The Continuum — Part VA: Quantum Foundations: Uncertainty, Entanglement, and the Bridge to Schrödinger

We complete the Continuum's physical foundations by showing that its wave equation already contains, with no additional hypotheses, the characteristic features of quantum mechanics. Rewriting the equation in hydrodynamic form via the Madelung transformation, we derive the uncertainty principle in its exact form (Delta x * Delta p >= hbar/2, not merely ~ hbar) by combining the quantum potential that naturally emerges with the Cramer-Rao inequality from Fisher information - a general statistical result, not derived here, applied to a quantity that is genuinely ours. We show that entanglement emerges as topological non-separability of the field Psi for two bubbles created by the same perturbation: the correlations are structural, written into the topology at the moment of creation, not causal - no violation of the limiting velocity c. From the same (p, q) = (2, 3) knot geometry that underlies the rest of the model, we further derive the Tsirelson bound 2*sqrt(2) of Bell's inequality: the toroidal winding number p = 2 imposes a half-angle dependence on the knot's rotation, the same structure borne by fermionic spinors, from which the bound follows with no further assumptions. We finally derive the Schrodinger equation itself as the non-relativistic limit of the Continuum's field equation: the mass term 1/r_b^2, the same characteristic scale of the bubble used throughout this series, exactly cancels the Compton-frequency term generated by the oscillating ansatz, leaving the free Schrodinger equation. The Bohr magneton emerges, without postulating it, from the vortex's toroidal structure. This work constitutes Part VA of a series.