A Scale-Invariant 5D Manifold: Unifying Quantum Probability and Galactic Rotations via Dependent Chronological Coordinates

This paper presents a novel 5-dimensional spacetime framework (M5) wherein the fifth dimension is defined as a dependent Chronological Decision Coordinate (D). By establishing D as a functional product of the traditional 4D spacetime coordinates (x), we resolve the mechanism of quantum wave-function collapse into deterministic, energy-conserving geometric timeline branches. Furthermore, by scaling this metric to cosmological dimensions via a 9-tier structural hierarchy, we introduce the dimensionless Saif Invariant (K_S) and the Saif Mass-Radius Bound (Lambda_S). This framework eliminates the requirement for non-baryonic Dark Matter and Dark Energy, proving that galactic rotation anomalies are an illusion generated by scale-invariant internal non-relativity and macro-scale quantum entanglement anchors.