This question will be based roughly on the Bourgade Keating review on Zeta function and eigenvalue asymptotics (BK): https://link.springer.com/chapter/10.1007/978-3-0348-0697-8_4

To set up the question, we consider the riemann zeta function $\zeta(s)$ with zeros on the critical line $\frac{1}{2} + i t_n$. The unfolded zeros are defined as: $w_n = \frac{t_n}{2\pi} \log \frac{t_n}{2\pi}$. The name "unfolded" is justified by the fact that distribution of $w_n$ is asymptotically uniform along the critical axis. In terms of these unfolded zeros, we can define the following integral: \begin{equation} R_{2,\zeta}(f,W) = \int_{-\infty}^{\infty} f(x) \frac{1}{W} \sum_{\substack {j\neq k \\ w_j,w_k \leq W}} \delta(x-w_j+w_k) dx \end{equation}

Suppose the $w_n$'s had no pair correlation, the integral above would simply equal to $\int f(x) dx$. However, $w_n$'s clearly have pair correlations! So deviation of the integral from $\int f(x) dx$ roughly measures the pair correlations between the unfolded zeros $w_n$, up to some cutoff $W$. For $\zeta$ functions Montgomery proved an important theorem about this integral stated in KS as follows:

Theorem 1(Montgomery): Assume the Riemann Hypothesis. Then for test functions f(x) such that: $$ \hat f(\tau) = \int_{-\infty}^{\infty} e^{2 \pi i x \tau} f(x) dx$$ has support in $(-1,1)$, the following limit exists: $$ \lim_{W \rightarrow \infty} R_{2,\zeta}(f,W) = \int_{-\infty}^{\infty} f(x) R_2(x) dx $$ with $$R_2(x) = 1 - (\frac{\sin(\pi x)}{\pi x})^2 $$

The confusion I have is: **this $R_2(x)$ seems to carry unnecessary information.** To show that, suppose I Fourier transform the integral:
\begin{equation}
\int_{-\infty}^{\infty} f(x) R_2(x) dx = \int_{-\infty}^{\infty} \hat f(\tau) K(-\tau) d \tau
\end{equation}
Where $K(\tau)$ is the Fourier transform of the pair correlation function, sometimes referred to as the spectral form factor. Then since $\hat f(\tau)$ only has support on $(-1,1)$, I can restrict the $\tau$ integral to $(-1,1)$. But that would mean the integral is only sensitive to $K(\tau)$ for $\tau \in (-1,1)$. Within that domain, there are other functions that produce the same $K(\tau)$. For example, following equation (26) in BK, we can define:
$$ \tilde R_2(x) = 1 - \frac{1}{2 (\pi x)^2} \quad R_2(x) = 1 - (\frac{\sin(\pi x)}{\pi x})^2 $$
These functions have the same Fourier transform within $(-1,1)$.

Montgomery's Theorem is usually stated as a connection between random matrix theory and zeta function because $R_2(x)$ is precisely the pair correlation of Wigner random matrix ensembles. However, the calculation above would suggest that $\tilde R_2(x)$ would do just as well within the domain of interest. Hence, I feel that $R_2(x)$ carries unnecessary information about the pair correlations, and it now seems rather artificial that it matches the random matrix results.

This seems like a simple enough question. But I haven't found any explanation of it in various review articles on RMT-zeta function connections. So I would like some help from the experts: **Why did Montgomery put the RMT correlation function in his theorem if it carries unnecessary information? Was it just an inspired guess or is there something deep I am missing?**

*Note: The referenced review article seem to be on the boundary between math and physics. But since my question is more about the mathematical side, I thought it would be best to pose it here. If the moderators feel this is more fitting for other Physics SE or Math SE, please help me move the question to the right place. Thanks*