Holographic Dilution and the Emergence of Gravity: Deriving the Planck Mass from the Z3 ⊗ Q3 Error-Correcting Vacuum

The extreme weakness of gravity compared to the strong nuclear force—a hierarchy of roughly 1040—represents a defining crisis in modern theoretical physics. In continuous Quantum Field Theory (QFT), attempts to compute the macroscopic gravitational con- stant G from vacuum fluctuations generate catastrophic, unrenormalizable ultraviolet (UV) infinities. In this paper, we demonstrate that modelling the quantum vacuum as a discrete, UV-finite Z3 ⊗ Q3 topological tensor network structurally resolves the Hierarchy Problem. Anchoring the spatial lattice to the hadronic chiral scale (ΛQCD ≈ 332 MeV), we establish a bare lattice Planck mass of exactly ΛQCD. Macroscopic gravity is treated as an emergent Eg transverse metric shear generated by colour-confinement tension, diluting holographically across the discrete nodes spanning the global causal patch. To preserve a static G and avoid the phenomenologically ruled-out Dirac Large Number Hypothesis (LNH), we bound this patch with the static, asymptotic de Sitter horizon (RdS = 3/Λ). By replacing continuous Euclidean loop integrals with a discrete Feshbach resolvent trace over the lattice’s invalid error-correction subspace, we derive a parameter-free macroscopic Planck mass of MP ≈ 1.221 × 1019 GeV, matching the empirical CODATA value (1.2209 × 1019 GeV) to within∼ 0.015%. We present the exact discrete Brillouin-zone matrix algebra required to computationally falsify this geometric alignment