I suppose, having written
Even if that doesn't lead to a proof of impossibility, it certainly means it's not surprising that a search for general integer-separation point sets would have a hard time finding ones with a unit distance: almost none of the sets you consider will meet the distance pairing constraint.
, that I was more or less obligated to do the simple search that takes that constraint into account. With 10 mintues each of coding time (I think I got all the properness constraints in there) and run time, I find the following two examples: [1, 24, 23, 16, 24, 23, 16, 46, 32, 30] [1, 25, 25, 2, 25, 25, 2, 42, 24, 24] That second one has a second plane of mirror symmetry, apart from the necessary symmetry which just swaps the two distance-1 points. The search had no cleverness involved, beyond the planarity constraint that I mentioned this morning. Brute-force search over the 5-dimensional space where first you specify the distance from the two sep-1 points to the other three points (say x,y,z, constraining each of them to lie on one of three circles, all coplanar and concentric) and further specify d(x,y) and d(x,z). You can write down explicit coordinates for x, y WLOG, and there are only two choices for z, and the two possible distances between y and z can be written with square roots nested only two deep. Then just search in lex order until you happen across an integer for the one unspecified separation. Er, since I'm searching first based on the three circle radii, it's still possible that there is a solution with maximal edge length smaller than the 42 above. By the way, in this morning's I forgot to mention the observation that this problem (two points at unit separation) is exactly the same as looking for a single proper simplex with integer edge-lengths and an altitude equal to 1/2, which is perhaps a nicer formulation than the one about the perpendicular bisector plane. --Michael Kleber -- It is very dark and after 2000. If you continue you are likely to be eaten by a bleen.