Quantum Entanglement as the Origin of the Gravitational Constant: Deriving the Planck Scale from Holographic Tensor Networks in the Selection-Stitch Model

In most approaches to quantum gravity, Newton’s gravitational constant (GN ) and the Planck length (lP ) are treated as fundamental, irreducible inputs. In the Selection-Stitch Model (SSM) [1], however, the 3D Face-Centered Cubic (FCC) vac- uum is not a foundational background, but rather an emergent structure projected holographically from a 2D hexagonal boundary tensor network [2]. In this paper, we demonstrate that GN is not a fundamental constant at all. Instead, it is a dy- namically derived parameter originating entirely from the topological entanglement entropy of the boundary network. By explicitly solving the exact Ryu-Takayanagi (RT) relation for the SSM’s discrete geometry [3, 4], we geometrically derive New- ton’s constant as GN = 2/3L2/(4 ln 2), where L is the physical lattice spacing. By equating this emergent GN to the standard Planck area, we mathematically fix the fundamental physical lattice spacing of the universe at L= 2√6 ln 2 lP ≈1.84lP. This result establishes gravity within the SSM purely as a consequence of quantum information mechanics, rigorously defining the scale of the discrete vacuum from first principles without appealing to classical thermodynamic approximations.