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.
Yunus Emre Tikbaş (Tue,) studied this question.