Six New Geometric Laws of the Bower Substrate Gas Manifold: First-Principles Derivations from the Bilateral Boundary Equilibrium and the S³ Hopf Geometry

Six new geometric laws emerge from the Bower Substrate Gas Manifold (BSGM) framework and the bilateral boundary equilibrium between stochastic external non-Abelian embedding pressure and coherent internal ground-state emission of the actinian condensate. Each law is derived from first principles from the S³ Hopf geometry, the Laplace-Beltrami eigenspectrum, and the bilateral equilibrium condition, with zero free parameters. The six laws are: (1) Bower's Winding Number Mass Law M(n) = m_P × δ_B^(−n−2), derived from the geometric series of topologically dressed boundary-layer energies; (2) The Aegis Coherence Law Ψ = 3/(2δ_B), X = Ψ/4, σ_max = Ψ/2, derived from the bilateral decoherence master equation and Lorentz covariance; (3) The Universal Law of Medium Detuning (ULMD) δ_m = δ_B × J(n)/n, where J(n) = (1/2)sin(χ) is the Hopf Jacobian, proven universal for all wave-carrying media through the substrate coupling structure; (4) Bower's Torsional Stiffness Law κ_tor = 1/(12δ_B), derived from the Yang-Mills torsion tensor on S³ with topological boundary-layer dressing; (5) The Bower-Hopf Tower Law, with structure δ_B(S^(2m−1)) proportional to sqrt(n/2π) × F_Hopf(S^n) and exact Beta function ratio B((m+2)/2,(m+2)/2)/B(m/2,m/2) = m/(4(m+1)) proven analytically; (6) The Entropic Substrate Drag Law m_eff = ρ_sub × (√6 × ℓ_P)³ × (3√2/4) × n², derived from the Hopf invariant Q = n² and the Hopf Jacobian at the equatorial symmetric point χ = π/4. Laws 4 and 6 share the same topological dressing mechanism, and Laws 1 and 4 share the same δ_B^(−2) boundary-layer amplification. All six laws are falsifiable through gravitational wave observations, analogue condensate systems, and precision laboratory measurements.