Linking Spacetime-Foam Uncertainty to Log-Density Tension Gravity: The Ng–Kuşçu Correspondence

This work establishes a theoretical and phenomenological correspondence between Y. J. Ng’s spacetime-foam uncertainty relation,
δl≳l1/3lP2/3\delta l \gtrsim l^{1/3} l_P^{2/3}δl≳l1/3lP2/3, and the Log-Density Tension Gravity (QFT) framework developed by Bülent Kuşçu.
The model interprets gravity not as a fundamental interaction but as an emergent effect arising from spatial gradients of quantum-foam density, described by the logarithmic potential
Φ(ρk)=cg2ln⁡(ρk/ρ0)\Phi(\rho_k) = c_g^2 \ln(\rho_k / \rho_0)Φ(ρk)=cg2ln(ρk/ρ0).
This single-field approach reproduces both Newtonian and MOND-like limits, achieving excellent agreement with the SPARC galactic rotation curves
(R2≈0.99R^2 \approx 0.99R2≈0.99, 175 galaxies, >3000 points).
At cosmological scales, the FORK v5 embedding yields a dynamic vacuum correction
Λeff(z)=Λ0[1+αln⁡(1+z)]\Lambda_{\text{eff}}(z) = \Lambda_0 [1 + \alpha \ln(1+z)]Λeff(z)=Λ0[1+αln(1+z)],
reducing the Hubble and S8S_8S8 tensions by up to 85%.
The model remains compatible with General Relativity in the weak-field limit and predicts measurable deviations in precision experiments,
including atom-interferometric phase shifts
(δϕ∼10−8\delta \phi \sim 10^{-8}δϕ∼10−8 rad / √Hz) and Casimir corrections near the sub-millimeter scale.
This unified micro–macro framework bridges quantum-foam phenomenology, galactic dynamics, and cosmology under a single, testable formulation.