Alright. Got the Blu-Ray. Decided I'd pause it during the scenes where Q analysed the ring. Here's what I've got:
Q finds three prints on the ring - Sciarra, White and a partial unknown print.
He analyses the unknown partial print and finds an unknown organic compound in it.
This compound turns out to be reidite (which, after a Google search, is a compound only found in certain impact craters on Earth).
The same compound, reidite, is also found in the toxicology reports of Silva, Le Chiffre, Greene and Sciarra.
Back in the hotel room with Bond and Swann, Q's computer screen reveals the unknown partial print belongs to Franz Oberhauser.
Alas, it's not the case that they all wore the same ring. In fact, we know White touched it in the film so it makes sense that the other two prints would belong to Sciarra and Oberhauser. The leader of SPECTRE handing it out to one of his members.
And if this compound reidite is super-rare, then it's not a massive leap to assume that everyone who has a trace of it is connected. What is a leap though is the fact that we don't know about the crater or the meteorite until much later in the film. Doh!
Edited by bonds_walther, 26 February 2016 - 03:41 PM.