Ah, I see. I saw that but disregarded it because if it's meant be an actual proof and not just a back of the envelope argument, it seems to be missing a few steps. On the face of it, the blanket assertion that at least two faces must be stable is clearly contradicted by these current results. To be valid, Goldberg would needed at least to have established that his argument was applicable to all tetrahedra of uniform density, and ideally to have also conceded that it may not be applicable to tetrahedra not of uniform density, don't you think?

This piqued my curiosity, which Google so tantalizingly drew out by indicating a paper (dissertation?) entitled "Phenomenal Three-Dimensional Objects" by Brennan Wade which flatly claims that Goldberg's proof was wrong. Unfortunately I don't have access to this paper so I can't investigate for myself. [Non working link: https://etd.auburn.edu/xmlui/handle/10415/2492 ] But Gemini summarizes that: "Goldberg's proof on the stability of tetrahedra was found to be incorrect because it didn't fully account for the position of the tetrahedron's center of gravity relative to all its faces. Specifically, a counterexample exists: A tetrahedron can be constructed that is stable on two of its faces, but not on the faces that Goldberg's criterion would predict. This means that simply identifying the faces nearest to the center of gravity is not sufficient to determine all the stable resting positions of a tetrahedron." Without seeing the actual paper, this could be a LLM hallucination so I wouldn't stand by it, but does perhaps raise some issues.