R C-R

From that panel it looks like the 'MTX' object is in a group that is not part of either artboard -- it is simply above both of them.

How did you manage to do that? The closest I can get to it is to add an object that is not part of any artboard.

Handy or not, it does not seem an intended function of the Add (or Divide?) boolean, so I wonder if it is something we can count on in future revisions. I also do not understand why these operations close open curves -- seems like a 'feature' nobody asked for.

I am not even sure why using Add on a single open curve that intersects with itself creates the extra nodes at the intersection.

For me, at its original rotation either Add or Divide leaves it as a "Curve" object (with 2 new nodes & a straight segment connecting the 2 ends). At 120° Add gives me a Curves object & Divide give me 2 Curve objects. At other angles from what I can tell I also get similar results to yours. I don't think any of this is the correct behavior -- surely the rotation should not matter! FWIW, I get similar results using anon2's file -- Curve or Curves depending on rotation angle & Divide or Add.

What OS version are you running on your Mac?

I am saying that either Add or Divide creates the 2 new nodes where the curve crosses itself. Then selecting both of those new nodes & using "Break Curve" results in 2 curves. IOW, with resect to your original set of steps, in your step 2 either Add or Divide works for me but your step 3 is still needed to produce 2 curves.

For me, it was about 14KB. For me, both Add & Divide work with the file & with this super simple loop.afdesign version made with the Pen Tool & with just 3 nodes. I also tried a few variations using the Pencil Tool but they all worked as expected with both the Add & Divide options. I did not notice any difference if the shape was drawn vertically or horizontally either. Puzzling.

For how big?.afphoto artboard 1 is 301 x 301 px but artboards 2 & 3 are not. To see this, you need to select each artboard in turn on the Layers panel & then check their dimensions in the Transform panel. Do that & you will see that Artboard 2 is about 212 x 150 px & Artboard 3 is about 286 x 141 px. For how big 2?.afphoto, it is even easier to see the different sizes because the artboards were converted to pages in APub before being saved in AP. So reopen that file & use the page navigator at the bottom left of the workspace window to move from page to page. With the Pan tool active you will see that the dimensions are different for each page. For how big?.afphoto (not how big 2?.afphoto) with the Pan tool selected 600 x 301 px is the overall workspace area used by the 3 artboards, but if for example you rearrange the artboards, those numbers will change -- they are not the dimensions of the artboards or anything in them. how big? 3.afphoto is identical to how big?.afphoto except I moved artboard 3 down below artboard 1. Note that with the Pan tool selected the numbers now read 527 x 448 px. Drag Artboard 3 somewhere else in the workspace & the numbers will change accordingly. This is because as @Callum explained, native format Affinity files are just containers & can include more than one item.

AP can open native format documents created in AD or APub, which means they can include multiple artboards or pages. For example, what would be the "set dimension" of how big?.afphoto or how big 2?.afphoto?

Maybe this is a change related to Big Sur? I am still running Catalina & like with all previous macOS versions I have used, the Affinity apps I have purchased from the MAS use the "com.seriflabs" Container style folder pathnames, the same style as used with the many "com.apple" folders in ~/Library/Containers/.

That's what I have always seen on my Macs, never anything like ~/Library/Containers/Affinity Photo/Data/*. I wonder if @Dan C was thinking about the Affinity Store versions or maybe the betas, although I think they should be in the ~/Library/Application Support folder.

If AP came from the Mac App Store, I think the file path should be: ~/Library/Containers/com.seriflabs.affinityphoto/Data/Library/Application Support/

I reported this as a bug years ago but nothing ever was done about it. 😒

Maybe I am just not looking hard enough but where does one find the "Open Lens Profile Folder ..." item?