Embedding.

It has to do with topology.

The concept is so vitally important, it's impossible to overstate its importance.

I will illustrate by example.

The quantum model is a DISCRETE space embedded into a CONTINUOUS space.

Energy is discrete because it is quantized. And we imagine SpaceTime to be entirely continuous.

Question: what new feature or phenomenon does this embedding confer?

The answer: COUNTABILITY. You can "count" the quanta.

When we "measure" something, the very first thing we have to do is overlay a number system onto whatever space embeds the object were trying to measure. In continuous space, we arbitrarily assign a unit, could be an inch, a millimeter, or a mile. But in the discrete space the units are chosen for us, one Quantum is indivisible.

Consider in the abstract, the idea of embedding a discrete space into a continuous space. If we count discrete items 1, 2, 3 and so on, what is "between" the points? The answer is, nothing. That is to say, more precisely, that our discrete assignment of numbers has not completely "filled" (the technical term is "covered") the underlying space. In our example, the discreet embedding only covers the integers, and there are no negative numbers since you cannot count Less Than Zero.

In quantum mechanics we can envision a PROCESS (a "generator", as in random stochastic process), that embeds the discrete system into the continuous space. In this example, it's as simple as placing the points, so like, you're saying "number one goes here, and number two goes there" and etc. In the process of so doing, you are "filling space". Not "all" of the space, just the points corresponding to the integers.

And, how "much" of the space is that? How much of a continuous space will a discrete integer embedding cover? We know it must be greater than 0%, and less than 100%. It is some fraction, maybe a third, maybe 1/5 or 1/10.

It turns out we can calculate this. Topological embeddings conform to the algebra of group Theory, and we can predict which symmetries will work in which contexts and exactly how they will work.

The Brilliance of the embedding concept, has to do with what the calculus people call "transforms". For example, if you have a function that's a figure 8, it is undifferentiable in at least two places, the two places where the slope is vertical. However what you can do is you can "lift" the Figure 8 into three dimensions, by multiplying it by another function, which makes it differentiable everywhere in three dimensions (because by doing this you've expanded the figure 8 into a helix) - and you can then do the math, and project the results back down into two dimensions. The only reason the original function was undifferentiable, is because of the choice that was made in terms of embedding it into the continuous analytical space. In some other space, the function becomes perfectly differentiable.

The same is true of stochastic generators. Generally speaking, they can be integrated and differentiated just like any other functions. The math for doing this is called Ito's calculus. The math for calculating coverage is simple fractal geometry.

Underneath all this, is the simple concept of countability. The attribute of countability, surprisingly enough, cannot be directly derived from the topology. It's something independent, something different.