Deriving the Einstein Equivalence Principle from the Fibonacci Cascade: Temporal Depth and Local Gravity in Fractal Mechanics

Abstract

The Einstein Equivalence Principle (EEP) — the equality of gravitational time dilation and gravitational acceleration — is a foundational postulate of General Relativity (GR), not derived from a deeper principle. We show that within Fractal Mechanics (FM), the EEP is a mathematical consequence of a single cascade parameter Im(𝜑), where 𝜑 = (1+√5)/2 is the golden ratio. In FM, the local flow of time and the local gravitational coupling both depend on Im(𝜑) with the same coefficient 𝛽 = 𝜑, derived analytically from the FM Lagrangian inter-level coupling term. The resulting formula, 𝐺_𝑙𝑜𝑐𝑎𝑙 ≈ 𝐺₀ (1 + 𝜑 · 𝛿𝐼𝑚(𝜑)/𝐼𝑚(𝜑)₀ ) = 𝜏_𝑙𝑜𝑐𝑎𝑙/𝜏₀ , directly encodes the EEP without postulating it. We derive two testable predictions: (i) residual correlations between the measured value of Newton’s constant G and local crustal geological density, unaccounted for by existing Cavendish experiments; and (ii) a small but specific correction 𝛿𝐹𝑀 to the gravitational redshift near compact objects, testable with future pulsar timing and atomic clock experiments.