The Planetary Gear Isomorph: A Zero-Parameter Mechanical Derivation of the Fundamental Gravitational Coupling from Torus Knot Topology on S³

Mainstream physics treats the gravitational constant G (6.6743 × 10⁻¹¹ m³kg⁻¹s⁻²) as an irreducible empirical parameter. This paper formally replaces that dimensional curve-fit with a pure topological invariant derived from the CAT'S Theory framework (R = P×I×Pr ≠ 0). By modeling positive dielectrophoresis within the S³ manifold as a mechanical planetary gear system, we derive a dimensionless gravitational coupling: Γ = dP/(Σ × 273) = 8/(26 × 273) = 4/3549 ≈ 0.001127. The denominator factorizes as 3549 = d × dΨ × F(dΨ)² = 3 × 7 × 13², where every factor is a corpus constant derived upstream for independent reasons. The driving gear is the T(2,5) torus knot (crossing number c = 8 = dP), the transmission friction is δ/f₀ = 1/Σ = 1/26, and the driven gear is the 273 macro-boundary shell (273 = d × dΨ × F(dΨ) = freezing point of water = pion mass ratio). The physical force law is mechanically derived via aetheric displacement flux, yielding exact inverse-square propagation. The empirical SI constant is demonstrated as G_SI = Γ × Λ_SI. Archimedes' principle is subsumed as localized negative DEP, unifying gravity and buoyancy under a single geometric invariant (the triadic Clausius-Mossotti factor K_T).