Is there any physics theory which is similar to these analogies?

Question

Since I am doing this little "research" project on my spare time and in my physical neighborhood there are not many people to discuss these ideas, I wanted to share with you a small point of view, in hope that someone with more experience in physics/number theory, can bring in some ideas to the discussion:

Let $k=s$ be a positive definite symmetric function and a simililarity over the natural numbers such that $k(a,a) = 1, k(ac,bc)=k(a,b) $ for all natural numbers $a,b,c$.

A similarity $s:X\times X \rightarrow \mathbb{R}$ is defined in the Encyclopedia of Distances as:

1. $s(x,y) \ge 0 \forall x,y \in X$

2. $s(x,y) = s(y,x) \forall x,y \in X$

3. $s(x,y) \le s(x,x) \forall x,x \in X$

4. $s(x,y) = s(x,x) \iff x=y$

Example: $k(a,b) = \frac{2 \gcd(a,b)}{a+b}$ or $k(a,b) = \frac{\gcd(a,b)^2}{ab}$

By the Moore-Aronszajn-Theorem there exists a Hilbert space $H$ such that $$ \langle\phi(x),\phi(y)\rangle = k(x,y), $$ where $\phi: \mathbb{N} \rightarrow H$ is a "feature" mapping.
By assumption, we have $k(a,a)=1$ for each natural number $a$.
We consider vectors $x,y$ in the Hilbert space of the form:

$$x = a_1 \phi(x_1) + \cdots a_n \phi(x_n)$$

$$y = b_1 \phi(x_1) + \cdots b_n \phi(x_n)$$

where $a_i$ are real numbers and $x_i$ are natural numbers.
Let $G = ( k(x_i,x_j) )_{1\le i,j \le n}$ be the gram matrix and let $T = ( a_i b_j )_{1 \le i,j \le n}$.
Then we have:

$$\langle G,T\rangle_F = \langle x,y\rangle_H$$

where the subscript $F$ denotes the Frobenius inner product of $G$ and $T$.

Furthermore we have:

$$\langle Gx,y\rangle = \langle x,Gy\rangle$$

Now the Gramian matrix lives in $S^+_n$ the space of SPD matrices. Here different Riemannian metrics are discussed on this space.

Let me try to give a "phyiscal interpretation / analogy" of this situation here:

1. The natural numbers would correspond, since $k(a,a) = 1$ to pure states in quantum mechanics.

2. For each set of $n$ pure states / natural numbers, we have an "observable" / Gramian matrix $G$: $$ \langle Gx,y\rangle = \langle x,Gy\rangle $$ and each measurement of this observable gives an eigenvalue of $G$ as a result.

3. Each observable $G$ lives in a symmetric (if the Riemannian metric is the affine invariant) one Riemannian space $S_n^+$, which would be interpreted as a point in space.

4. Not every point $X \in S_n^+$ space has measureable observables, since there might be some $X$ which we can not write as a Gramian matrix $G$.

5. One could think of a "triangulation" of the space $S_n^+$ based on simplicex formed by sets of pure states / sets of natural numbers.

The "drawback" of these analogies is that

1. the space would have dimensions $$1,3,6,\cdots,\frac{n(n+1)}{2},\cdots$$

2. I have not said anything about time

and those dimensions would depend on what dimension (number of distinct eigenvalues) the observables in the quantum level has.

I am not familiar with the many theories of physics, so my question is:

1) Physics question: Is there any physics theory which is similar to these analogies?

Thanks for your help.

Edit: The following point of view seems to be more promising:

1. To each observable (hermitian matrix) $A$ of a $n$ dimensional Hilbert subspace, we can associate $\exp(A)=G$ which is a Gramian Matrix of dimension $n$ and corresponds to a basis of the subspace.(This is not unique).

2. To each basis $B = (\phi(x_1),\cdots,\phi(x_n))$of a finite subspace, we can associate the Gramian matrix $G_B$ and the Hermitian matrix $\log(G_B)$ which we could interpret as an observable.

3. In the question above I wrote about sets of natural numbers, which is incorrect and should be replaced with $n$-tuples of natural numbers (which correspond to a basis of a Hilbert subspace).

4. Hence since permutations of the basis / natural numbers, will give in general a distinct Gramian matrix $G_1$ and $G_2$, but the measurements will be the same, since the eigenvalues $\sigma(G_1) = \sigma(G_2)$ are the same, and we could have the case that the eigenvectors are the same, meaning that the quantum state is the same of the two observables $\log(G_1)$ and $\log(G_2)$ then we conclude that in the distinct "spatial" space points $G_1$ and $G_2$ there are two identical quantum states (or to better imagine it, particles).

possibly related question: A RKHS interpretation of the Rydberg formula for hydrogen and an application for physics?