Part IV — Entropy, Regularity, and Bounds

• Hypocoercive Entropy: From Kinetic Mixing to the Full Swarm
• Mass, Hellinger, and Transport Convergence
• C³ Regularity of the Fractal Gas Fitness
• Smooth and Analytic Regularity of the Fractal Gas Fitness
• Logarithmic Sobolev inequalities and entropy convergence
• The Geometric Gas: Regularity, Stability, and Entropy
• Quantitative Error Bounds
• Parameter Constraints and Tuning