Fermi wrote:
> In a spherical structure having the reproduction factor 1 for
infinite dimensions the activation of a detector placed at the center
due to the natural neutrons is proportional to the square of the radius.

I cannot exactly say what does mentioned by "natural neutrons" but it is looked as the most realistic case to be spontaneous fission neutrons produced by the reactor nuclear materials. Then, in terms of the point kinetic, the ratio $N_{\text{source}}/N_{\text{total}}$ is actually limiting to $0$ like $R^2_{\text{eff}}/A$ in the report if multiplication coefficient $k_m = \left(N_{\text{total}} - N_{\text{source}}\right)/{N_{\text{total}}}$ is limiting to $1$ on fixed $N_{\text{source}}$. So, the main question is about nature of relation $R^2_{\text{eff}}$ and $N_{\text{source}}$.

Accordingly to the neutron diffusion theory, solution of one-speed transport equation for point source on distance $l$ with intensity $S_0$ in infinity media with coefficient of diffusion $D$ (see, for example, [here][1]):
$$ \Phi\left(l\right) = \frac{S_0 e^{-l\sqrt{\frac{\Sigma_a}{D}}}}{4\pi l D} $$

We will systematically overestimated flux in the sphere center if we will use it for a spherical model with leakage relates to the problem. However, for that case total flux at the reactor core center will be represented by an integral sum of concentric elemental spherical layers volumes $4\pi r^2 \mathrm{d}r$ over sphere of radii $R$:
$$ \Phi\left(0, R\right) = \int_{0}^{R} \frac{S e^{-r\sqrt{\frac{\Sigma_a}{D}}}}{4\pi r D} 4\pi r^2 \mathrm{d}r = \frac{S}{D} \int_{0}^{R} r e^{-r\sqrt{\frac{\Sigma_a}{D}}} \mathrm{d}r = \frac{S}{\Sigma_a} \left( 1 - \left( R\sqrt{\frac{\Sigma_a}{D}} + 1 \right) e^{-R\sqrt{\frac{\Sigma_a}{D}}} \right) $$

As we can see now, growing rate of primary neutron flux is much more less than $R^2$. Moreover, on large $R$ function $\Phi\left(0, R\right)$ becomes asymptotic with constant $S/\Sigma_a$ what is confirmed by computation experience when every space-strengh source has a plateau in flux or dose rates distribution at plots.

For small values of $R$ the exponent can be expanded by the Taylor series about the point $R=0$:
$$ \exp\left( {-R\sqrt{\frac{\Sigma_a}{D}}} \right) \approx 1 - R\sqrt{\frac{\Sigma_a}{D}} + O\left(R^2\right) $$

Substitution it to the expression gives the following expression for spontaneous fission neutron flux in the center of radii $R \rightarrow 0$ sphere
$$ \Phi\left(0, R\right) \approx \frac{S}{\Sigma_a} \left( 1 - \left( 1 - R \sqrt{\frac{\Sigma_a}{D}} \right) \left( 1 + R \sqrt{\frac{\Sigma_a}{D}} \right) \right) = \frac{S R^2}{D} $$

It is proportional to $R^2$.

So, we can conclude that "the rule of $R_{\text{eff}}^2$" relatively quite approximates changing of $N_{\text{source}}$ for initial stages of the experiment with small sizes of the reactor and high rate of neutron leakage so it can be used in approximation of the $N_{\text{source}}/N_{\text{total}}$ ratio changing. For example, the first two lines in the table 1 shows growing of flux in almost 2 times — from 42 to 78. However, the main reason of that is not growing of neutron multiplication but quantity of nuclear material in the system what is seen from changing of $R^2_{\text{eff}}/A$ just in 1,2 times — from 390 to 320. On the last subcritical stages, the $R^2_{\text{eff}}/A$ rate get closer to the commonly used today the $1/A$ ratio ("$1/M$ Plot").

[1]: https://www.nrc.gov/docs/ML1214/ML12142A086.pdf