of Psychophysical Correlation

The empirical study of consciousness has matured significantly in recent decades, most notably with regard to the production of increasingly fine-grained experimental data. However, the foundations of theoretical cognitive science remain unsettled, creating a "crisis of falsifiability." We argue that the crisis of falsifiability in consciousness studies ultimately stems from insufficient logical and mathematical rigor: leading theories of consciousness such as Integrated Information Theory (IIT) and Global Workspace Theory (GWT) are fundamentally incapable of interfacing with standard mathematical physics, making such theories either empirically untestable in principle or structurally under-constrained—lacking the invariant geometric structure necessary to fix the system's embodied reference frame. Thus, drawing on a rigorous formalization of phenomenological structure, we propose a formal axiomatic system which is fully integrated with fundamental physics from the outset. Specifically, we posit an orthogonal phase-space geometry: the standard "physical Hilbert space" of textbook quantum mechanics, linked to an orthogonal "phenomenal Hilbert space" by mathematically well-defined Awareness (A) and Volition (V) operators. Crucially, this orthogonal phase-space geometry yields a novel and falsifiable empirical prediction: a precise π/2 (90-degree) phase offset between the neural correlates of sensory awareness and the neural correlates of volitional action in a sensorimotor loop. The "Awareness Operator" or A|Ω⟩ formalism thus offers a comprehensive resolution to the crisis in the foundations of cognitive science: a mathematically precise, physically grounded, and empirically testable model.