  The exceptional Lie group E8 plays a prominent role in both mathematics and
theoretical physics. It is the largest symmetry group associated with the most
general possible normed division algebra, namely, that of the non-associative
real octonions, which --- thanks to their non-associativity --- form the only
possible closed set of spinors (or rotors) that can parallelize the 7-sphere.
By contrast, here we show how a similar 7-sphere also arises naturally from the
algebraic interplay of the graded Euclidean primitives, such as points, lines,
planes, and volumes, which characterize the three-dimensional conformal
geometry of the ambient physical space, set within its eight-dimensional
Clifford-algebraic representation. Remarkably, the resulting algebra remains
associative, and allows us to understand the origins and strengths of all
quantum correlations locally, in terms of the geometry of the compactified
physical space, namely, that of a quaternionic 3-sphere, S3, with S7 being its
algebraic representation space. Every quantum correlation can thus be
understood as a correlation among a set of points of this S7, computed using
manifestly local spinors within S3, thereby extending the stringent bounds of
+/-2 set by Bell inequalities to the bounds of +/-2\/2 on the strengths of all
possible strong correlations, in the same quantitatively precise manner as that
predicted within quantum mechanics. The resulting geometrical framework thus
overcomes Bell's theorem by producing a strictly deterministic and realistic
framework that allows a locally causal understanding of all quantum
correlations, without requiring either remote contextuality or backward
causation.
