On the Geometric Origin of Time, Uncertainty, and the Dark Sector: From the Quaternionic Vacuum to Observable Cosmology

Quantum Geometrodynamics (QGD) presents a framework in which the primordial vacuum is modelled by the flat quaternionic space (ℍ, δ) with Euclidean isometry group ISO(4). A localisation event breaks ISO(4) → SO(4)ₚ; the massless Goldstone modes expand at the characteristic vacuum speed c, producing the wavefront Στ ≅ S³, identified with the spatial universe. The algebraic structure of ℍ forces the holonomy of the frame bundle to SU(2)_L via SO(4) ≅ (SU(2)_L × SU(2)_R)/ℤ₂, making the vacuum hyper-Kähler and therefore a self-dual gravitational instanton. The Gibbons–Hawking classification selects the self-dual Taub–NUT (SDTN) instanton as the unique nucleation topology; the BPS condition M = n follows from W⁻ ∝ (M − n) = 0. Scale-invariance of (ℍ, δ) forces both radial and tangential speeds to equal c; the virial theorem applied to the closed isoclinic orbits uniquely determines Rᵤ = GMᵤ/c² and Rg = GM/c² without external input. From this single geometric object the framework derives, without adjustable parameters:(i) time as the Aut(ℍ)-invariant Goldstone mode;(ii) the Minkowski metric from Re(q²) = w² − |v|², with signature fixed by i² = −1 and no analytic continuation;(iii) the uncertainty principle from the symplectic geometry of T*S³ with Hopf quantisation c₁ = 1;(iv) spatial flatness Ωk = 0 from the Gauss–Codazzi identity at null expansion speed;(v) the trade-off law rc·Rg = ℓP² from isoclinic angular momentum conservation;(vi) ΩΛ = 2/3 from the BPS Back-EMF mechanism; Ωg = 1/3 by subtraction; Ωm = 1/π from the universal background acceleration a₀ = c²/(2πRᵤ). All results agree with Planck 2018 to within 3%. The principal falsifiable prediction is a₀(z) = a₀(1 + z), testable with JWST.