Effective Field Theory of a Fractal Vacuum: Anisotropic Metric and Statistical Signatures in High-Resolution Interferometry

We present a scalar-tensor framework where gravity and electrodynamics emerge from the discrete information-exchange dynamics of a fractal vacuum network. By introducing an effective refresh rate Γ and a renormalized roughness amplitude Af, we derive an anisotropic effective metric that recovers the Schwarzschild limit to first post-Newtonian order. To reconcile the model with local gravity tests, we demonstrate a Chameleon-like screening mechanism where the fractal field mass m_φ scales with local density, ensuring consistency with Cassini PPN limits (γ ≈ 1) and Eöt-Wash equivalence principle tests. The core of this work proposes a novel Multi-frequency Differential (MD) Estimator for interferometric closure phases. We demonstrate that while standard General Relativity-plus-plasma models predict a convergent unit ratio for this estimator, the fractal vacuum introduces a persistent geometric noise floor scaling with baseline length as B5/3. This signature serves as a critical benchmark for next-generation VLBI observations (e.g., ngEHT) to distinguish emergent fractal gravity from standard astrophysical noise.