### A.10. GRG lobe volume-filling fraction

Assuming self-similar growth, the combined proper volume $V$ of an RG's lobes and its intrinsic proper length $l$ obey $V \propto l^3$. The constant of proportionality varies per RG and depends on the shape of the lobes; we treat it as an RV $\Upsilon := \frac{V}{L^3}$. Then the proper VFF of GRG lobes $VFF_{GRG}(z) = VFF_{GRG}(z=0) \cdot (1+z)^3$, where

$$ 
\begin{aligned}
VFF_{GRG}(z=0) &:= n_{GRG} \cdot \mathbb{E}[V | L_p > l_{p,GRG}] \\
&= n_{GRG} \cdot \mathbb{E}\left[\frac{V}{L^3} \cdot L^3 \mid L_p > l_{p,GRG}\right] \\
&= n_{GRG} \cdot \mathbb{E}[\Upsilon \cdot L^3 \mid L_p > l_{p,GRG}].
\end{aligned} 
\quad (A.96) $$

Assuming that RGs grow self-similarly, so that shape does not reveal length, $\Upsilon$ and $L^3$ are conditionally independent given $L_p > l_{p,GRG}$: $\Upsilon \perp L^3 \mid L_p > l_{p,GRG}$. As a result,

$$ 
\begin{aligned}
VFF_{GRG}(z=0) &= n_{GRG} \cdot \mathbb{E}[\Upsilon | L_p > l_{p,GRG}] \cdot \mathbb{E}[L^3 | L_p > l_{p,GRG}] \\
&= n_{GRG} \cdot \mathbb{E}[\Upsilon] \cdot \mathbb{E}[L^3 | L_p > l_{p,GRG}].
\end{aligned} 
\quad (A.97) $$

To obtain this last line, we once more exploit self-similarity: $\Upsilon \mid (L_p > l_{p,GRG}) = \Upsilon$. We can approximate $\mathbb{E}[\Upsilon]$ by taking the mean of some $\frac{V}{F}$ deduced from observations. A technical complication arises from the fact that $\mathbb{E}[L^3 \mid L_p > l_{p,GRG}]$ does not exist for $\xi \ge -4$ under our model. This is an artefact of the Pareto distribution assumption for $L$, which unrealistically features support over an infinitely long part of the real line: $\{l \in \mathbb{R} \mid l > l_{min}\}$. This causes the expectation value integral to diverge for $\xi \ge -4$. An approximation to $VFF_{GRG}(z=0)$ that works for all $\xi$ is the lower bound

$$ VFF_{GRG}(z=0) > n_{GRG} \cdot \mathbb{E}[\Upsilon] \cdot \mathbb{E}^3[L | L_p > l_{p,GRG}], \quad (A.98) $$

which follows from Jensen's inequality. Here

$$ \mathbb{E}[L | L_p > l_{p,GRG}] = l_{p,GRG} \frac{\Gamma(-\frac{\xi}{2} - 1) \Gamma(-\frac{\xi}{2} + 1)}{\Gamma(-\frac{\xi}{2} - \frac{1}{2}) \Gamma(-\frac{\xi}{2} + \frac{1}{2})}. \quad (A.99) $$

Alternative approximation formulae, which use $\Upsilon_p := \frac{V}{L_p^3}$, are

$$ VFF_{GRG}(z=0) > n_{GRG} \cdot \mathbb{E}[\Upsilon_p] \cdot \mathbb{E}^3[L_p | L_p > l_{p,GRG}] \quad (A.100) $$

and

$$ VFF_{GRG}(z=0) > n_{GRG} \cdot \mathbb{E}[\Upsilon_p] \cdot m_{L_p^3} \mid L_p > l_{p,GRG}, \quad (A.101) $$

where

$$ m_{L_p^3} \mid L_p > l_{p,GRG} = l_{p,GRG}^3 \cdot 2^{-\frac{3}{(\xi+1)}} \quad (A.102) $$

is the median of the cubed projected proper length for giants. An advantage of these latter expressions is that there are more data available to estimate $\mathbb{E}[\Upsilon_p]$ than there are to estimate $\mathbb{E}[\Upsilon]$.

### A.11. Unification model constraints from quasar and non-quasar giants

The quasar GRG probability $p_Q$ is

$$ 
\begin{aligned}
p_Q &:= \frac{\mathbb{P}(L_{p,obs} \ge l_{p,GRG}, \sin \Theta \le \sin \theta_{max})}{\mathbb{P}(L_{p,obs} \ge l_{p,GRG})} \\
&= \mathbb{P}(\sin \Theta \le \sin \theta_{max} \mid L_{p,obs} \ge l_{p,GRG}) \\
&= \int_0^{\sin \theta_{max}} \frac{\mathbb{P}(L_{p,obs} \ge l_{p,GRG} \mid \sin \Theta = x) f_{\sin \Theta}(x) \, dx}{\mathbb{P}(L_{p,obs} \ge l_{p,GRG})} \\
&= \int_0^{\sin \theta_{max}} \frac{\mathbb{P}(L_p \ge l_{p,GRG} \mid \sin \Theta = x) f_{\sin \Theta}(x) \, dx}{\mathbb{P}(L_p \ge l_{p,GRG})} \\
&= \int_0^{\sin \theta_{max}} \frac{(1 - F_L(\frac{l_{p,GRG}}{x})) f_{\sin \Theta}(x) \, dx}{1 - F_{L_p}(l_{p,GRG})}.
\end{aligned} 
\quad (A.103) $$

Now using Eqs. A.3 and A.8 and $f_{\sin \Theta}(x) = \frac{x}{\sqrt{1-x^2}}$ over the domain of integration,

$$ p_Q = \frac{4\Gamma(-\frac{\xi}{2} + 1)}{-(\xi + 1) \sqrt{\pi} \Gamma(-\frac{\xi}{2} - \frac{1}{2})} \int_0^{\sin \theta_{max}} \frac{x^{-\xi} \, dx}{\sqrt{1-x^2}}. \quad (A.104) $$

### A.12. Extreme giants in a sample

An interesting feature of the model is its ability to predict the occurrence of giants with extreme projected proper lengths in a GRG sample of, say, size $N$. Now consider some $l_p > l_{p,GRG}$ — what is the probability $p_{>l_p}$ that an observed GRG will have a projected proper length exceeding $l_p$? Proceeding as in the derivation of Eq. A.12, we find

$$ 
\begin{aligned}
p_{>l_p}(l_p) &:= \mathbb{P}(L_{p,obs} > l_p \mid L_{p,obs} > l_{p,GRG}) \\
&= \frac{1 - F_{L_{p,obs}}(l_p)}{1 - F_{L_{p,obs}}(l_{p,GRG})}.
\end{aligned} 
\quad (A.105) $$

In the absence of selection effects, $p_{>l_p}(l_p)$ is given by Eq. A.15. The number of giants with extreme projected proper lengths $N_{>l_p} \sim \text{Binom}(N, p_{>l_p}(l_p))$.

Interesting questions can be answered readily. For example, the probability that the sample contains at least one GRG with projected proper length $l_p$ or larger, is

$$ \mathbb{P}(N_{>l_p} \ge 1) = 1 - \mathbb{P}(N_{>l_p} = 0) = 1 - (1 - p_{>l_p}(l_p))^N. \quad (A.106) $$

## Appendix B: Additional images

In this appendix, as in Fig. 12, we show newly discovered giants. These cover the projected length range $l_p \in [0.7 \text{ Mpc}, 4 \text{ Mpc})$.
<figure><img src="image_25.png" /><figcaption>**Fig. B.1.** Details of the LoTSS DR2-estimated specific intensity function $I_v(\hat{r})$ at central observing frequency $v_{\text{obs}}$ = 144 MHz and resolutions $\theta_{\text{FWHM}} \in \{6'', 20'' \ 90''\}$, centred around the hosts of newly discovered giants. Row-wise from left to right, from top to bottom, the projected proper length $l_p$ is 3.9 Mpc, 3.5 Mpc, 3.3 Mpc, 3.3 Mpc, 3.3 Mpc, and 3.2 Mpc; in the same order, $\theta_{\text{FWHM}}$ is 90'', 20'', 20'', 6'', 6'', and 20''. The giants in the top-left and middle-left panels appear larger in the sky than the Moon. Contours signify 3, 5 and 10 sigma-clipped standard deviations above the sigma-clipped median. For scale, we show the stellar Milky Way disk (with a diameter of 50 kpc) generated using the Ringermacher & Mead (2009) formula, alongside a 5 or 10 times inflated version.</figcaption></figure>
<figure><img src="image_26.png" /><figcaption>**Fig. B.2.** Details of the LoTSS DR2-estimated specific intensity function $I_ν(\hat{r})$ at central observing frequency $v_{\text{obs}} = 144$ MHz and resolutions $\theta_{\text{FWHM}} \in \{6''$, $20''\}$, centred around the hosts of newly discovered giants. Row-wise from left to right, from top to bottom, the projected proper length $l_p$ is 2.8 Mpc, 2.6 Mpc, 2.6 Mpc, 2.2 Mpc, 2.1 Mpc, and 2.1 Mpc; in the same order, $\theta_{\text{FWHM}}$ is 6'', 20'', 6'', 20'', 20'', and 6''. The GRG in the bottom-left panel appears larger in the sky than the Moon. Contours signify 3, 5, and 10 sigma-clipped standard deviations above the sigma-clipped median. For scale, we show the stellar Milky Way disk (with a diameter of 50 kpc) generated using the Ringermacher & Mead (2009) formula, alongside a 5 or 10 times inflated version.</figcaption></figure>
<figure><img src="image_27.png" /><figcaption>**Fig. B.3.** Details of the LoTSS DR2-estimated specific intensity function $I_v(\hat{r})$ at central observing frequency $v_{\text{obs}} = 144$ MHz and resolutions $\theta_{\text{FWHM}} \in \{6''$, $20''\}$, centred around the hosts of newly discovered giants. Row-wise from left to right, from top to bottom, the projected proper length $l_p$ is 1.6 Mpc, 1.5 Mpc, 1.3 Mpc, 1.2 Mpc, 1.1 Mpc, and 0.7 Mpc; in the same order, $\theta_{\text{FWHM}}$ is 6'', 6'', 20'', 6'', 6'' and 6''. Contours signify 3, 5, and 10 sigma-clipped standard deviations above the sigma-clipped median. For scale, we show the stellar Milky Way disk (with a diameter of 50 kpc) generated using the Ringermacher & Mead (2009) formula, alongside a 3 or 5 times inflated version.</figcaption></figure>
## Appendix C: Stellar and supermassive black hole masses

In Fig. C.1, we present the SDSS-derived relations between host stellar mass and projected proper length, and between host supermassive black hole mass and projected proper length, for all giants in our final catalogue. We obtained the data as in Section 3.7 of Oei et al. (2022a).

<figure><img src="image_28.png" /><figcaption>**Fig. C.1.** Observed relations between host stellar mass $M_*$ and GRG projected length $l_p$ (top) and between host SMBH mass $M_*$ and GRG projected length $l_p$ (bottom). Our LoTSS DR2 sample confirms that luminous giants typically have $M_* \in 10^{11}-10^{12} M_\odot$ and $M_* \in 10^8-10^{10} M_\odot$. The sample effects an almost fivefold increase in the number of giants with SDSS-derived host stellar masses and SMBH masses. We do not show or count giants for which only a nearest host candidate could be determined.</figcaption></figure>

## Appendix D: Surface brightness prior

Consider a radio galaxy at cosmological redshift $z$ of intrinsic proper length $l$ bounded by spherical lobes of radius $R$ and spectral index $\alpha$. If a fraction $f_l$ of the radio galaxy's central axis lies within the lobes, then

$$R = \frac{1}{4} f_l \cdot l. \qquad (D.1)$$

If a fraction $f_L$ of the radio galaxy's total luminosity density $L_v$ comes from the lobes, then the monochromatic emission coefficient

$$j_v = \frac{L_v f_L}{4\pi^2 \frac{1}{2} V}, \qquad (D.2)$$

where the lobe volume $V = \frac{4}{3}\pi R^3$. A formula of practical value features projected proper length $l_p$ instead of $l$. Given the approximate nature of our approach, we therefore simply assume $l \approx \mathbb{E}[D](\eta(f_l)) \cdot l_p$, with the deprojection factor expectation $\mathbb{E}[D]$ given in Eq. A.29; $\eta(f_l) = \frac{f_l}{2-f_l}$. The maximum surface brightness of the lobes as seen by an observer is

$$b_{v,\max} = \frac{3 f_{L_v} \cdot L_v}{\pi^2 \cdot \mathbb{E}[D](\eta(f_l)) \cdot f_l^2 \cdot l_p^2 \cdot (1+z)^{3-\alpha}}, \qquad (D.3)$$

valid for the line of sight that pierces through a lobe along a diameter. The average line of sight length $d$ within a lobe is smaller than $d_{\max} = 2R$, though, and given by

$$\langle d \rangle := \frac{\int_0^R 2 \sqrt{R^2 - x^2} 2\pi x \, dx}{\pi R^2} = \frac{4}{3} R. \qquad (D.4)$$

Therefore, the mean surface brightness of the lobes as seen by an observer is

$$\langle b_v \rangle = \frac{\langle d \rangle}{d_{\max}} b_{v,\max} = \frac{2}{3} b_{v,\max}. \qquad (D.5)$$

## Appendix E: Likelihood function

In Table E.1, we present maximum likelihood and likelihood mean and standard deviation estimates for the inference described in Section 4.1.4; one may compare the results to those in Table 4. In Fig. E.1, we visually summarise the likelihood function; one may compare to Fig. 16.

The strong degeneracy between $\xi$ and $b_{v,ref}$, directly apparent from the central two-parameter marginal in the leftmost column of Fig. E.1, translates to a ridge of essentially constant likelihood that extends from $\xi \approx -4$ to $\xi \approx -2$. Compared to a non-degenerate case, this makes the maximum likelihood parameters both intrinsically less meaningful and more prone to numerical approximation errors.

<table><caption>**Table E.1.** Maximum likelihood estimate (MLE) and likelihood mean and standard deviation (SD) estimates of the free parameters in intrinsic GRG length distribution inference.<fnref n="1" /></caption><thead><tr><th colspan="3" style="text-align:center;">z<sub>max</sub> = 0.5</th></tr><tr><th>parameter</th><th>MLE</th><th>likelihood mean and SD</th></tr></thead><tbody><tr><td>ξ</td><td>-2.15</td><td>-3.4 ± 0.5</td></tr><tr><td>1/2(φ<sub>max</sub> - φ<sub>min</sub>)</td><td>1.7'</td><td>1.9 ± 0.3'</td></tr><tr><td>b<sub>v,ref</sub></td><td>90 Jy deg<sup>-2</sup></td><td>660 ± 400 Jy deg<sup>-2</sup></td></tr><tr><td>σ<sub>ref</sub></td><td>1.35</td><td>1.2 ± 0.2</td></tr></tbody></table><table><thead><tr><th colspan="3" style="text-align:center;">z<sub>max</sub> = 0.25</th></tr><tr><th>parameter</th><th>MLE</th><th>likelihood mean and SD</th></tr></thead><tbody><tr><td>ξ</td><td>-2.2</td><td>-3.5 ± 0.4</td></tr><tr><td>1/2(φ<sub>max</sub> - φ<sub>min</sub>)</td><td>2.25'</td><td>1.7 ± 0.3'</td></tr><tr><td>b<sub>v,ref</sub></td><td>150 Jy deg<sup>-2</sup></td><td>1050 ± 560 Jy deg<sup>-2</sup></td></tr><tr><td>σ<sub>ref</sub></td><td>1.25</td><td>1.4 ± 0.4</td></tr></tbody></table>

<footnote n="1">¹ The model assumes $\xi$ is constant for $z \in [0, z_{\max}]$. We determined the likelihood function twice: for $z_{\max} = 0.5$, using 1473 giants, and for $z_{\max} = 0.25$, using 811 giants. We caution that, in this case, the MLE parameters are a poor measure of central tendency.</footnote>
<figure><img src="image_29.png" /></figure>

**Fig. E.1.** Joint likelihood function over $\xi$ — the parameter of interest — and $\frac{1}{2}(\phi_{\text{max}} - \phi_{\text{min}})$, $b_{\nu,\text{ref}}$ and $\sigma_{\text{ref}}$ — the selection effect parameters, based on 1473 projected lengths of LoTSS DR2 giants up to $z_{\text{max}} = 0.5$. We show all two-parameter marginals of the likelihood function, with contours enclosing 30% and 70% of total probability. We mark the maximum likelihood parameters (white circle) and the likelihood mean parameters (white cross). The single-parameter marginals again show the estimated mean, now marked by a vertical line, alongside shaded median-centred 80% credible intervals. The likelihood function is the posterior for a uniform prior. To compare the likelihood function to the posterior actually chosen, see Fig. 16.

## Appendix F: Properties of newly discovered giants

Table 2 provides properties of the 50 projectively longest giants discovered during this work’s LoTSS DR2 search campaign. We share these data, alongside those for the other 2010 (98%)

giants in our sample, in Flexible Image Transport System (FITS) format through the Centre de Données astronomiques de Strasbourg (CDS).

For our final catalogue, which also includes literature giants, please contact the authors.