In it, he references a fascinating article by Lorenzo Iorio from the National Institute of Nuclear Physics in Pisa, Italy who does the math in a paper titled Constraints on Planet X and Nemesis from Solar System’s inner dynamics, Lorenzo Iorio, 2009. arXiv:0904.1562v1 [gr-qc] and sets the minimum distances to objects of various masses based on the orbital dynamics of the inner solar system. If these objects were closer then they would be causing detectable perturbations in the motions of the inner planets.
Iorio’s conclusions: A planet with the mass of Mars would be no closer than 62 AU. A planet with the mass of Earth would be no closer than 430 AU. A planet with the mass of Jupiter would be no closer than 886 AU. An object with the mass of a brown dwarf (75-80 Jupiters) would be no closer than 3,736 to 3,817 AU (about 0.6 light years). A red dwarf (0.075-0.5 solar masses) would be at least 3,793 to 7,139 AU (or 0.6 to 0.11 light years).
This is yet another nail in the coffin of the 2012 hoaxers’ claims. By this point the mass of iron nails exceeds the mass of the coffin by about 10:1.
UPDATE: Ian O’Neil has corrected some of the distances ( and I have reflected those corrections here). Also, Ian cites another a paper by David Jewitt, at the Institute for Astronomy, University of Hawaii, which indicates that anything as big as Jupiter would be detectible out to 2140 AU.