Since the value at $\lambda =0$ is exactly 1 by construction of the dataset, the standard ${\chi }^2$ is infinite without regularization. Therefore, we modify the data covariance to attribute an uncorrelated standard deviation of ${10}^{-3}$ to that point. In contrast, applying a simple SVD cut would effectively exclude the $\lambda =0$ point from contributing to the ${\chi }^2$.

We only consider positive values of $x$ since we study separately the real and imaginary parts of the matrix elements, which correspond to definite parities in $x$.

For the RBF kernel, we use $\sigma =10$, $l=0.4$ and a prior mean of $f(x)=10$. For the logarithmical RBF, we use $\sigma =1$, $l=0.7$ and a prior mean $f(x)=x^{-0.15}$.

For the RBF kernel in $x$-space, we use $\sigma =50$, $l=0.4$ and prior mean of $f(x)=3$. For the logarithmical RBF kernel in $x$-space, we use $\sigma =4$, $l=0.7$ and prior mean of $f(x)=x^{-0.15}$. For an exponential decay in $\lambda$ of $r=1.0\mathrm{fm}$, we use a kernel in $\lambda$-space with $\mathrm{\Sigma }=0.5$ and $L=5$. For an exponential decay of $r=0.6\mathrm{fm}$, $\mathrm{\Sigma }=1$ so that the prior decay uncertainty coincides at the beginning of the extrapolation region.

In fact, if one uses the criterion proposed in Ref. [23] that Fourier components at large $\lambda$ do not matter to the reconstruction at a given value of $x$ if the derivative of the Fourier transform of the uncertainty is much smaller than $x$ (with their notations, $|\delta {\tilde{h}}^{\prime }(\lambda )|\ll x$), one finds a much more sensible depiction of uncertainty propagation in the framework of short-distance factorization. Using their example Eq. (B22) for a non-exponential algebraic decay of the large Fourier components as ${\lambda }^{-1/2}$, corresponding to a phenomenological PDF which diverges as $x^{-1/2}$ at small $x$, the asymptotic error could be approximated by $|\delta \tilde{h}(\lambda )|\lesssim \tilde{h}(\lambda )\approx 0.1(10/\lambda {)}^{1/2}$. Already if $\lambda =10$, the derivative of this bound equals 0.005, so well below any value of $x$ of phenomenological relevance. Unlike what Ref. [23] concludes, their own reasoning would give that short-distance factorization has a controlled asymptotic expansion, resolving the contradiction between their statements and the empirical evidence they provide themselves that the exponential decay does not matter crucially. But the reasoning is also fragile because it relies on asymptotic forms in a regime where they are not established and very approximate bounds: We find that our more in-depth study shows sensitivity beyond $\lambda =10$ because the approximation by a simple asymptotic model is inaccurate.