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I would bet that a subtle error in the measured distance or time is more likely than that their ratio – the inferred speed – exceeds Einstein's speed limit.
This was my first thought when I read about the results, too. I find it hard to believe, however, that they would report such results without checking and rechecking their sources of error many times. If so, shame on them. Its an obvious thing to think; I know if I thought of it, then scientists who work in the field thought of it, too.
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They have rechecked their calculations more than 15000 times, but to my knowledge, they have not yet repeated the experiment (I could be wrong). It doesn't matter if all 15000 calculations are right if the measurements are wrong. That just means that one small error in their procedure or measurements spawned 15000 misleading calculations.
I think that, to sway the skeptics, they would have to repeat this at least a few dozen times (ideally a few thousand times) and describe their methodology in great detail. Although I imagine that the reason they haven't done repeated trials is because such an experiment is likely to require much preparation and be quite expensive.
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I read that they had a similar result a few years ago at Fermi, although the experimental uncertainty in those trials was apparently quite large. I am of the belief right now that these results are probably erroneous, but by the same token I hope they aren't. After all, it was the measurement of discrete light emissions from black bodies that led to the entire field of modern physics, which I've been told was looked on at the time with skepticism. If true, this could be an equally important observation.
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I've read the same thing about the Fermilab experiment. The problem here is that CERN has the top of the line equipment, so no other current research lab can even come close to trying to recreate this. That presents probably the biggest hurdle in trying to independently verify their findings.