Scale-Invariant Fractal Spacetime: A Unified Geometric Theory of Gravity and Quantum Mechanics

We present Scale-Invariant Fractal Spacetime (SIFS), a unified geometric field theory based on a 5-dimensional Randall-Sundrum bulk in which the fifth coordinate S is identified with the logarithm of the physical energy scale. Our 4D universe emerges as a 3-brane embedded in this bulk. The warping parameter k is treated as a phenomenological constant whose numerical value k approx 1/pi² (in units of MPl) is determined by requiring simultaneous consistency of three independent constraints: the observed electroweak/Planck hierarchy, the KK graviton mass gap, and Goldberger-Wise radion stabilisation. We note explicitly that this is a numerical consistency condition, not an analytic derivation. The theory predicts: (i) evolving dark energy driven by brane drift along S, consistent with DESI DR2 observations; (ii) Kaluza-Klein graviton excitations at 1-10 TeV accessible at FCC-hh; (iii) a log-periodic power-law tail with beta = 2pi/ (kL) approx 0. 165 superimposed on the standard gravitational-wave ringdown, detectable by LISA and Einstein Telescope. Quantitative comparisons with DESI DR2 BAO data and EHT M87* polarisation variability are presented at a qualitative level; precision numerical fits are deferred to companion work. We identify the principal open problems: a first-principles derivation of k, a complete 5D treatment of the particle horizon model, and full waveform templates for GW data pipelines.