There is a mathematically based definition of a perfect market. It’s only an approximation of real life, but it’s heavily used: https://www.investopedia.com/terms/p/perfectcompetition.asp
"Companies earn just enough profit to stay in business"
which contradicts your comment:
"in a “perfect” market there cannot be profits"
Besides, who would run a business that could not make a profit?
That makes as much sense as people working for nothing. I don't see any explanation for profits tending to zero, either.
I wonder why "perfect" markets are even a thing. It has no applicability to actual markets. It offers no insight on actual markets, and offers no useful predictions about markets. It's about as useful a model as the Monopoly game is - i.e. none.
I've seen many people claim that a "perfect" market is a requirement for free markets to work well. This isn't remotely true. "Perfect" markets and free markets have nothing in common.
"Perfect" markets just seem to be a strawman put forth by people who argue that since "perfect" markets are unachievable, therefore free markets are no good.