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Could the Heisenberg Uncertainty Principle turn out to be false?

While investigating the EPR Paradox, it seems like only two options are given, when there could be a third that is not mentioned - Heisenberg's Uncertainty Principle being given up. The setup is this ...
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Relativity and Entanglement

Say we have two particles which are entangled so that they have opposite spins. If one is up, the other is down. They are sent off to two spatially separated observers A and B. Both observers can ...