Paper III: Topological Stability of Membrane Condensates, the Origin of Spin, and the Eective Equation of State

Within the Quantum Geometrodynamics (QGD) framework, we establish that particles are topologically stabilised condensates of the S³ membrane. The expansion of the universe at speed c exerts a dilution pressure on every localised energy concentration; we prove that only configurations carrying quantised isoclinic angular momentum Liso = nℏ (n ∈ ℤ, n ≠ 0) resist this dilution, with the minimum stable condensate at |n| = 1. The stabilisation mechanism is topological: the winding number on the Hopf fibre S¹ ↪ S³ → S² cannot change under continuous deformations, providing an exact analogue of vortex stability in superfluid helium. From this foundation we derive: The identification of mass with the energetic cost of resisting cosmological dilution, yielding E = mc² as a geometric identity; The classification of excitations by winding number (n = ±1: stable particles/antiparticles; |n| > 1: unstable resonances); A corrected ontology for the Compton wavelength rc as a symplectic radius in T*S³ rather than a spatial extent on S³; The Higgs field as the depth-determining field of the condensate, with the Mexican-hat potential derived from the spectral action on S³; The resolution of the equation-of-state tension: the intrinsic value w = −1/3 produces a luminosity–distance relation dL(z) = (c/H₀)(1+z)ln(1+z) that, when fitted with ΛCDM templates, yields an apparent wfit ≈ −1.0, consistent with Planck 2018; CP violation as a geometric consequence of quaternionic anti-commutativity (ij ≠ ji), proving that matter–antimatter asymmetry requires exactly three generations; The Yukawa coupling formula y₃ = (1 + e−1/α)/2 from spectral overlap integrals, reducing the mass hierarchy to a single computable parameter. All results follow from the QGD axioms with zero free parameters. The principal falsifiable prediction is the high-redshift luminosity distance: QGD deviates from ΛCDM by ~2.3% at z = 3, testable with LSST/Rubin and the Roman Space Telescope. Keywords: topological stability, Hopf fibration, winding number, particle spectrum, Higgs mechanism, CP violation, baryogenesis, Yukawa couplings Related papers: Foundational axioms in Paper I. Kerr metric in Paper II. Thermodynamic gauge theory in Paper IV. Dimensional foundation in Paper 0.