Argh, bad news then...
A spinning cursor and a Windows Explorer hang means it's likely something to do with your Windows setup (damn Win 10 :/). Cogmind itself is actually quite simple from a technical standpoint, and if there are no log files, then the game didn't actually even start... so my hands are tied on that end.
The best I can do in this case is help search for generic causes, since it looks like some people have been having similar trouble with various games and software in Win 10. (I don't use Win 10 myself--currently avoiding it like the plague, but I will no doubt have to get it one day...)
Just spent an hour Googling for possibilities, and the best I've come up with so far is that sometimes games can be run in windowed mode to start up properly when they won't otherwise. I'm skeptical of this solution with regards Cogmind, since it should have at least created a log file before getting to the fullscreen initialization point (unless you're not running as an admin, but you are). Worth a shot, though. Take the attached .cfg file and put it in your [Cogmind]/user/ directory. That should start it up in windowed mode (assuming it can start up at all).
Another possibility: It's got to be something on the Windows end, so maybe the necessary Windows DLLs were corrupted? Unlikely since you already attempted to install the redistributable and I imagine it checks for corruption on existing files, but this is the only other possibility I've found so far.
Dynamic-link library files have a file extension of .dll. These are collections of routines that are stored until a program uses them. If these files are corrupt or out of date, they can cause problems with your computer. To fix this problem, re-register the .dll files by:
Follow this Official Microsoft Fix It Tool link
Click Run in the File Download dialog box
Follow the steps in the wizard
Note: This wizard may be in English only; however, the automatic fix also works for other language versions of Windows.
Probably unrelated: If you perchance have a High-DPI monitor, Cogmind still doesn't support those natively and requires a settings change, but that shouldn't cause it to fail startup, just cause it to not fit the screen (which can be resolved via Windows settings). So I doubt that's it.
Note: If you have no other computer to run it on, and this ends up not being solvable, I'm happy to refund your purchase!
Thanks for the help so far in being as clear as possible about what's going on
(Sorry for the non-rapid replies on this--you're active when I'm asleep on this side of the world
)