CAUSIUM: A Four-Layer Framework for Emergent Time via SCGF Geometry, Renormalization Flow, and Field Embedding

CAUSIUM is a four-layer theoretical framework in which time emerges as a geometric invariant of stochastic field dynamics rather than a primitive background parameter. The framework constructs an operational time functional τ from an Ornstein–Uhlenbeck-driven scalar field and induces a Fisher–Rao geometry on the associated parameter manifold, characterized by constant scalar curvature R₀ = −2. Deviations from the Gamma regime arise through higher-order cumulants, producing curvature anomalies quantified by A = |R + 2|. A renormalization group flow on the space of scaled cumulant generating function (SCGF) jets is then defined, exhibiting contraction toward the Gamma manifold as an infrared fixed point under Ornstein–Uhlenbeck dynamics. A conjectural physical embedding is introduced via the Scalar Geometric Field Theory (SGFT) coupling hypothesis Φ ≡ ρE/S̃, under which the Newtonian gravitational limit is recovered at the fixed-point regime. Layers I–III are formulated within established stochastic process theory and information geometry. Layer IV remains exploratory and hypothesis-dependent, requiring independent empirical validation. This work is presented as a public research preprint within an exploratory theoretical program and has not yet undergone peer review.