Paper posits ultradense asteroid 33 Polyhymnia has transuranic elements

How about just doing some more precise measurements first? I'm 99.99% certain that the high density estimate stops existing afterwards

Also, how exactly do you want to get an object without guidance or comms near enough to a main-belt asteroid?

This is all just ridiculous. Some theoretical physicist does some calculation around the Island of Stability, comes up with a high density, and tries to give his paper more relevance by finding an improbably dense astronomical object, and then wildly speculates that the two are related.

What are the chances that some rando asteroid consists of 100% of an element that we

1) don't know if it actually is stable

2) don't actually know the density of, besides a single paper about it

3) don't know any natural, plausible mechanism for synthesis

as opposed to it just being not well studied?

From the abstract of the article, it appears that it is simply presenting some numerical calculation that show that the element 164 would have a density in a certain range (36-68 g/cm3) given some models of the nucleus and of the electron shells. The numerical methods could be innovative enough that the article is worth publishing.

But 33 Polyhymnia being rich in this element is complete nonsense. There is no plausible scenario where an asteroid would be composed exclusively out of such an element. Heavy elements are created in supernova explosions. They are relatively rare. You can find them in small proportion among more frequent elements, such as iron, nickel, lead, etc. There is no scenario where you'd find only a superheavy element in isolation. Uranium has an average concentration of 2.8 parts per million on Earth.

Much, much more likely is that the estimation of the mass of 33 Polyhymnia was erroneous. Here's why: you can't measure the mass of a celestial object by how far it moves around the Sun. All objects will move with the same speed on the same orbit. You can however determine the mass of the Sun by observing a single planet. Similarly, you can measure the mass of Jupiter by observing how fast its satellites go around it. You only need one satellite, but if you have more, it will only increase your confidence. You can estimate like this the mass of all the planets, with a very high accuracy.

But you can't estimate the mass of small celestial objects that don't have satellites. Instead you look for n-body interactions, and there are such in the asteroid belt. But for any asteroid there, the main forces are the gravitational attraction to the Sun and Jupiter. A distant third will be Mars, and occasionally asteroids feel the gravitational influence of their neighbors. But not very often, because distances between asteroids are actually astronomical.

So, the most likely explanation (virtually certain) is that whoever estimated the density of 33 Polyhymnia to be 75 g/cm3 was simply wrong. There was no subsequent confirmation. 75 g/cm3 is such an extraordinary claim, the it requires truly extraordinary evidence, and such evidence is entirely lacking.

One don't need to touch material, but need to hang close to it and measure acceleration with some conventional instrument (for asteroid distance will be about few meters and acceleration around millimeters/s^2).

Most known appearance in Clarke Odyssey 2001.

Popping the core out of the centre of a planet intact would be your bigger challenge, I think. A collision with enough force to crack open a planet down to its core is probably not going to leave the core crystal intact.

Question is, that at the moment not found any similar asteroids, so this is not common for them, to appear such way.

So, when we see something so unusual, we have two options - they could be just from more heavy elements, or their history was unusual.