A Single Field Generates Both General Relativity and Quantum Mechanics: Unified Foundation from the Scalar Vibrational Field

We demonstrate that a single fundamental scalar field Sₕ(x,t) generates both Einstein's General Relativity and quantum mechanical behavior as complementary limits. Starting from the master equation Sₕ + ∂ V/∂ Sₕ = 0, we derive the emergence of classical spacetime geometry and canonical quantization identically from the same field. The theory reproduces fundamental constants (α⁻¹, G, and mW) with relative errors <10⁻⁴ and eliminates ghost degrees of freedom via a degeneracy condition. This work provides the unified foundation for the Unified Vibrational Force Theory (UVFT).