The Universal Law of Medium Detuning (Bower Principle) A Universal Principle for Wave Propagation in Diverse Media

Abstract The Universal Law of Medium Detuning (Bower Principle) states that every wave-carrying medium possesses a characteristic dimensionless detuning constant δₘ governed by the master equation δₘ = Σ αᵢ · (Xᵢ − X₀ᵢ). All sensitivity coefficients are derived from the single geometric Bower Constant δ_B = 0.01502435758831411516… of Thyra-Hopf Detuning Theory. This Version 3.0 provides complete first-principles derivations of three geometric overlap factors from the SU(2)ₗ × SU(2)ᵣ structure of the l = 1 eigenspace: C_geom = 1 + 4δ_B, C_Cas = 1 + 8δ_B, and C_fiber = 1 + 6δ_B. The exact Hopf Jacobian J_H = ½ sin χ is used to derive the second-order detuning tensor β_{ij} = −αᵢαⱼ/δ_B and the coordinate map χ = arcsin(δₘ/δ_B). Lorentz invariance is established as a rest-frame scalar. The framework is explicitly linked to the Six Geometric Laws and extended to the Vacuum Catastrophe resolution via the terminal conformal factor Ω² = 1/16. Explicit projections are given for seawater, atmosphere, biological tissue, solids, plasma, and vacuum. All results follow from the single irreducible topological invariant δ_B with zero free parameters.