Consciousness as the Generative Dimension of Spacetime: A Fiber-Bundle and Noncommutative-Geometry Formalization within the Extended Reality Theory (TRE)

This paper presents a mathematical formalization of the Extended Reality Theory (Teoria della Realtà Estesa, TRE), a theoretical framework in which consciousness is proposed as the fundamental dimension from which spacetime and the physical universe are generated, rather than as an epiphenomenon or as an additional coordinate superimposed on a pre-existing spacetime background. Building on an initial vector formulation, in which a conscious state is represented as a point C = (φ, α, η, ε) in a four-component space added to spacetime as a degenerate fifth coordinate, we develop two successive, nested extensions. First, the consciousness vector is promoted to a full differentiable manifold M_C, over which ordinary four-dimensional spacetime M₄ is realized as the fiber of a principal bundle equipped with a connection ω; the associated curvature F = dω + ω∧ω generates spacetime curvature as a calculable consequence of the geometry of M_C, and a variational action S[C, g] is proposed from which Einstein-type field equations for M_C and Yang–Mills-type equations for ω can, in principle, be derived. Second, motivated by the historical necessity of abandoning simultaneously well-defined coordinates at the most fundamental level of physical description, M_C is itself reinterpreted as the commutative limit of a noncommutative spectral triple (A_C, H_C, D_C) in the sense of Connes, from which distance, geometry, and dynamics — via the spectral action S = Tr f(D_C/Λ) — can be derived rather than postulated. Within this framework, the informal Creation Formula C → E → M is given explicit dynamical content as a renormalization-group-type flow of a noncommutativity parameter θ_C toward a commutative fixed point. We conclude by identifying the coupling parameter λ between consciousness and brain dynamics as the theory's central falsifiable prediction, and by listing the concrete steps — explicit construction of the finite algebra A_F,C, computation of the associated KO-dimension, and specification of the beta function β(θ_C) — required to move the proposal from a coherent mathematical scaffold to a genuinely testable physical theory.