Why does the Verhulst logistic equation dp/dt = αp(1−p) appear across population ecology, evolutionary genetics, quantum condensation, social diffusion, and neural dynamics? This paper identifies four minimal axioms—probability simplex (A1), quadratic separable interaction (A2), supercritical regime (A3), and dissipative continuous dynamics (A4)—and proves that any system satisfying all four must exhibit logistic growth for its dominant mode (Theorem 1, exact in the symmetric case with bounded correction otherwise). The free-energy functional has a unique minimum (Theorem 2), each axiom is independently necessary (Theorem 3: four axiom-removal families, six explicit counterexample cases), and the results are structurally stable under perturbation (Theorem 4). Of the four axioms, three are inherited from the Ω₀/Ω₁ framework of Papers 8–10; only A2 (quadratic separability) is the sole non-trivial condition. This paper serves as the mathematical corridor connecting the consciousness theory (Papers 8–10) to the genetic code theory (Evolution 2.0 Prize submission), explaining why both necessarily produce logistic dynamics. Paper 11 in the Information Physics series.
Taekyung Lee (Mon,) studied this question.