I always spray, roll or brush a bonding agent (Link or Plasterweld) into cracks or old plaster edges before taping. I embed the tape (paper for corners) in a setting-type compound, like Durabond 90 (sets in 90 mins or less). So the second coat can go on the same day. For second coats you can use another setting-type compound, Easysand (setting time @ 90, 45, etc), or regular green-lid drywall mud. Finishing touches with regular mud, for ease of sanding. Since it's a ceiling corner, it would be all right to use the dust-reduced type of mud, which I do not recommend for general work as it is very soft, but at a ceiling it wouldn't see any rough handling. It could, of course, all be done with ordinary green-lid mud, but I have had a great track record with durabond for old plaster repairs. Casey

Always caulk any changes in plane. So because your granite and backsplash change planes (from horizontal to vertical), you need to caulk rather than grout. (The same goes for any corners where you've changed planes -- such as where two walls meet at a 90 degree angle.) This allows for movement of the surfaces without cracking. You can get sanded caulk which matches grout and so it should not be noticeable that it's caulk not grout. I would certainly defer to a tile and/or granite expert, but unless the tiles are meant to butt directly up against each other, you should at least follow the minimum grout joint width requirement of the tile. This allows the caulk to get between the tiles and isolate any movement of either surface and also allows for any variations in the tile or granite. Stand up to your tile installer and insist. If he has a problem with it, you can do the caulking yourself. Just make certain to mask (with tape) the granite and the tile before caulking to protect your investment. (Nothing worse than having to clear smeared caulk off a new surface.) In my (minimal) experience it would be easier to just smoosh grout into every crevice since then you don't have to be as careful, but it is the wrong thing to do. In my opinion, either the installer is unaware (maybe inexperienced? or just improperly educated on grouting?) or is simply being lazy. This is your home and your investment and the installer needs to respect that.

It's nothing structural, so no need for worry. I suspect it is just a cracked corner that happened when the carpet installer was banging the carpet in. If you are concerned about the appearance, you should be able to fix it with a little painters caulk and matching paint. However, if it is very loose and ready to fall out, you might need to remove it and build the corner back up with a patching compound.

I’m not arguing because he’s offered a solution that may work however, the comment by GN Builders suggests a sealant. That’s basically caulk. Our home has vaulted ceiling’s on the 2nd floor. They showed serious cracks during the new home warranty period. The builder applied some caulking to repair. It wasn’t nicely “finished”, wouldn’t hold paint and during the following winter movement caused it to tear free. Dynaflex 230 may not have those characteristics but I’d apply to a small section and wait to see. I removed all the original tape and the caulk to retape with a corner bead for vaulted ceilings similar to the earlier link. It’s lasted over 20 years.