I had composed a PM to you and was just about to send it to you when I looked at the second nudge again and saw that there were eight black squares along the outer perimeter (which is the largest square). Plus I had already seen that I could generate STR for the first three of these*, so then it just fell to me to prove that the remaining squares completed to STRAIGHT, which was my backsolve guess. Which worked, and miraculously I finally saw that all the new words were synonyms of SQUARE, which had eluded me for days.
Nicely constructed. Your title made me recall algebra class where we learned to complete the square before we learned the Pythagorean Theorem, which is just a condensation of that method. (Just as I later determined on my own years later that the brute force algorithm for extracting a square root to x-number of places is just a condensation of a binomial expansion.)
*Though I still don't see that MATCH is a synonym, which threw me off track early on. I won't get into the rabbit holes that ensured...