With quantum, you can go further: even if you’re instantiated in a universe that starts out perfectly rotationally symmetric, as long as we’re not modifying physics to guarantee the symmetry is preserved over time, then the vastly overwhelming majority of quantum behavior will not have spurious correlation between rotationally symmetric positions, meaning that the rotational symmetry will almost instantly be lost at the nanoscale, and then it’s just some chaos away from having macroscopic effects. You can think about this under the many worlds interpretation: there’s a vanishingly small fragment of the wavefunction where every time an atom collides with another atom in one position, the quantum random portion of the collision’s outcome is exactly matched in the rotationally symmetric position. From the outside, this means that even picking which classical slice of the wavefunction is still symmetric requires you to specifically encode the concept of symmetry in your description of what slice to take. (I don’t quite know the quantum math necessary to write this down.) From the inside, it means an overwhelmingly high probability of nanoscale asymmetry, and from there it’s only chaos away from divergence.
I expect it would still take at least a few seconds for the chaos in your brains to diverge enough to have any detectable difference in motor behavior. If we were instead running the same trial twice rather than putting two instances in the same room across from each other, I’d expect it would take slightly longer to become a macroscopic difference, because the separate trials don’t get to notice each others’ difference in behavior to speed up the divergence.
Importantly, assuming many worlds, then the wavefunction will stay symmetric—for every classical slice of the wavefunction (“world”) where person on side A has taken on mindstate A and side B has mindstate B, there’s another world in which person on side A has mindstate B and vice versa. This is because if many worlds is correct then it’s a “forall possible outcomes”, and from the outside, it’s not “randomized” until you pick a slice of this forall. It’s unclear whether many worlds is physically true, so this might not be the case.
With quantum, you can go further: even if you’re instantiated in a universe that starts out perfectly rotationally symmetric, as long as we’re not modifying physics to guarantee the symmetry is preserved over time, then the vastly overwhelming majority of quantum behavior will not have spurious correlation between rotationally symmetric positions, meaning that the rotational symmetry will almost instantly be lost at the nanoscale, and then it’s just some chaos away from having macroscopic effects. You can think about this under the many worlds interpretation: there’s a vanishingly small fragment of the wavefunction where every time an atom collides with another atom in one position, the quantum random portion of the collision’s outcome is exactly matched in the rotationally symmetric position. From the outside, this means that even picking which classical slice of the wavefunction is still symmetric requires you to specifically encode the concept of symmetry in your description of what slice to take. (I don’t quite know the quantum math necessary to write this down.) From the inside, it means an overwhelmingly high probability of nanoscale asymmetry, and from there it’s only chaos away from divergence.
I expect it would still take at least a few seconds for the chaos in your brains to diverge enough to have any detectable difference in motor behavior. If we were instead running the same trial twice rather than putting two instances in the same room across from each other, I’d expect it would take slightly longer to become a macroscopic difference, because the separate trials don’t get to notice each others’ difference in behavior to speed up the divergence.
Importantly, assuming many worlds, then the wavefunction will stay symmetric—for every classical slice of the wavefunction (“world”) where person on side A has taken on mindstate A and side B has mindstate B, there’s another world in which person on side A has mindstate B and vice versa. This is because if many worlds is correct then it’s a “forall possible outcomes”, and from the outside, it’s not “randomized” until you pick a slice of this forall. It’s unclear whether many worlds is physically true, so this might not be the case.