Period coincidence of Planet Nine with furthest Kuiper belt objects A lead scientist of the Planet Nine theory said that "The period with which it goes around the sun is a rational multiple of the periods of all the furthest Kuiper Belt objects"

To what degree of precision can Konstantin Batygin make that kind of measurement? Has it been backed up by other physicists with a degree of certainty?

He also says that they think it's near it's aphelion "because it hasn't been seen, so it's far away", not because of the above period coincidence, why can't they say where it is more convincingly?

Does that mean that they have very strong mathematical evidence for Planet Nine?
I did a little digging and I found this article.

It's worth noting that the initial publication wasn't about resonances but the fact that all 6 objects came from the same side of the sky, from somewhat similar directions, which, given that they were the 6 most distant at the time, is statistically unusual and it was both similar direction and similar inclination to the plane, so the statistical improbability was high enough to investigate.  Orbital resonance wasn't the primary argument, it was tacked on later as a secondary.

From the article:


  Their analysis also offered suggestions as to what kind of resonance
  the planet has with the KBOs in question. Whereas Sedna’s orbital
  period would have a 3:2 resonance with the planet, 2010 GB174
  would be in a 5:2 resonance, 2994 VN112 in a 3:1, 2004 VP113
  in 4:1, and 2013 GP136 in 9:1. These sort of resonances are
  simply not likely without the presence of a larger planet.
  
  “For a resonance to be dynamically meaningful in the outer Solar
  System, you need one of the objects to have enough mass to have a
  reasonably strong gravitational effect on the other,” said the
  research team. “The extreme Kuiper belt objects aren’t really massive
  enough to be in resonances with each other, but the fact that their
  orbital periods fall along simple ratios might mean that they each are
  in resonance with a massive, unseen object.”
  
  But what is perhaps most exciting is that their findings could help to
  narrow the range of Planet 9’s possible location. Since each orbital
  resonance provides a geometric relationship between the bodies
  involved, the resonant configurations of these KBOs can help point
  astronomers to the right spot in our Solar System to find it.


now, I want to point out that it's them, not me who did the typos on 2 of the objects, I'm pretty sure (based on Wikipedia and the chart below) that it should be 2004 VN112 (not 2994 VN112) that has the 3:1 and 2012 VP113 (not 2004 VP113) that has the 4:1.

This chart below also has a typo, it should be 2010 GB174 not 2012 GB174.  I guess nobody proofreads letter/number assignments.   Correct names will help if anyone wants to run the numbers see how see how close the resonances come to each other.



As to the degree of statistical certainty?   If you're a good enough statistician, you can run the numbers.   3/2, it seems to me, would be the most common resonance based on Jupiter's abundant Hildas and the Plutinos and only 1 of the 5 potentially resonant objects has a 3/2, but 3/2 is also very far from the sun and hardest to detect, so that's not a deal breaker.  

I'd guess that the degree of certainty from these orbital resonances is OK but not great - but that's entirely a guess.   I also don't know enough about astronomy to know how close numbers have to be to provide some evidence of orbital resonance.  Jupiter's resonant moons, for example, if you check their orbits, are off by 1 or 2 percentage points from each other.