PaperVII: Gravity as Elastocapillarity on the S3 Membrane

An exact structural identity is established between the gravitational radius R_g = GE/c⁴ and the elastocapillary length L_c = γ_s/E_s of continuum mechanics on the S³ membrane of Quantum Geometrodynamics (QGD). The surface tension is defined as γ_s = E/ℓ_P² (condensate energy per Planck area, fixed by Hopf quantisation with first Chern number c₁ = 1); the elastic modulus is E_s = F_P/ℓ_P² = ρ_P c² (Planck stiffness, fixed by the BPS vacuum condition); their ratio yields L_c = E_s/F_P = GE/c⁴ = R_g with no free parameters. From this identification the Kerr–Schild metric is derived as a mechanical necessity through three steps: (i) the condensate's point-like topology forces the elastocapillary stress tensor to be rank-1, σ_μν = A(r) u_μ u_ν; (ii) the null-confinement axiom requires u_μ ∝ k_μ with k_μ k^μ = 0; (iii) Hooke's law h_μν = 2σ_μν/E_s then yields the exact Kerr–Schild form h_μν = (2Φ/c²) k_μ k_ν. The null condition k_μ k^μ = 0 annihilates all nonlinear corrections to the Einstein equations, providing a mechanical explanation for why the Kerr–Schild ansatz generates exact solutions of nonlinear General Relativity. A duality theorem resolves the apparent tension between the elastocapillary framework and the Tikbon gauge theory (Paper V): the Tikbon field A_μ is the strain-rate potential of the S³ membrane; its field strength F_μν is the strain-rate tensor; the Tikbon Lagrangian −(1/4)F_μν F^μν is the elastic energy density (1/2)E_s(∇w)²; and the Dirac–Born–Infeld extension (Paper II) is the exact nonlinear elasticity action. The gauge-theory and continuum-mechanics descriptions are mathematically dual formulations of a single physical structure. The graviton is identified with the quantised phonon of the S³ elastic membrane. The framework is extended to rotating condensates: the ring-source geometry produces an anisotropic surface tension γ_s(r,θ) ∝ r/Σ, and the complete Kerr metric is assembled component by component from Hooke's law applied to the resulting null stress tensor. The horizon function Δ = r² − R_S r + a² is derived from the membrane velocity budget; Δ = 0 corresponds to the surface on which the normal velocity component vanishes. Four condensed-matter phenomena acquire precise gravitational counterparts: (i) frame-dragging is the gravitational Marangoni effect, driven by an angular surface-tension gradient; (ii) the gravitational capillary number Ca_g ∼ v²/c² classifies Newtonian (Ca_g ≪ 1) versus gravitational-wave (Ca_g ∼ 1) regimes; (iii) the MOND transition radius r_MOND = √(GM/a₀) emerges as a capillary-length crossover; (iv) cosmic censorship (M ≥ m_P) follows from Plateau–Rayleigh stability of the ring condensate. The black-hole formation threshold E ≥ E_P is identified as the elastocapillary instability of the S³ membrane.