PaperVII: Gravity as Elastocapillarity on the S3 Membrane

An exact structural identity is established between the gravitational radius Rg = GE/c⁴ and the elastocapillary length Lc = γₛ/Eₛ of continuum mechanics on the S³ membrane of Quantum Geometrodynamics (QGD). The surface tension is defined as γₛ = E/ℓP² (condensate energy per Planck area, fixed by Hopf quantisation with first Chern number c₁ = 1) ; the elastic modulus is Eₛ = FP/ℓP² = ρP c² (Planck stiffness, fixed by the BPS vacuum condition) ; their ratio yields Lc = Eₛ/FP = GE/c⁴ = Rg 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/Eₛ 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) Eₛ (∇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 γₛ (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 − RS 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 Cag ∼ v²/c² classifies Newtonian (Cag ≪ 1) versus gravitational-wave (Cag ∼ 1) regimes; (iii) the MOND transition radius rMOND = √ (GM/a₀) emerges as a capillary-length crossover; (iv) cosmic censorship (M ≥ mP) follows from Plateau–Rayleigh stability of the ring condensate. The black-hole formation threshold E ≥ EP is identified as the elastocapillary instability of the S³ membrane.