The Survival Law: How a Single Dimensionless Number Governs Every Interface in Living Systems

Every living system is a network of impedance interfaces. At each interface, a single dimensionless number — the reflection coefficient Γ = (Z₂−Z₁)/(Z₂+Z₁) — determines how much energy passes through and how much is wasted. Starting from the First Law of Thermodynamics and the Maximum Power Transfer Theorem, combined with the evolutionary postulate that biological systems are under selection pressure to minimise interfacial energy waste, we derive three inviolable constraints: C1 (Energy Conservation), C2 (Phase-Boundary Law), and C3 (Geometric-Mean Law). These constraints, applied through Γ alone with no fitted parameters, are sufficient to predict:• The emergence of all biological interfaces as impedance boundaries• The existence of transition tissues as quarter-wave bridges• The unified fibre-bundle rule N∝f(Z) governing nerves, muscles, skin collagen, and blood vessels• The bilateral brain topology as a C1 supply-chain constraint• The energy ceiling Zac ≈ 1.60 MRayl that forces the all-or-nothing action potential• Aging, osteoporosis, phantom limb pain, and dementia as consequences of Γ-drift All results are verified by a companion physics engine bodyenginev2.py — 60 computational consistency checks yield a 100% pass rate. Keywords: Reflection coefficient, Impedance matching, Minimum Reflection Principle, Thermodynamics, Biophysics, Neural impedance