Maybe I'm just completely stupid and absolutely don't understand Fermat's last theorem but can't you prove that there is a solution for n>2 as follows:
1^2 + 0^2 = 1^2.
Now this holds true for any n element of the real numbers.
There any many other psyops in the history of science. One example was that special relativity demonstrated long contraction and time dilation which in fact a few years after Einstein offered his theory of special relativity two phycists actually proved time dilation and length contractions occur at high speeds even if the speed of light is NOT constant from every frame of reference. All that is required to prove time dilation and length contraction is the notion of that there is no universal reference frame but only inertial reference frames which galileo made a point of. Another is that Galileo discovered the earth revolves around the sun. He didn't prove it. It wasn't prove and accepted by the scientific community until the 1700s after Newtonian physics was discovered. Bug the biggest of biggest of biggest psyops was awarding the major proponent and pioneer of transgender theory the highest award in science: an award given my the royal society of scientists in England, which few received historically including newton. THAT is a real disgusting politically corrupt and politically motivated psyop.
The real psyop in Mathematical history is the belief that infinite processes can be completed and the real numbers, in particular irrational numbers, are sensible mathematical objects. The psyop encompasses the acceptance of cantor's theory of infinite sets of which many philosophers of mathematics demonstrated was full of logical paradoxes. This fact was largely accepted by the mathematical community until... they realized you need this theory to define the real numbers. No one has ever provided a valid definition of real numbers: dedekind cuts and Cauchy sequences are full of vagueness, circular reasoning, and the assumption of infinite irrational numbers as valid. In other words, they assume real numbers are valid and defined when trying to prove real numbers are valid and offer a definition. I've been reflecting on the meaning of the square root of 2 for years, and it was the marvellous work of math professor Norman Wildberger on YouTube which demonstrated the logical nonsense of the real numbers.
Your take on infinities is insightful. Maybe I can add to it.
To know something is to know a category (or as the historical philosophers say, a unifying idea; whether that idea unifies an objects identity through time or is the category of multiple objects through space.)
Instead of thinking of the universal definitions of the natural numbers as an infinity set, maybe we can take a clue (roughly speaking) from special relativity and say the size of the set of natural numbers is relative according to the category in a given circumstance. So you can never have a set of natural numbers without first defining an environment with objects which could be categorised (this links to special relativity in that there is no single entity which is the universe but rather only particular individual-entities; this is also Aristotelian in nature).
So if I'm in a park and there are three tree, two dogs, one human. Then under the category of tree, the set of natural numbers is 1, 2, 3 (a finite set). Under the category of dog, the natural numbers is the set of 1, and 2. Under the category of human, the natural numbers is 1. You only have finite sets. What about under the category of hammer? The natural numbers is 0.
Under this perspective, you can only have an infinite set of natural numbers if there's some individual-entity which embodies a category (given there are many of this entity) in a circumstance or reality as a whole of which theres an infinite number and, technically is a property or part of every being. If this is true, you can have an infinite set, if this is false you cannot have an infinite set. Hence, defining the size of a set of natural numbers begins with an environment of categorizable objects. This also has the benefit of ensuring your mathematics is always tethered to describing something in reality as a whole whether that object is physical or not.
Anyways, this is just my random ponderings and is most likely full of problems because I'm actually quite stupid and a slow learner.