Unicity Theorem for Structural Modifications of Cosmological Evolution

"This work provides a rigorous mathematical proof for the Unicity Theorem regarding structural modifications to the cosmological evolution equation, establishing a formal foundation for the Structural Time Framework (TFT). By systematically evaluating cosmological candidate terms against general physical constraints—such as dimensional consistency, causality, well-posedness (absence of Ostrogradsky instabilities), the absence of new fundamental scales, and the correct asymptotic limit—we demonstrate through exhaustive elimination that the only admissible structural modification to the second Friedmann equation is a term proportional to H² (α H²). As a direct consequence of this theorem, late-time cosmic acceleration naturally emerges from causal and dimensional geometry rather than as an arbitrary phenomenological addition like Dark Energy. Furthermore, this unique modification predicts a universal acceleration scale (a₀ ~ cH₀) that structurally connects global expansion to galactic rotation dynamics without requiring dark energy or local modifications to General Relativity. The framework remains fully falsifiable through precision cosmological observations."