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No, the density of matter needed to exert enough gravity to create an event horizon, the "black hole" part of the black hole, will in herently exceed the saidatters Swarzchild radius causing it to collapse into a singularity

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If it has an event horizon it's by definition a black hole. That's what defines a black hole, not the singularity. According to general relativity any black hole inevitably forms a singularity (under some pretty safe assumptions) so any object that forms an event horizon will also result in a singularity. But we believe that general relativity isn't accurate near the center of a black hole and needs to be replaced by a theory of quantum gravity, and that this will get rid of the singularity

But the Swarzchild radius increases with mass linearly, while the radius of a circular body only increases with the cube root of mass, so could a neutron star get big enough to overcome the swarzchild radius without collapsing into a singularity? What is collapsing and when does it happen? How do we know if it’s a singularity?

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So how do we know if something is a singularity? So it’s kind of an assumption that this happens every time by general relativity? And what part of it leads to that?

If I’m hearing right, yes some neutron stars could gain enough mass to become black holes, and we’re not sure yet if all black holes are singularities because not everything adds up right and a new rule could be discovered?

> So how do we know if something is a singularity?

We do not. It's a strong prediction of general relativity because inside a black hole, all possible trajectories lead towards the center (or a ring for rotating black holes): Trying to stay at a finite distance is as impossible as trying to stay in Sunday.

> If I’m hearing right, yes some neutron stars could gain enough mass to become black holes

Yes, because they collapse above a certain mass. They are not neutron stars any more if that happens.

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Its more about density than just mass, every mass has S radius that it tips over into blackhole territory but will not trap light before that point.

Only singularities can warp space time enough to trap light making a blackhole it also doesn't help that we lack the math to fully explain blackholes entirely, all values involved end at either 0 or infinite or some impossible arithmetic involving the two extremes

The Schwarzschild radius would be a good thing to look up to learn about when and why a black hole forms. And the event horizon is basically the point where we have to rely on math rather than observation to be able to tell what’s going on. But the same used to be true for black holes themselves, and the math held up for that

Neutron stars, like other astronomical bodies made of degenerate matter, shrink in radius as their mass increases.

You don't really need the density either.

If you had enough air at standard Earth atmospheric density, then that would create a black hole.

The calculation works out that 3.8 Billion kg of air at standard Earth density will form a black hole.

What is meant by the event horizon? It's the distance at which anything moving slower than c, the speed of light, will not escape the object's gravitational field. In other words, the object has an escape velocity, v_e, that is greater than c. And v_e depends on the mass and radius of the object. If you have enough mass contained within a small enough radius, then v_e > c. That's why they're called "black" holes, because light cannot reflect off them and come back to our sensors (eye, telescope, etc.) But the only event that can bring that much mass within that small a radius is a (massive) collapsing star. Once the nuclear fuel burns out, there is no more outward force, and the star's massive gravitational field starts pulling everything inward. If the mass is high enough, protons and electrons are smashed together to form neutrons, a supernova occurs, and you have a neutron star left (itself, extremely dense, but still no event horizon - light can still escape from even something as dense as a neutron star. There just isn't enough mass within a small enough space to make the escape velocity high enough.) But if the star's original mass is large enough, even the "quantum degeneracy pressure" of the neutrons isn't enough to keep all that mass at a certain distance. The star continues to collapse. So we end up with a huge amount of mass in a very tiny space --> enormous gravitational field --> so big that c < v_e --> light cannot escape --> black hole.