Can we fully describe the physical world as a collection of fields?

I’m new to the Physics SE, coming from a pure math background. I think my question is full of incomprehension and lack of basic knowledge, but here goes.

I’m trying to wrap my head around the modern description of our physical world. My basic understanding is that there are $n$ known elementary particles (that might be 17/24/25/31 depending on what counts, but the details are not so important to me). Particles can be understood as ‘ripples’ or ‘excitations’ of their underlying fields. I assume the nature of this field might change depending on the particle (scalar/vector/tensor?), but what does excitation mean? Above/below some kind of threshold that brings the field to a ‘non-ground’ state?

Now assuming that each elementary field is given by $F_i : mathbb{R}^4 to mathbb{R}^{k_i}$ for each $i in {1, ldots, n}$, does this mean that the physical world we live in can be fully described as the collection ${F_i}$ which takes in a point in spacetime $(x,y,z,t)$ and returns the ‘excitation’ level of each elementary field at that point?

Edit: following G.Smith’s comment, quantum fields map a point in spacetime to an operator on a Hilbert space. Is there a finite number of such quantum fields for each elementary particle, and so can we describe fully the physical universe as a finite collection of such maps in principle?