The Rozier Foam Model: A Geometric Framework for Gravity, Dark Matter, and the Cosmological Constant

Title The Rozier Foam Model: A Geometric Framework for Gravity, Dark Matter, and the Cosmological Constant Summary This preprint presents a unified classical framework in which spacetime is modeled as a 3-dimensional interface embedded in a 4-dimensional bulk fluid with surface tension and quantized vorticity. The cosmological constant Λ, Newton's constant G, and dark matter phenomenology emerge from a single geometric principle. Key Results 1. Zero-Parameter Prediction of Λ Derives critical dimension d_H = 2 + 1/(8π) ≈ 2.0398 from Einstein's coupling constant 8πG/c⁴ Identifies d_H with the optimal D₄ lattice packing in 4D Predicts Λ = d_H H₀²/c² to 98.8% accuracy with zero adjustable parameters 2. Derivation of Newton's Constant G emerges from fractal adhesion mechanics between vortex-induced surface deformations Achieves 95% agreement with CODATA value through multifractal corrections Coherence area A_quantum ~ 1 m² derived from d_H (not fitted) 3. Hubble Tension Resolution Proposes geometric phase transition from disordered (φ ≈ 0.46) to ordered (φ_D₄ ≈ 0.617) foam Accounts for ~4-6% of the observed ~8% H₀ discrepancy Predicts power-law H(z) ∝ (1+z)^0.48 at z < 2 4. Dark Matter from Topological Flux Galactic rotation curves from codimension-2 vortex sheets Derives Donato's universal core-halo relation ρ₀r₀ = 2σ/c² Predicts sharp subhalo cutoff at ~10⁶ M_☉ 5. Connection to Quantum Gravity d_H ≈ 2.04 matches critical dimensions from Loop Quantum Gravity, Causal Set Theory, Asymptotic Safety, and Causal Dynamical Triangulations Proposes d_H = 2 + 1/(8π) as universal critical dimension for emergence of classical spacetime Falsifiable Predictions (5-10 year timeline) H(z) power-law scaling (Euclid 2028) ξ(z) evolution from ~2.03 to ~2.04 (Euclid 2028) Subhalo mass cutoff at 10⁶ M_☉ (LSST 2027) Material-dependent G at 0.1-0.5% level (white dwarf timing 2030) Primordial tensor ratio r < 10⁻⁴ (CMB-S4 2030) Discrete black hole ringdown overtones (LISA 2035+) Entanglement decoherence scaling τ ∝ ℓ^(5/3) (satellite QKD 2027-30) If any single prediction fails decisively, the framework is ruled out. Added Section 3.8 and Supplementary Material providing numerical validation of critical dimension d_H = 2.060 ± 0.014.