Gravity from Temporal Gradients: The Fractal-Spectral Framework

We present a physical-frame scalar-tensor theory in which the fundamental scalar is the local temporal flow rate τ (x) = e^σ, coupled to a fractal texture tower Φₙ with geometric mass hierarchy (√2) ^−n. The theory is formulated in the physical frame where matter couples minimally, with a non-constant kinetic factor K (σ) = 6 − αe^−2σ and a confining temporal potential U₀ (σ) = Λ₀e^−λσ + Λb e^νσ. The central structural result is that the theory admits a stabilized constant-σ branch on which the field equations reduce exactly to Einstein gravity with an effective cosmological constant. On this branch, the local vacuum geometry is exactly Schwarzschild–de Sitter, all four classical GR tests are passed exactly, and the PPN parameter γ = 1. Away from the stabilized branch, the non-constant profile τ (r) = √ (1 − rₛ/r) satisfies □̃ (ln τ) = 0 on a Schwarzschild background (compatibility) ; the Newtonian acceleration can be heuristically written as g ≃ −c²∇ln τ in the weak-field regime. The full coupled proof that the non-constant branch admits exact Schwarzschild remains open. The fractal corrections from the texture tower produce three categories of testable deviations from general relativity. First, composition-dependent gravitational frequency shifts: different atomic species couple differently to texture modes, producing species-dependent clock rates measurable with differential atomic clock comparisons at the ~10⁻¹⁹ level — at the frontier of current optical clock capability. Second, log-periodic gravitational wave dispersion: modulations in GW propagation with the universal log-frequency ωf = 2π/ln √2 ≈ 18. 1, testable through stacking of LIGO/Virgo events. Third, element-resolved gravitational spectroscopy: atomic transition frequencies shifted by the local τ-gradient in species-dependent ways. The framework provides a microphysical mechanism — collective vibrational modification of local temporal flow — while reducing to standard GR on the stabilized branch. It is compatible with all currently measured gravitational phenomena. Open problems include: the full coupled Schwarzschild proof on the non-constant branch, the strong-field regime (Kerr), the first-principles determination of fractal coefficients βₙ, and the connection to quantum gravity.