Thermodynamics of the Fibrous Medium: The Fundamental Dynamical Equation of the Al-Ani Fabric Theory

We present the complete thermodynamic foundation of the Al-Ani Fabric Theory. Starting from the principle of free energy minimization, we derive the fundamental dynamical equation governing spacetime deformations: ∂δ/∂t = D∇²δ - Γδ + S, where D is the diffusion coefficient, Γ is the damping coefficient, and S represents source terms from visible matter. This single equation unifies five fundamental phenomena: (1) nebulae as steady-state solutions, (2) dark matter density profiles, (3) the correlation length scaling λ ∝ M⁰·³⁵, (4) black hole shadows, and (5) quantum fluctuations yielding wave-particle duality, measurement collapse, and entanglement correlations S = 2√2. Four testable predictions are presented for future observations.