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: (i) time as emergent from uniform motion along a fourth spatial dimension at speed c; (ii) the uncertainty principle from the symplectic geometry of the cotangent bundle T*S³; (iii) the Minkowski metric as a consequence of 4D momentum conservation rather than a postulate; and (iv) 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 Ω_k = 0. Dark energy emerges as vacuum strain energy with equation of state w = -1/3, predicting Ω_Λ = 2/3. Dark matter effects arise from a universal background acceleration a₀ = c²/(2πR_U) = cH₀, predicting Ω_m = 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) = cH₀(1+z).

This is a condensed companion paper to the full QGD monograph (DOI: 10.5281/zenodo.18274078).