Building upon the pioneering work of J. Bell [1] and an incredible result due to L. Hardy [2] it was shown that the probability of quantum entanglement of two particles is a maximum of 9.0169945 percent [2]. This happens to be exactly the golden mean to the power of five (?5) [3-7]. Although it has gone largely unnoticed for a long time, this result was essentially established independently in a much wider context by the present author almost two decades ago [3-6]. The present work gives two fundamentally different derivations of Hardy’s beautiful result leading to precisely the same general conclusion, namely that by virtue of the zero measure of the underlying Cantorian-fractal spacetime geometry the notion of spatial separability in quantum physics is devoid of any meaning [7]. The first derivation is purely logical and uses a probability theory which combines the discrete with the continuum. The second derivation is purely geometrical and topological using the fundamental equations of a theory developed by the author and his collaborators frequently referred to as E-infinity or Cantorian spacetime theory [3-7].

L. Hardy, “Nonlocality for Two Particles without Inequalities,” Physical Review Letters, Vol. 71 No. 11, 1993, pp. 1665-1668. doi:10.1103/PhysRevLett.71.1665

M. S. El Naschie, “A Review of E-Infinity Theory and the Mass Spectrum of High Energy Particle Physics,” Chaos, Solitons & Fractals, Vol. 19, No. 1, 2004, pp. 209-236.

M. S. El Naschie, “The Theory of Cantorian Spacetime and High Energy Particle Physics (an Informal Review),” Chaos, Solitons & Fractals, Vol. 41, No. 5, 2009, pp. 2635-2646. doi:10.1016/j.chaos.2008.09.059

J.-H. He, et al., “The Importance of the Empty Set and Noncommutative Geometry in Underpinning the Foundations of Quantum Physics,” Nonlinear Science Letters B, Vol. 1, No. 1, 2001, pp. 15-24.

M. S. El Naschie, “Quantum Collapse of Wave Interference Pattern in the Two-Slit Experiment: A Set Theo-retical Resolution,” Nonlinear Science Letters A, Vol. 2, No. 1, 2011, pp. 1-9.

J.-H. He, et al., “Quantum Golden Mean Entanglement Test as the Signature of the Fractality of Micro Spacetime,” Nonlinear Science Letters B, Vol. 1, No. 2, 2011, pp. 45-50.