The π-Periodic 22/7ths Dimension: A Quantum Gravity Framework for Dark Energy

We propose a novel 4+1-dimensional quantum gravity framework incorporating a compactifiedextra dimension, τ , with a periodicity of π (to 22 decimal places), symbolicallytied to the rational approximation 22/7. Integrated into loop quantum gravity (LQG), theτ -dimension’s compactification radius R ≈ 5.6 × 10−11 m is derived to match the observeddark energy density (ρΛ ≈ 2.3 × 10−3 eV/cm3). A scalar field φ(τ ) with an exponential potentialdynamically generates this density, reducing fine-tuning compared to static vacuumenergy models. Numerical solutions using the Runge-Kutta method confirm a slow-roll behavior,yielding an equation of state w ≈ −0.999, consistent with cosmological observations(type Ia supernovae, CMB, BAO). We discuss implications for quantum gravity, particlephysics, and future observational tests with DESI and Euclid, offering a new perspective onthe cosmological constant problem.