<table><caption>**Table C4.** As in Table 2, but now showing the results of the correlation analysis when the REXCOR results for the five observations with negative $\Delta\chi^2$ in Table C1 are used in the analysis (Fig. C1). The significance of the three relationships with $\lambda_{\text{obs}}$ presented in the top half of the table are relatively unaffected with this change. Evidence for the 'v'-shape transition in the $\tau$ and $h_f$ relationships with $\lambda_{\text{obs}}$ is lost after switching in the new values, but is recovered after removing the two low $\lambda_{\text{obs}}$ sources that exhibited the largest change in $h_f$ and $\tau$.</caption><thead><tr><th>Relationship with λ<sub>obs</sub></th><th>Kendall's τ</th><th>p-value</th></tr></thead><tbody><tr><td>f<sub>x</sub></td><td>-0.56 ± 0.06</td><td>2.8<sup>+31</sup><sub>-25</sub> × 10<sup>-6</sup></td></tr><tr><td>Γ</td><td>0.24 ± 0.06</td><td>0.050<sup>+0.097</sup><sub>-0.037</sub></td></tr><tr><td>(0.3–10) keV flux ratio</td><td>0.56 ± 0.03</td><td>2.8<sup>+6.6</sup><sub>-2.0</sub> × 10<sup>-6</sup></td></tr><tr><td>(0.3–10) keV flux ratio (no RE 1034+396)</td><td>0.53 ± 0.03</td><td>1.1<sup>+2.6</sup><sub>-0.8</sub> × 10<sup>-5</sup></td></tr><tr><td>τ and h<sub>f</sub> transition λ<sub>obs</sub> (i.e. λ<sub>obs,t</sub>)</td><td>Mean p-value</td><td>Mean p-value (no Mrk 590 and ESO 198–G24 [Obs ID 0305370101])</td></tr><tr><td>0.10</td><td>0.32<sup>+0.26</sup><sub>-0.18</sub></td><td>0.16<sup>+0.33</sup><sub>-0.13</sub></td></tr><tr><td>0.12</td><td>0.36 ± 0.14</td><td>0.062<sup>+0.199</sup><sub>-0.049</sub></td></tr><tr><td>0.15</td><td>0.10<sup>+0.11</sup><sub>-0.06</sub></td><td>0.007<sup>+0.024</sup><sub>-0.005</sub></td></tr><tr><td>0.20</td><td>0.16<sup>+0.14</sup><sub>-0.11</sub></td><td>0.12<sup>+0.08</sup><sub>-0.09</sub></td></tr></tbody></table>

## C2 The coronal height, h

For a specific $\lambda$ (either $\lambda = 0.1$ or 0.01), there are REXCOR grids calculated for two specific heights of the hot, X-ray emitting corona: $h = 5$ or $20\ r_g$ (Xiang et al. 2022). The purpose of these grids is not to make a measurement of $h$, per se, but to distinguish between a low lamppost (at $h = 5$) and a high lamppost (at $h = 20$), since they lead to different illumination patterns on the accretion disc (e.g. Fukumura & Kazanas 2007; Dauser et al. 2013; Ballantyne 2017).

Table A1 shows the value of $h$ for the REXCOR grid that gives the lowest $\chi^2$. In 13 of the 34 observations (those indicated with a † symbol by the Obs ID) fits with the alternative coronal height have a $\Delta\chi^2 < 6$, which, based on the $\Delta$BIC criterion described above, means that one value of $h$ cannot be preferred over the other. In 9 out of the 13 observations, the warm corona parameters found with the alternative $h$ are consistent with the ones shown in Table A1. For UGC 3973 (Obs ID 040070301) the fit with $h = 20$ gives $\tau = 23.3^{+0.4}_{-3.3}$ (rather than $16.5 \pm 0.3$), which is still consistent with the observed trends seen in Fig. 3. A value of $h = 5$ when fitting observation (Obs ID 0601390701 of Mrk 509 pushes $h_f$ to 0.76, $\tau$ to 16.8, and $f_x$ down to 0.02. When $h = 20$ observation (Obs ID 030248501) of Mrk 279 yields $\tau = 23.8^{+5.8}_{-3.8}$, inconsistent with the value of $\tau = 14^{+4.1}_{-1}$ found when $h = 5$. This new value of $\tau$ would pull this observation away from the 'v' shape seen in Fig. 3, reducing its significance. However, all the other observations of Mrk 509 and Mrk 279 give consistent REXCOR parameters when fit with either coronal height. Although we cannot rule out variability intrinsic to the source, since all the observations of these objects have approximately the same $\lambda_{\text{obs}}$, we expect that the best-fitting values shown in Table A1 remain the most likely results.

Lastly, a coronal height of $h = 5$ decreases both $\tau$ and $h_f$ for UGC 3973 (Obs ID 0400070401) to $10.6^{+1.7}_{-0.6p}$ and $0.43^{+0.14}_{-0.11}$, respectively. These values are similar to the ones found in Obs ID 0400070201 of the same source, and are inconsistent with the trends

with $\lambda_{\text{obs}}$ seen in Fig. 3. The $h = 20$ and 5 fits both give a reduced $\chi^2 \approx 1$ and so it is difficult to distinguish the two cases. However, this is the only observation where a difference in $h$ may plausibly increase the scatter in the $\tau-\lambda_{\text{obs}}$ and $h_f-\lambda_{\text{obs}}$ planes.

## C3 The black hole spin, a

The REXCOR grids provide different choices for the black hole spin, $a = 0.9$ and 0.99 (Xiang et al. 2022). As with the corona height, the purpose is not to measure a black hole spin with these models, but to test if the data require a maximal spin as compared to a black hole that is simply rapidly spinning. In the REXCOR model, a higher spin not only increases the relativistic blurring and light-bending effects, but it also increases the flux dissipated into the coronae (which are fractions of the dissipation rate $D(r, \lambda)$; Section 2). This effect can be compensated for by reducing $f_x$, so there is a moderate degeneracy between $a$ and $f_x$ in the REXCOR model (Xiang et al. 2022).

Each AGN in the sample was fit with REXCOR grids using both values of $a$ with the one associated with lowest $\chi^2$ model listed in Table A1. If an AGN has more than one observation, then the value of $a$ found in the first observation was used for subsequent fits. For four AGNs (HE 1029–1401, IRASF 12397+3333, Mrk 509, and NGC 4593; indicated with a '*' symbol by the Obs ID in Table A1), assuming the alternative spin when fitting gives a $\Delta\chi^2 < 6$, indicating that either value of $a$ may be considered acceptable. In the cases of HE 1029–1401, IRASF 12397+3333, and NGC 4593, the resulting values of $f_x$, $h_f$, $\Gamma$, and $\tau$ are all consistent irrespective of the choice of $a$. For Mrk 509, assuming $a = 0.9$ increases $f_x$ by $\approx 0.02$ in each of the observations compared to the values listed in Table A1 with the other parameters consistent with the previous results. According to Fig. 4, such a modest change in $f_x$ for the Mrk 509 observations would not alter the observed trend between $f_x$ and $\lambda_{\text{obs}}$.
<figure><img src="image_12.png" /></figure>

**Figure C1.** As in Figs 1–4, but now using the results of Table C1 for the five observations that give a negative $\Delta\chi^2$ when using the $\lambda = 0.1$ REXCOR grids. The most significant changes in the REXCOR parameters occur in two of the lowest $\lambda_{obs}$ in our sample (Mrk 590 and ESO 198–G24 [Obs ID 0305370101]). The resulting statistical analysis is shown in Table C4, which shows that the $f_X-\lambda_{obs}$, $\Gamma-\lambda_{obs}$, and the flux ratio-$\lambda_{obs}$ correlations remain significant, but the 'v'-shaped $\tau-\lambda_{obs}$ and $h_f-\lambda_{obs}$ relationships are retained only when omitting Mrk 590 and ESO 198–G24 [Obs ID 0305370101].

This paper has been typeset from a TeX/LATEX file prepared by the author.
<figure><img src="image_13.png" /></figure>

**Figure C1.** As in Figs 1–4, but now using the results of Table C1 for the five observations that give a negative $\Delta\chi^2$ when using the $\lambda = 0.1$ REXCOR grids. The most significant changes in the REXCOR parameters occur in two of the lowest $\lambda_{\text{obs}}$ in our sample (Mrk 590 and ESO 198–G24 [Obs ID 0305370101]). The resulting statistical analysis is shown in Table C4, which shows that the $f_X-\lambda_{\text{obs}}$, $\Gamma-\lambda_{\text{obs}}$, and the flux ratio-$\lambda_{\text{obs}}$ correlations remain significant, but the 'v'-shaped $\tau-\lambda_{\text{obs}}$ and $h_f-\lambda_{\text{obs}}$ relationships are retained only when omitting Mrk 590 and ESO 198–G24 [Obs ID 0305370101].

This paper has been typeset from a TeX/LATEX file prepared by the author.