Entropy, magnitude, diversity of finite metric spaces in number theory

Question

I was reading the article by Tom Leinster, (Maximizing diversity in biology and beyond, arXiv link), and find it very interesting.

Since I was searching for entropies of finite metric spaces I found this article.

Consider subsets $X_n := \{1,\ldots,n\}$ of natural numbers with the "abc"-metric

$$d(a,b) = 1-\frac{2\gcd(a,b)^3}{(a \cdot b \cdot (a+b))}$$

Then we have a sequence of finite metric spaces $X_1 \subset X_2 \subset X_3 \subset \ldots \subset X_n \subset \ldots$

I did some experiments based on this article and computed the matrix $ Z = \exp(-d(a,b))$ for some of these finite metric space. It seems that $Z_n$ is positive definite and invertible hence the maximum diversity is given by $|Z_n|$ the magnitude and the maximum probability is $w_n/|Z_n|$ , where $w$ is the unique weighting vector.

Is it possible to prove that:

1. $Z_n$ is always positive definite and invertible.

2. Given $w_n/|Z_n|$, it seems that its Shannon entropy is very well approximated by $n \log(n)$.

3. Is there any number theoretic interpretation of this entropy and the fact that adding one number / point to the metric space will increase its (Shannon) entropy?

4. In physics increasing entropy is related to irreversible processes, so how can one view this process of adding one point to the metric space as irreversible?

Thanks for your help!