  Using the powerful language of geometric algebra, we present an
observationally symmetric derivation of the strong correlations predicted by
the entangled singlet state in a deterministic and locally causal model,
usually also referred to as a local-realistic model, in which the physical
space is assumed to be a quaternionic 3-sphere, or $S^3$, available as the
spatial part of a solution of Einstein's field equations of general relativity,
and compare it in quantitative detail with Bell's local-realistic model for the
singlet correlations set within a flat Euclidean space ${\mathrm{I\!R}^3}$.
Since the quantitatively detailed expressions of relative-angle-dependent
probabilities of observing measurement outcomes for Bell's local model do not
seem to have been fully articulated before, our novel analysis exploiting the
non-commutative properties of quaternions, in addition to allowing the
comparison with the quaternionic 3-sphere model, may also provide useful
comparisons for other less compelling local-realistic models, such as those
relying on retrocausality or superdeterminism. Apart from the conservation of
zero spin angular momentum, the key attribute underlying the strong singlet
correlations within $S^3$ in comparison with Bell's local model turns out to be
the spinorial sign changes intrinsic to quaternions that constitute the
3-sphere. In addition, we also discuss anew a macroscopic experiment that can,
in principle, test our 3-sphere hypothesis.
