“I have a titanium bracelet with germanium stones embedded in it. Is it possible to test its genuinity and purity without damaging the bracelet or the metal?”
The term “possible” covers a lot of ground. What came to my mind first was a simple mass spectrometer; part of any electron microscope. You’ll need to know somebody who works for a university or industry that has one, and is or knows a microscopist in residence. But it is a quick and simple test, if the chamber is big enough for the bracelet. Back when I worked in a radiological dating lab, I knew the right people and did such testing on meteorite and geological samples. That was around 1980.
Laser spectrography also can do the job. There are hand held devices that can tell you the exact content of the metal and stones without visibly damaging them. But you still need to know someone with the tool. Here’s an article about how that works at HowStuffWorks.com
In brief, both mass-spec and laser-spec ionize surface atoms (chase away their electrons) and as the atoms relax and recover their charges, they give off a fingerprint of colors. Every element has it’s own distinct spectrum. So these gadgets count how many atoms of each element the beam (electron or laser) stimulates, letting one know exactly what proportions are there.
I’ve already listed simpler tests in my post: How to tell if a piece of metal is really titanium, but many of those would leave bigger marks on the piece.