The Bulk-Constrained Structural Law of Observables: Complete Derivation from General Relativity and Holography

The Structural Law states that physical observables are modulated by local vacuum structure through a universal dimensionless coupling constant ϕ. In the semiclassical regime, this vacuum structure correlates with spacetime curvature, allowing the structural index S(x) to be modeled as a function of the Ricci scalar. This paper presents a complete derivation of the Structural Law from General Relativity, causal coherence, and holographic entropy bounds. The coupling constant ϕ = 1/2√2 ln 2 emerges from first principles via entropy saturation and geometric projection at the Planck scale. No empirical parameters are introduced, and all derivation steps are made explicit. This theory provides a unified foundation for previously unexplained observational anomalies using a single parameter-free law. The fundamental coupling is to the entanglement structure of the vacuum; spacetime curvature serves as a semiclassical proxy.