>>4401628 (lb)
Looking into what actions the Senate might take in the transition between the outgoing and incoming sessions of Congress, here's a page from the Senate website:
https://www.senate.gov/artandhistory/history/common/contested_elections/procedures_contested_elections.htm
"As it considered election cases over the years, the Senate developed a series of informal precedents to guide its actions. For example, if a senator-elect arrived with credentials that appeared valid on their face and were signed by the proper state authorities, that individual typically would be permitted to take his seat even if a challenge to the election had been filed with the Senate."
"Generally, a senator with the proper credentials against whom a challenge had been filed would be seated "without prejudice." Under this arrangement, if a committee investigation later determined that for some reason the individual was not entitled to a seat, he could be "excluded" from the Senate by a simple majority vote, as opposed to having to be expelled, which required a two-thirds majority."
From a process standpoint, I draw attention to the fact that "expulsion" is a constitutional provision for removing a senator. So when the article talks about expulsion, the context is a scenario where a senator is seated, and then later removed. The article specifically contrasts this scenario with the common practice of letting a contested election winner serve "without prejudice" where the senate retains the option of removing the member by a simple majority; in effect, this is deferring the original "yes or no, do we accept this member?" decision until after a senate committee has reviewed the case and made a recommendation (while letting the "senator" serve in a provisional basis in the interim.)
The implication is that the Senate can refuse to seat an incoming member by a simple majority vote (NOT requiring a two-thirds vote, as would be required for explusion.) This is restricted to the constitutional scope of the Senate judging the returns and qualifications of its members; in contrast, expulsion is a wide-open freedom of the Senate not restricted to any particular cause (which accounts for the higher two-thirds standard, the same threshold as a conviction of impeachment.)
For reference, from Constitution Art I Section 5: "Each House shall be the Judge of the Elections, Returns and Qualifications of its own Members … Each House may determine the Rules of its Proceedings, punish its Members for disorderly Behavior, and, with the Concurrence of two-thirds, expel a Member."
I took anon's idea of looking at the membership of the Senate counting only continuing classes (that is, assuming all Class I seats that expire in January are initially vacant), and it looks like about 43 R - 24 D. This is just short of two-thirds, but that doesn't matter, since it appears the constitutional standard for judging returns and qualifications of the incoming class is only a simple majority. So if the Senate had a good reason for viewing some of the election returns as invalid, the Senate could simply refuse to fill those seats. (Without a good reason accepted by the public, this would appear to be partisan bad behavior, but with public evidence …)
This only applies for the narrow issue of seating senators. A two-thirds majority of the Senate would be much more exciting for removing cabal members in the federal judiciary (would take first an impeachment vote by the house, then two-thirds majority in the Senate to convict.) Unless the outgoing House pulls impeachment articles out of a hat in the next couple of weeks, that's not on the near-term agenda. And even so, the transition senate looks like it's just short of two-thirds R.