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

We present a unified geometric framework — Quantum Geometrodynamics (QGD) — in which time, the Heisenberg uncertainty principle, Minkowski spacetime structure, and the cosmological dark sector all emerge as necessary consequences of a single postulate: the primordial vacuum possesses quaternionic (ℍ) structure. The Skolem–Noether theorem then uniquely determines spacetime geometry as a self-dual gravitational instanton with S³ spatial topology. From this foundation, we derive: Time as emergent from uniform motion along a fourth spatial dimension at speed c; The uncertainty principle from the symplectic geometry of the cotangent bundle T*S³; The Minkowski metric as a consequence of 4D momentum conservation rather than a postulate; The quantum-gravitational trade-off law rc · Rg = ℓP² as a consequence of isoclinic angular momentum conservation (Liso = ℏ); The complete dark sector with zero free parameters. The "Theorem of Dynamic Flatness" demonstrates that the intrinsic curvature of S³ is exactly cancelled by the extrinsic curvature of light-speed expansion, yielding Ωk = 0. Dark energy emerges as vacuum strain energy with equation of state w = −1/3, predicting ΩΛ = 2/3. Dark matter effects arise from a universal background acceleration a₀ = c²/(2πRU) = cH₀/(2π), predicting Ωm = 1/π ≈ 0.318. All predictions agree with Planck 2018 observations to within 1%, achieved with zero adjustable parameters. The framework resolves the 10¹²⁴ cosmological constant problem and provides falsifiable predictions, most notably the redshift evolution of the MOND acceleration scale: a₀(z) = a₀(1+z). Keywords: quaternionic geometry, emergent time, uncertainty principle, dark energy, dark matter, MOND, cosmological constant problem, S³ topology Related papers: Dimensional foundation in Paper 0. Kerr metric derivation in Paper II. Particle spectrum in Paper III. Thermodynamic gauge theory in Paper IV. Gravitational mediator in Paper V. Dark matter resolution in Paper VI.