Topological Origin of Interaction Hierarchy and Vacuum Energy: A Background-Independent Elastic Framework

We present a unified geometric framework that simultaneously addresses the Hierarchy Problem and the Cosmological Constant Problem. By treating the vacuum as a background-independent medium belonging to the universality class of elastic Cosserat solids, we demonstrate that fundamental interactions emerge as collective excitations of the vacuum microstructure. We rigorously distinguish between the "hardware" of the vacuum (high-energy stiffness scales Y, D) and the "software" of physical fields (topological defects). Within an Effective Field Theory (EFT) framework, we show that: The huge ground-state energy of the vacuum (~ 10¹¹³ J/m³) does not gravitate because gravity couples only to curvature (strain gradients), not to the background energy density. This acts as a volume-preserving constraint (Lagrange multiplier), effectively decoupling the bulk energy from the equations of motion. The weakness of gravity (G/ke ~ 10⁻⁴⁰) arises naturally from the geometric scale separation between the Planck length (ℓP) and the Compton wavelength (L), scaling as (ℓc/L)², where ℓc is the Cosserat characteristic length identified with ℓP. This framework provides a non-perturbative explanation for the stability of matter and the immense disparity of forces without fine-tuning.