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The abstract says 4e-15, not 4-12?

So does the paper. The paper also says this, right before explaining that it's a component of the ~4e-15 number, though:

>A conservative estimate of the height difference between clock and experiment contributes a gravitational red shift uncertainty of 7.7 × 10−17.

So, it appears the 4e-15 is not an actual measured value, but a synthetic value based on estimation and measurement.

Yeah that's right, the red shift is just one contribution to the total uncertainty.

They're all listed in table I and the 4.2 * 10^-15 just comes from adding them in quadrature.

When physicists say "estimate" in this context, they mean "take into account this fourth-order tiny effect by using an approximate theory of said effect with accuracy of a few percent, and then use 1.2 meters when it's really 1.0 so you overestimate by 10%".

It's like when you take a caliper and measure the width of an object, you typically neglect the error due to thermal expansion of the caliper since it's hotter/colder than when it was marked with gradations. If you're really anal about uncertainty (like these guys), you'd estimate the effect of this expansion and say "a conservative estimate says this contributes an uncertainty of 0.0003 mm".

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