Observer Scales and the Arrow of Time in the Dynamic Rigid Fractal Spacetime Manifold Theory: A Geometric Unification of the Quantum-Thermodynamic Paradox

This paper proposes a geometric model that integrates the Dynamic Rigid Fractal Spacetime Manifold Theory with a multi-scale observer framework to address the conflict between quantum mechanics and thermodynamics regarding the arrow of time. By introducing the dynamic fractal dimension Df(t) and the observer scale parameter ℓ, we redefine entropy production mechanisms and quantum irreversibility: 1. Fractal Entropy Coupling Model: The coupling between global informational entropy (Hglobal) and local observational entropy (Hlocal) is unified through an entropy flow equation in fractal measure space: ∂S ∂t =∇Γ·(D(Γ)∇ΓS)+σ(Γ). 2. Observer Scale Effects: A rigorous mathematical definition of a ”small observer” (dimH ≤ 103) is proposed, revealing a relationship between thermalization time τthermal ∼ (dimH)β (β ≈ 0.6) and the growth of fractal phase-space trajectory dimensions. 3. Experimental Validation: Theoretical predictions are verified through enhanced fractal superconducting critical currents (Ic(Df)/Ic(3) ≈ 1.23) and cross-scale entanglement entropy propagation speeds (vE ∝ T3/2).