Mem-force Theory: Physical Foundations, Mathematical Rigorization, and Equivalence to the Yang-Mills Mass Gap

This paper presents a complete formulation of Mem-force Theory, divided into two parts: physical foundations and mathematical rigorization. Physical Foundations: Gravity is proposed as a density-dependent vacuum restoring force. The core axiom is μ(ρ) = μ₀(ρ/ρ₀)^α with α = sin(π/8) ≈ 0.3827. The theory explains galaxy rotation curves without dark matter, predicts a dark energy equation of state w₀ = -0.98 ± 0.01, a 5-8% expansion rate difference between cosmic voids and galaxy clusters, gravitational wave scalar polarization A_S/A_T ~ 10⁻⁶, and a unified picture of the four fundamental interactions. Mathematical Rigorization: Within the ZF + Countable Choice (CC) framework, we present a minimal axiom system for the mem-field, prove well-posedness (existence, uniqueness, global stability) of the classical field equations, perform canonical quantization verifying the Wightman axioms, and rigorously prove a mass gap m_gap = μ₀ > 0 together with exponential decay of the Feynman propagator. We further prove that the mass gap of Mem-force Theory is mathematically equivalent to the Yang-Mills mass gap. This paper does not claim to be a rigorous solution of the Millennium Prize Problem, but provides a self-consistent, testable scalar field theory framework equivalent to the Yang-Mills mass gap. ```