with the following covariance,

$$C_{\text{SF}}^{\text{LRG2}}(0.71) = 10^{-4} \begin{pmatrix} 762.457717 & 39.781004 & -1.068968 & -62.812849 \\ - & 36.344098 & -9.777053 & -8.333900 \\ - & - & 8.938416 & 4.998214 \\ - & - & - & 47.624520 \end{pmatrix}. \quad (A.6)$$

For the LRG3 sample at $z_{\text{eff}} = 0.92$ we find,

$$D_{\text{SF}}^{\text{LRG3}}(0.92) = \begin{pmatrix} D_V/r_d(0.92) \\ D_H/D_M(0.92) \\ f\sigma_{s8}(0.92) \\ (m+n)(0.92) \end{pmatrix} = \begin{pmatrix} 19.676985 \\ 0.844996 \\ 0.215641 \\ -0.024690 \end{pmatrix}, \quad (A.7)$$

with the following covariance,

$$C_{\text{SF}}^{\text{LRG3}}(0.92) = 10^{-4} \begin{pmatrix} 847.499793 & 26.038900 & 1.663841 & -37.016091 \\ - & 16.251088 & -5.130513 & -3.698790 \\ - & - & 5.836787 & 3.303604 \\ - & - & - & 34.883220 \end{pmatrix}. \quad (A.8)$$

For the ELG2 sample at $z_{\text{eff}} = 1.32$ we find,

$$D_{\text{SF}}^{\text{ELG2}}(1.32) = \begin{pmatrix} D_V/r_d(1.32) \\ D_H/D_M(1.32) \\ f\sigma_{s8}(1.32) \\ (m+n)(1.32) \end{pmatrix} = \begin{pmatrix} 23.861806 \\ 0.470709 \\ 0.162741 \\ 0.059960 \end{pmatrix}, \quad (A.9)$$

with the following covariance,

$$C_{\text{SF}}^{\text{ELG2}}(1.32) = 10^{-4} \begin{pmatrix} 2342.506886 & 26.159601 & 5.841073 & -95.336060 \\ - & 10.309303 & -2.874815 & -5.183903 \\ - & - & 2.612264 & 3.935355 \\ - & - & - & 43.575710 \end{pmatrix}. \quad (A.10)$$

For the QSO sample at $z_{\text{eff}} = 1.49$ we find,

$$D_{\text{SF}}^{\text{QSO}}(1.49) = \begin{pmatrix} D_V/r_d(1.49) \\ D_H/D_M(1.49) \\ f\sigma_{s8}(1.49) \\ (m+n)(1.49) \end{pmatrix} = \begin{pmatrix} 25.708520 \\ 0.426508 \\ 0.175769 \\ 0.064550 \end{pmatrix}, \quad (A.11)$$

with the following covariance,

$$C_{\text{SF}}^{\text{QSO}}(1.49) = 10^{-4} \begin{pmatrix} 3013.788566 & -2.205101 & 14.685439 & -98.167826 \\ - & 5.845806 & -2.727191 & -1.913326 \\ - & - & 3.232534 & 2.165520 \\ - & - & - & 26.266260 \end{pmatrix}. \quad (A.12)$$
We now report the results for the ShapeFit fits in combination with the BAO reconstructed fits from [73],

For the BGS sample at $z_{\text{eff}} = 0.30$ we find,

$$ D_{\text{SF+BAO}}^{\text{BGS}}(0.30) = \begin{pmatrix} D_V/r_d(0.30) \\ D_H/D_M(0.30) \\ f\sigma_{s8}(0.30) \\ (m+n)(0.30) \end{pmatrix} = \begin{pmatrix} 7.920703 \\ 3.014389 \\ 0.274892 \\ -0.049030 \end{pmatrix}, \qquad (\text{A.13}) $$

with the following covariance,

$$ C_{\text{SF+BAO}}^{\text{BGS}}(0.30) = 10^{-4} \begin{pmatrix} 239.901232 & -61.419576 & 16.258820 & -33.006574 \\ - & 726.794608 & -101.118203 & -17.577288 \\ - & - & 38.931424 & 6.105473 \\ - & - & - & 167.972620 \end{pmatrix}. \qquad (\text{A.14}) $$

For the LRG1 sample at $z_{\text{eff}} = 0.51$ we find,

$$ D_{\text{SF+BAO}}^{\text{LRG1}}(0.51) = \begin{pmatrix} D_V/r_d(0.51) \\ D_H/D_M(0.51) \\ f\sigma_{s8}(0.51) \\ (m+n)(0.51) \end{pmatrix} = \begin{pmatrix} 12.621670 \\ 1.574992 \\ 0.340079 \\ 0.020350 \end{pmatrix}, \qquad (\text{A.15}) $$

with the following covariance,

$$ C_{\text{SF+BAO}}^{\text{LRG1}}(0.51) = 10^{-4} \begin{pmatrix} 185.794784 & -2.289289 & 5.946084 & -13.518142 \\ - & 30.722651 & -9.075587 & -2.167703 \\ - & - & 13.607072 & 8.087907 \\ - & - & - & 42.091060 \end{pmatrix}. \qquad (\text{A.16}) $$

For the LRG2 sample at $z_{\text{eff}} = 0.71$ we find,

$$ D_{\text{SF+BAO}}^{\text{LRG2}}(0.71) = \begin{pmatrix} D_V/r_d(0.71) \\ D_H/D_M(0.71) \\ f\sigma_{s8}(0.71) \\ (m+n)(0.71) \end{pmatrix} = \begin{pmatrix} 15.924595 \\ 1.181734 \\ 0.270365 \\ 0.022320 \end{pmatrix}, \qquad (\text{A.17}) $$

with the following covariance,

$$ C_{\text{SF+BAO}}^{\text{LRG2}}(0.71) = 10^{-4} \begin{pmatrix} 310.004892 & 6.627447 & 5.569427 & -28.045518 \\ - & 15.261852 & -4.619876 & -1.747683 \\ - & - & 7.330424 & 3.991968 \\ - & - & - & 45.717820 \end{pmatrix}. \qquad (\text{A.18}) $$

For the LRG3 sample at $z_{\text{eff}} = 0.92$ we find,

$$ D_{\text{SF+BAO}}^{\text{LRG3}}(0.92) = \begin{pmatrix} D_V/r_d(0.92) \\ D_H/D_M(0.92) \\ f\sigma_{s8}(0.92) \\ (m+n)(0.92) \end{pmatrix} = \begin{pmatrix} 19.772506 \\ 0.825195 \\ 0.223962 \\ -0.024940 \end{pmatrix}, \qquad (\text{A.19}) $$
with the following covariance,

$$C_{\text{SF+BAO}}^{\text{LRG3}}(0.92) = 10^{-4} \begin{pmatrix} 300.214114 & -2.141102 & 4.208062 & -8.800301 \\ - & 5.531413 & -2.070961 & -0.329386 \\ - & - & 4.438534 & 2.092917 \\ - & - & - & 26.960240 \end{pmatrix}. \quad (A.20)$$

For the ELG2 sample at $z_{\text{eff}} = 1.32$ we find,

$$D_{\text{SF+BAO}}^{\text{ELG2}}(1.32) = \begin{pmatrix} D_V/r_d(1.32) \\ D_H/D_M(1.32) \\ f\sigma_{s8}(1.32) \\ (m+n)(1.32) \end{pmatrix} = \begin{pmatrix} 24.247096 \\ 0.486334 \\ 0.160989 \\ 0.041720 \end{pmatrix}, \quad (A.21)$$

with the following covariance,

$$C_{\text{SF+BAO}}^{\text{ELG2}}(1.32) = 10^{-4} \begin{pmatrix} 1205.501318 & -15.845487 & 12.762868 & -41.167094 \\ - & 4.516231 & -1.445459 & -1.151721 \\ - & - & 2.204277 & 3.612122 \\ - & - & - & 38.465670 \end{pmatrix}. \quad (A.22)$$

For the QSO sample at $z_{\text{eff}} = 1.49$ we find,

$$D_{\text{SF+BAO}}^{\text{QSO}}(1.49) = \begin{pmatrix} D_V/r_d(1.49) \\ D_H/D_M(1.49) \\ f\sigma_{s8}(1.49) \\ (m+n)(1.49) \end{pmatrix} = \begin{pmatrix} 25.768648 \\ 0.425629 \\ 0.176574 \\ 0.061950 \end{pmatrix}, \quad (A.23)$$

with the following covariance,

$$C_{\text{SF+BAO}}^{\text{QSO}}(1.49) = 10^{-4} \begin{pmatrix} 2601.484225 & -2.427625 & 16.302570 & -77.850995 \\ - & 5.679769 & -2.869848 & -2.754462 \\ - & - & 3.435620 & 2.825103 \\ - & - & - & 26.733090 \end{pmatrix}. \quad (A.24)$$

## B Consistency tests on the full-modelling approach

In this appendix, we display the battery of tests performed on unblinded data that demonstrate that the results presented in section 7 are robust. These tests are complementary to those shown both in section 4.7 about the baseline selection, and those on section 6.1 about the robustness of unblinded results, and just re-confirm the same conclusions drawn from those sections.

Figure 19 displays the performance of different setup options for the priors of counter-terms and stochastic terms, analogously to figure 2, but for the DESI DR1 data instead. We focus only on the Full-Shape combined with BAO data, but the results are very similar to the Full-Shape only datasets. Changing the counter-terms prior can moderately affect some parameters by fractions of the statistical 68% confidence region, similarly to what
<figure><img src="image_22.png" /><figcaption>**Figure 19.** Constraints on ΛCDM parameters from the joint fit of the six tracers of DESI DR1 data with the BAO reconstruction signal included. We employ the Full-Modelling approach using different choices of priors on counterterms (CT) and stochastic terms (ST), as displayed, and described in section 4.7. The filled(empty) symbols represent the mean(MAP) value. The rows correspond to the same baseline models tested in figure 2.</figcaption></figure>

was displayed already in figure 2. We therefore re-confirm with the actual data that our baseline choice produces robust results.

We also have explored some extra model choices, such as the choice between the Eulerian and Lagrangian PT modes. The results on the DESI DR1 data are very similar to those already shown in figure 4 and we do not show them for conciseness. The highest deviation we observe in this choice is on the $\Omega_m$ parameter, with a change of $\Delta\Omega_m \sim 0.002$ towards higher values of $\Omega_m$ when the Lagrangian PT is used compared to the Eulerian PT, when using both full-modelling and reconstructed BAO information. Additionally, we have checked that by adding the hexadecapole signal to the DESI DR1 data does not produce any significant changes in the contours, as anticipated by ref. [100] in figure 10.

In figure 20 we display the impact of different minimum cutoff scales, $k_{max}$ for the power spectrum perturbation theory model on the DESI DR1 data. Refs. [71, 72, 100] already validated using mocks that the cutoff scale was around $0.20 h\,\text{Mpc}^{-1}$, which is part of the baseline model. We display two additional conservative cuts, at $k_{max} = 0.16, 0.18 h\,\text{Mpc}^{-1}$. We find barely no difference in the position of the maximum of the posteriors nor in their width, validating the robustness of results and statistical errors under changes of the exact $k_{max}$ choice. We have performed a similar analysis changing the $k_{min}$ choice to vary from $0.02 h\,\text{Mpc}^{-1}$ to $0.04 h\,\text{Mpc}^{-1}$ with no significant changes, as displayed in figure 21.

We also aim to validate the effect of the mock-based covariance in the final results. We compare the baseline results with those obtained when using an analytical-based covariance as described in section 4.6. As described before, the mock-based covariance is generated
<figure><img src="image_23.png" /><figcaption>**Figure 20.** Effect of changing the minimum scale cutoff, $k_{max}$ on the posteriors of the ΛCDM model for the Full-Modelling type of analysis in combination with the BAO reconstructed data on DESI DR1 data. We test two larger cutoff scales, $k_{max} = 0.16, 0.18 \text{ hMpc}^{-1}$, in addition to the baseline choice of $0.20 \text{ hMpc}^{-1}$.</figcaption></figure>

from 1000 EZmocks, whereas the analytical approach is based on Gaussian approximation in combination with non-linear clustering methods. Figure 22 shows very good agreement between the posteriors obtained from the mock-based covariance (the baseline choice) and the alternative analytical covariance choice. We observe that the analytical approach slightly under-estimates the posteriors obtained from the mock-based covariance, likely due to the non-inclusion of all non-linear terms. Thus, we conclude that both methods retrieve very consistent posteriors, and that the baseline method is the most conservative option as it retrieves slightly larger errors.

For a full description of the effect of the imaging weight correction on the cosmology results for the full-modelling approach we refer the reader to ref. [85]. In short, we find that dropping the correction terms of equation (5.1) produces no change for the LRG and BGS samples as expected. Surprisingly, we also find that the effect of these terms is very small for the QSO sample. The largest effect is therefore for the ELG sample (as displayed on the left panel of figure 23), although the overall effect is to modify the position of the
<figure><img src="image_24.png" /><figcaption>**Figure 21.** Effect of changing the maximum scale cutoff, $k_{\min}$ on the posteriors of the ΛCDM model for the Full-Modelling type of analysis in combination with the BAO reconstructed data on DESI DR1 data. We test the cutoff scales, $k_{\max} = 0.03, 0.04 \, h\,\text{Mpc}^{-1}$, in addition to the baseline choice of $0.02 \, h\,\text{Mpc}^{-1}$.</figcaption></figure>

posteriors of $\Omega_m$ and $\sigma_8$ by around 1/4 of a sigma at the most. Although not shown, the impact on $\Omega_b h^2$ and $n_s$ is negligible. The panels of figure 23 show the posteriors when the reconstructed BAO information is also added, although the results look very similar if the BAO reconstructed data is not used. However, when all samples are combined (right panel of figure 23), the overall effect of ignoring these correction terms is again negligible, as the signal is mainly driven by BGS, LRGs and QSO.

Finally, we also test here the effect of changing the prior on the physical baryon density, $\Omega_b h^2$ from the BBN one to the Planck prior, for the DESI DR1 data in combination with the BAO reconstructed signal. Since this choice mainly affects the determination of $H_0$, we display in figure 24 the contours only for this parameter. We see that the change in prior on $\Omega_b h^2$ does not affect the constraints on $H_0$, as expected given the very good consistency between the physical baryon density inferred from BBN and Planck data.
<figure><img src="image_25.png" /><figcaption>**Figure 22.** Effect of the choice of the covariance when analyzing the DESI DR1 data. The orange contours display the EZmock-based covariance which is the baseline method employed for the main results of this work. The blue contours display the results for the analytical covariance, as described in section 4.6.</figcaption></figure>

### C Fiducial cosmology values for the ShapeFit compressed parameters

Here we report the table with the fiducial BAO distances and $\sigma_{s8}$ value employed in this paper for the ShapeFit type of analysis. The details of this fiducial cosmology model, based on Planck-ΛCDM, are reported in the first row of table 6.

### D Pre-reconstructed two-point clustering of DESI DR1

Figure 25 and figure 26 show the three multipoles of the power spectrum of the four DESI samples with the LRG sample being split into three redshift bins for the unblinded (solid) and blinded (dashed) catalogues. The shaded region corresponds to the $1\sigma$ error from the diagonal terms of the EZmocks covariance matrix. We notice that the blinding procedure successfully modified the BAO position and the amplitude of the quadrupole at large scales. The biggest effect can be observed on the amplitude of the monopole and quadrupole at large
<figure><img src="image_26.png" /><figcaption>**Figure 23.** Effect of not applying the imaging systematic correction described in equation (5.1), on the ELG sample (left panel) and on all combined galaxy and quasar samples (right panel). Although $n_s$ and $\Omega_b$ are also varied, here we do not display them for conciseness, as the imaging systematic correction impact on those variables is totally negligible. For both panels we display the posteriors obtained when combining the full-shape analysis with the BAO reconstruction information, although only considering the full-shape information does not change the significantly the relative effect.</figcaption></figure>

<table><caption>**Table 11.** Fiducial values on the angular and radial BAO distances, $D_M$ and $D_H$, in units of the sound horizon scale, for the six DESI DR1 samples. Additionally, we report the isotropic BAO distance, $D_V$ and the Alcock-Paczynski related parameter, $D_H/D_M$. For all redshift bins, we have that the sound horizon scale at drag epoch is $r_d = 99.0792$ [Mpc/h]. We also report the amplitude of cold+baryon matter fluctuations smoothed in units of 8 Mpc/h of the fiducial cosmology, $\sigma_{s8}$.</caption><thead><tr><th>sample (redshift)</th><th>D<sub>M</sub>(z)/r<sub>d</sub></th><th>D<sub>H</sub>(z)/r<sub>d</sub></th><th>D<sub>V</sub>(z)/r<sub>d</sub></th><th>D<sub>H</sub>(z)/D<sub>M</sub>(z)</th><th>σ<sub>s8</sub>(z)</th><th>fσ<sub>s8</sub>(z)</th></tr></thead><tbody><tr><td>BGS (0.295)</td><td>8.2908</td><td>25.8506</td><td>8.0663</td><td>3.1180</td><td>0.6936</td><td>0.4723</td></tr><tr><td>LRG1 (0.510)</td><td>13.4928</td><td>22.7462</td><td>12.8269</td><td>1.6858</td><td>0.6210</td><td>0.4733</td></tr><tr><td>LRG2 (0.706)</td><td>17.6976</td><td>20.1727</td><td>16.4597</td><td>1.1399</td><td>0.5638</td><td>0.4608</td></tr><tr><td>LRG3 (0.919)</td><td>21.7238</td><td>17.7321</td><td>19.7356</td><td>0.8162</td><td>0.5108</td><td>0.4398</td></tr><tr><td>ELG2 (1.317)</td><td>28.0276</td><td>14.0956</td><td>24.4318</td><td>0.5029</td><td>0.4320</td><td>0.3944</td></tr><tr><td>QSO (1.491)</td><td>30.3606</td><td>12.8359</td><td>26.0292</td><td>0.4228</td><td>0.4042</td><td>0.3750</td></tr></tbody></table>

scales for QSO. For further validation tests of the blinding procedure for DESI DR1 BAO and Full-Shape galaxy two-point clustering, we refer the reader to our supporting paper [93].

## E Gaussian priors on counterterms

Here we explicitly show how we define the priors on counterterms based on their contribution (at $1\sigma$) on the total linear signal.

The galaxy bias term at leading order can be written as,

$$ \delta^{(1)}(k, \mu) = (b + f\mu^2)\delta_L(k) \qquad (E.1) $$
<figure><img src="image_27.png" /><figcaption>**Figure 24.** Effect of modifying the physical density baryon prior on the $H_0$ parameter inferred from DESI DR1 data in combination with the BAO reconstructed signal. We choose the BBN-informed prior (as our baseline choice) and the Planck measured value, as labelled. Due to the high consistency between BBN and Planck, the impact on $H_0$ is minor. We do not display the rest of ΛCDM parameters, as their dependence on $\Omega_b h^2$ (as expected) is negligible.</figcaption></figure>

where $\delta_L$ is the linear matter$^{25}$ field in real space, $b$ is the (linear) bias, $f$ the logarithmic growth factor and $\mu$ the cosine of the angle between the galaxy pair vector and the line-of-sight. The counterterm contribution at the $\delta$-level is given by,

$$ \delta_{\text{CT}}(k, \mu) = \frac{1}{2}(b\alpha_0 + f\mu^2\alpha_2 + \dots)k^2\delta_L(k) \quad (\text{E.2}) $$

where $\alpha_i$ are the counterterms, which for this case we only consider up to $\mu^2$ term.

We then write the linear galaxy signal in redshift space and the first counter-term contribution, respectively, as,

$$ P_{\text{lin,g}}(k, \mu) = \langle \delta^{(1)}(k) \delta^{(1)}(k)^* \rangle; \quad P_{\text{CT,g}}(k, \mu) = 2 \langle \delta^{(1)}(k) \delta_{\text{CT}}(k)^* \rangle \quad (\text{E.3}) $$

The Gaussian prior we impose on the counterterms $\alpha_i$ are such that the contribution of $P_{\text{CT,g}}$ is a fraction of $X$ of $P_{\text{lin,g}}$ at $k = k_{\text{max}}$ for $1\sigma$,

$$ XP_{\text{lin}}(k_{\text{max}}, \mu) \ge |P_{\text{CT}}(k_{\text{max}}, \mu)| \quad (\text{E.4}) $$

<footnote>$^{25}$In fact, only baryons and dark matter, ‘cb’.</footnote>
<figure><img src="image_28.png" /><figcaption>**Figure 25.** Power spectrum multipoles (monopole on the left columns, quadrupole on the middle column, hexadecapole on the right column) corresponding to the BGS (top row), ELG (middle row) and QSO (bottom row) data samples before (dashed lines) and after unblinding (solid lines).</figcaption></figure>

The $\mu^0$ term implies that $\alpha_0$ must satisfy,

$$ \frac{X}{k_{\text{max}}^2} \geq |\alpha_0| \qquad (\text{E.5}) $$

and similarly for the $\mu^2$ term on $\alpha_2$ (given that we now have fixed the prior on $\alpha_0$),

$$ \frac{X}{k_{\text{max}}^2} \geq |\alpha_2| \qquad (\text{E.6}) $$

For our baseline choice $X = 0.5$ (50% contribution) and $k_{\text{max}} = 0.20 h^{-1} \text{Mpc}$ we find the 1$\sigma$ Gaussian constraints used in table 4,

$$ \alpha_{0,2} = \frac{0.5}{0.20^2} \text{Mpc}^2 h^{-2} = 12.5 \text{Mpc}^2 h^{-2} \qquad (\text{E.7}) $$
<figure><img src="image_29.png" /><figcaption>**Figure 26.** Power spectrum multipoles before and after unblinding. Same notation as figure 25 but for the three LRG redshift bins.</figcaption></figure>

## F Debiasing the inverse of the sum of a sample covariance and an external covariance

We consider here the case of adding a sample-estimated covariance matrix and some external (e.g. systematic) covariance.

The standard sample covariance matrix estimate for an unknown mean is,

$$X_{ab} = \frac{1}{n_S - 1} \sum_{i=1}^{n_S} (x_{a,i} - \bar{x}_a)(x_{b,i} - \bar{x}_b), \quad (F.1)$$

with $\bar{x}_a \equiv \frac{1}{n_S} \sum_{i=1}^{n_S} x_{a,i}$.

The sample covariance estimate is unbiased, meaning that the expectation value is the true covariance: $\langle X \rangle = C_s$. We can write $X = C_s + \delta X$ with $\langle \delta X \rangle = 0$.
The covariance of sample covariance elements estimated in this way are,

$$ \operatorname{cov}(X_{ab}, X_{cd}) = \langle \delta X_{ab} \delta X_{cd} \rangle = \frac{C_{s,ac}C_{s,bd} + C_{s,ad}C_{s,bc}}{n_S - 1}, \qquad (F.2) $$

and the total covariance is,

$$ C = X + C_e, \qquad (F.3) $$

and we assume that $C_e$ is exact. Naturally, $\langle C \rangle = C_s + C_e$.

Let now $C^{-1}$ be a biased estimate due to the noise in $X$ and the non-linearity of the inversion operation. Hence, we can write $\langle C^{-1} \rangle \neq (C_s + C_e)^{-1} \equiv \Psi_0$, where the term in the r.h.s. of the non-equality is what we aim to compute.

Let us now expand the inverse of the total covariance,

$$ C^{-1} = (C_s + C_e + \delta X)^{-1} = [\mathbb{I} + (C_s + C_e)^{-1} \delta X]^{-1} (C_s + C_e)^{-1} \qquad (F.4) $$

$$ = (\mathbb{I} + \Psi_0 \delta X)^{-1} \Psi_0 = (\mathbb{I} - \Psi_0 \delta X + \Psi_0 \delta X \Psi_0 \delta X - \dots) \Psi_0. \qquad (F.5) $$

Then let us take the ensemble average (its expectation value),

$$ \langle C^{-1} \rangle \approx (\mathbb{I} - \underline{\Psi}_{\theta} \langle \delta X \rangle + \Psi_0 \langle \delta X \Psi_0 \delta X \rangle) \Psi_0. \qquad (F.6) $$

The linear term drops out because $\langle \delta X \rangle = 0$.

If we designate $\mathbf{D} \equiv \Psi_0 \langle \delta X \Psi_0 \delta X \rangle$ we then can write,

$$ \langle C^{-1} \rangle \approx (\mathbb{I} + \mathbf{D}) \Psi_0, \qquad (F.7) $$

$$ \Psi_0 \approx (\mathbb{I} + \mathbf{D})^{-1} \langle C^{-1} \rangle \approx (\mathbb{I} - \mathbf{D}) \langle C^{-1} \rangle. \qquad (F.8) $$

Hence, left-multiplication by $(\mathbb{I} - \mathbf{D})$ unbiases the estimate of the inverse of the total covariance to the next-to-leading nonvanishing order.

Now, to compute the $\mathbf{D}$ matrix we first consider its $ij$ element,

$$ D_{ij} = \Psi_{0,ik} \Psi_{0,lm} \langle \delta X_{kl} \delta X_{mj} \rangle = \frac{1}{n_S - 1} \Psi_{0,ik} \Psi_{0,lm} (C_{s,km} C_{s,lj} + C_{s,kj} C_{s,lm}). \qquad (F.9) $$

We can then cast this back to matrix operations:

$$ \mathbf{D} = \frac{1}{n_S - 1} \Psi_0 \mathbf{C}_s [\operatorname{tr}(\Psi_0 \mathbf{C}_s) \cdot \mathbb{I} + \Psi_0 \mathbf{C}_s]. \qquad (F.10) $$

In the special case of $C_e = 0$, $\Psi_0 = C_s^{-1}$ and we obtain the Hartlap correction $\mathbf{D} = \frac{N_{\text{bins}}+1}{n_S-1} \cdot \mathbb{I}$.

In the general case, we should use approximations instead of unknown $\Psi_0$ and $C_s$ in the expression for $\mathbf{D}$. It is natural to take $C_s \approx \mathbf{X}$ and $\Psi_0 \approx (\mathbf{X} + C_e)^{-1}$. Thus the practical estimates would be,

$$ \hat{\mathbf{D}} = \frac{1}{n_S - 1} (\mathbf{X} + C_e)^{-1} \mathbf{X} \{ \operatorname{tr}[(\mathbf{X} + C_e)^{-1} \mathbf{X}] \cdot \mathbb{I} + (\mathbf{X} + C_e)^{-1} \mathbf{X} \}, \qquad (F.11) $$

$$ \hat{\Psi} = (\mathbb{I} - \hat{\mathbf{D}})(\mathbf{X} + C_e)^{-1}. \qquad (F.12) $$

This unbiased (to next-to-leading order) precision matrix estimate is symmetric (theoretically, numerical errors may lead to small violations).
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The DESI collaboration

A.G. Adame$^{a}$, J. Aguilar$^{b}$, S. Ahlen$^{c}$, S. Alam$^{d}$, D.M. Alexander$^{e,f}$, M. Alvarez$^{b}$, O. Alves$^{g}$,
A. Anand$^{b}$, U. Andrade$^{h,g}$, E. Armengaud$^{i}$, S. Avila$^{j}$, A. Aviles$^{k,l}$, H. Awan$^{g}$,
S. Bailey$^{b}$, C. Baltay$^{m}$, A. Bault$^{n}$, J. Behera$^{o}$, S. BenZvi$^{p}$, F. Beutler$^{q}$, D. Bianchi$^{r}$,
C. Blake$^{s}$, R. Blum$^{t}$, S. Brieden$^{q}$, A. Brodzeller$^{b}$, D. Brooks$^{u}$, E. Buckley-Geer$^{v,w}$,
E. Burtin$^{i}$, R. Calderon$^{x}$, R. Canning$^{y}$, A. Carnero Rosell$^{z,aa}$, R. Cereskaite$^{ab}$,
J.L. Cervantes-Cota$^{ac}$, S. Chabanier$^{b}$, E. Chaussidon$^{b}$, J. Chaves-Montero$^{d,j}$, S. Chen$^{ad}$,
X. Chen$^{m}$, T. Claybaugh$^{b}$, S. Cole$^{f}$, A. Cuceu$^{ae,af}$, T.M. Davis$^{ag}$, K. Dawson$^{ah}$,
A. de la Macorra$^{ai}$, A. de Mattia$^{i}$, N. Deiosso$^{aj}$, A. Dey$^{t}$, B. Dey$^{ak}$, Z. Ding$^{al}$, P. Doel$^{u}$,
J. Edelstein$^{am,an}$, S. Eftekharzadeh$^{ao}$, D.J. Eisenstein$^{ap}$, A. Elliott$^{aq,ar}$, P. Fagrelius$^{t}$,
K. Fanning$^{as,at}$, S. Ferraro$^{b,an}$, J. Ereza$^{au}$, N. Findlay$^{y}$, B. Flaugher$^{w}$, A. Font-Ribera$^{u,j}$,
D. Forero-Sánchez$^{av}$, J.E. Forero-Romero$^{aw,ax}$, C. Garcia-Quintero$^{ap,ay,af}$,
L.H. Garrison$^{az,ba}$, E. Gaztañaga$^{bb,y,bc}$, H. Gil-Marín$^{bd,bb,be}$, S. Gontcho A Gontcho$^{b}$,
A.X. Gonzalez-Morales$^{bf,bg}$, V. Gonzalez-Perez$^{bh,a}$, C. Gordon$^{j}$, D. Green$^{n}$, D. Gruen$^{bi,bj}$,
R. Gsponer$^{y,av}$, G. Gutierrez$^{w}$, J. Guy$^{b}$, B. Hadzhiyska$^{b,an}$, C. Hahn$^{bk}$, M.M. S Hanif$^{g}$,
H.K. Herrera-Alcantar$^{bl,i,bg}$, K. Honscheid$^{ae,aq,ar}$, C. Howlett$^{ag}$, D. Huterer$^{g}$, V. Iršič$^{bm,bn,bo}$,
M. Ishak$^{ay}$, S. Juneau$^{t}$, N.G. Karaçaylı$^{ae,bp,aq,ar}$, R. Kehoe$^{bq}$, S. Kent$^{v,w}$, D. Kirkby$^{n}$,
H. Kong$^{j,br}$, S.E. Koposov$^{q,bs}$, A. Kremin$^{b}$, A. Krolewski$^{bt,bu,bv}$, Y. Lai$^{ag}$, T.-W. Lan$^{bw}$,
M. Landriau$^{b}$, D. Lang$^{bu}$, J. Lasker$^{bx,bq}$, J.M. Le Goff$^{i}$, L. Le Guillou$^{by}$, A. Leauthaud$^{bz,ca}$,
M.E. Levi$^{b}$, T.S. Li$^{cb}$, K. Lodha$^{x,cc}$, C. Magneville$^{i}$, M. Manera$^{cd,j}$, D. Margala$^{b}$,
P. Martini$^{ae,bp,ar}$, M. Maus$^{an}$, P. McDonald$^{b}$, L. Medina-Varela$^{ay}$, A. Meisner$^{t}$,
J. Mena-Fernández$^{ce}$, R. Miquel$^{cf,j}$, J. Moon$^{cg}$, S. Moore$^{f}$, J. Moustakas$^{ch}$, E. Mueller$^{ab}$,
A. Muñoz-Gutiérrez$^{ai}$, A.D. Myers$^{ci}$, S. Nadathur$^{y}$, L. Napolitano$^{ci}$, R. Neveux$^{q}$,
J.A. Newman$^{ak}$, N.M. Nguyen$^{g}$, J. Nie$^{cj}$, G. Niz$^{bg,l}$, H.E. Noriega$^{k,ai}$, N. Padmanabhan$^{m}$,
E. Paillas$^{bt,ck,bv}$, N. Palanque-Delabrouille$^{i,b}$, J. Pan$^{g}$, S. Penmetsa$^{bt}$, W.J. Percival$^{bt,bu,bv}$,
M.M. Pieri$^{cl}$, M. Pinon$^{i}$, C. Poppett$^{b,am,an}$, A. Porredon$^{q,aj,ar}$, F. Prada$^{au}$,
A. Pérez-Fernández$^{ai,cg}$, I. Pérez-Ràfols$^{cm}$, D. Rabinowitz$^{m}$, A. Raichoor$^{b}$, C. Ramírez-Pérez$^{j}$,
S. Ramirez-Solano$^{ai}$, M. Rashkovetskyi$^{ap}$, C. Ravoux$^{cn,i}$, M. Rezaie$^{o}$, J. Rich$^{i}$, A. Rocher$^{av,i}$,
C. Rockosi$^{bz,ca,co}$, F. Rodríguez-Martínez$^{ai}$, N.A. Roe$^{b}$, A. Rosado-Marin$^{cp}$, A.J. Ross$^{ae,bp,ar}$,
G. Rossi$^{cq}$, R. Ruggeri$^{s,ag}$, V. Ruhlmann-Kleider$^{i}$, L. Samushia$^{cr,o,cs}$, E. Sanchez$^{aj}$,
C. Saulder$^{cg}$, E.F. Schlafly$^{ct}$, D. Schlegel$^{b}$, M. Schubnell$^{g}$, H. Seo$^{cp}$, R. Sharples$^{cu,f}$,
J. Silber$^{b}$, A. Slosar$^{cv}$, A. Smith$^{f}$, D. Sprayberry$^{t}$, T. Tan$^{i}$, G. Tarlé$^{g}$, S. Trusov$^{by}$,
R. Vaisakh$^{bq}$, D. Valcin$^{cp}$, F. Valdes$^{t}$, M. Vargas-Magaña$^{ai}$, L. Verde$^{cf,be}$,
M. Walther$^{bi,bj}$, B. Wang$^{cw,cx}$, M.S. Wang$^{q}$, B.A. Weaver$^{t}$, N. Weaverdyck$^{b}$,
R.H. Wechsler$^{as,cy,at}$, D.H. Weinberg$^{bp,ar}$, M. White$^{cz,an}$, M.J. Wilson$^{f}$, J. Yu$^{av}$, Y. Yu$^{al}$,
S. Yuan$^{at}$, C. Yèche$^{i}$, E.A. Zaborowski$^{ae,aq,ar}$, P. Zarrouk$^{by}$, H. Zhang$^{bt,bv}$, C. Zhao$^{cx}$,
R. Zhao$^{y,cj}$, R. Zhou$^{b}$, H. Zou$^{cj}$

<footnote>a Instituto de Física Teórica (IFT) UAM/CSIC, Universidad Autónoma de Madrid,
Cantoblanco, E-28049, Madrid, Spain</footnote>

<footnote>b Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, U.S.A.</footnote>

<footnote>c Physics Dept., Boston University, 590 Commonwealth Avenue, Boston, MA 02215, U.S.A.</footnote>

<footnote>d Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India</footnote>

<footnote>e Centre for Extragalactic Astronomy, Department of Physics, Durham University,
South Road, Durham, DH1 3LE, U.K.</footnote>
f Institute for Computational Cosmology, Department of Physics, Durham University,
South Road, Durham DH1 3LE, U.K.

g Department of Physics, University of Michigan, Ann Arbor, MI 48109, U.S.A.

h *Leinweber Center for Theoretical Physics, University of Michigan,*
*450 Church Street, Ann Arbor, Michigan 48109-1040, U.S.A.*

i IRFU, CEA, Université Paris-Saclay, F-91191 Gif-sur-Yvette, France

j Institut de Física d'Altes Energies (IFAE), The Barcelona Institute of Science and Technology,
Campus UAB, 08193 Bellaterra Barcelona, Spain

k Instituto de Ciencias Físicas, Universidad Autónoma de México, Cuernavaca, Morelos, 62210, México

l Instituto Avanzado de Cosmología A. C.,
San Marcos 11 - Atenas 202. Magdalena Contreras, 10720. Ciudad de México, México

m Physics Department, Yale University, P.O. Box 208120, New Haven, CT 06511, U.S.A.

n Department of Physics and Astronomy, University of California, Irvine, 92697, U.S.A.

o Department of Physics, Kansas State University, 116 Cardwell Hall, Manhattan, KS 66506, U.S.A.

p Department of Physics & Astronomy, University of Rochester,
206 Bausch and Lomb Hall, P.O. Box 270171, Rochester, NY 14627-0171, U.S.A.

q Institute for Astronomy, University of Edinburgh, Royal Observatory,
Blackford Hill, Edinburgh EH9 3HJ, U.K.

r Dipartimento di Fisica “Aldo Pontremoli”, Università degli Studi di Milano,
Via Celoria 16, I-20133 Milano, Italy

s Centre for Astrophysics & Supercomputing, Swinburne University of Technology,
P.O. Box 218, Hawthorn, VIC 3122, Australia

t NSF NOIRLab, 950 N. Cherry Ave., Tucson, AZ 85719, U.S.A.

u Department of Physics & Astronomy, University College London,
Gower Street, London, WC1E 6BT, U.K.

v Department of Astronomy and Astrophysics, University of Chicago,
5640 South Ellis Avenue, Chicago, IL 60637, U.S.A.

w Fermi National Accelerator Laboratory, PO Box 500, Batavia, IL 60510, U.S.A.

x Korea Astronomy and Space Science Institute,
776, Daedeokdae-ro, Yuseong-gu, Daejeon 34055, Republic of Korea

y Institute of Cosmology and Gravitation, University of Portsmouth,
Dennis Sciama Building, Portsmouth, PO1 3FX, U.K.

z Departamento de Astrofísica, Universidad de La Laguna (ULL), E-38206, La Laguna, Tenerife, Spain

aa Instituto de Astrofísica de Canarias, C/ Vía Láctea, s/n, E-38205 La Laguna, Tenerife, Spain

ab Department of Physics and Astronomy, University of Sussex, Brighton BN1 9QH, U.K.

ac Departamento de Física, Instituto Nacional de Investigaciones Nucleares,
Carretera México-Toluca S/N, La Marquesa, Ocoyoacac, Edo. de México C.P. 52750, México

ad Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, U.S.A.

ae Center for Cosmology and AstroParticle Physics, The Ohio State University,
191 West Woodruff Avenue, Columbus, OH 43210, U.S.A.

af NASA Einstein Fellow

ag School of Mathematics and Physics, University of Queensland, 4072, Australia

ah Department of Physics and Astronomy, The University of Utah,
115 South 1400 East, Salt Lake City, UT 84112, U.S.A.

ai Instituto de Física, Universidad Nacional Autónoma de México, Cd. de México C.P. 04510, México

aj CIEMAT, Avenida Complutense 40, E-28040 Madrid, Spain

ak Department of Physics & Astronomy and Pittsburgh Particle Physics, Astrophysics, and Cosmology
Center (PITT PACC), University of Pittsburgh, 3941 O'Hara Street, Pittsburgh, PA 15260, U.S.A.

al Department of Astronomy, School of Physics and Astronomy, Shanghai Jiao Tong University,
Shanghai 200240, China

am Space Sciences Laboratory, University of California, Berkeley, 7 Gauss Way, Berkeley, CA 94720, U.S.A.

an University of California, Berkeley, 110 Sproul Hall #5800 Berkeley, CA 94720, U.S.A.

ao Universities Space Research Association, NASA Ames Research Centre, U.S.A.
ap Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, U.S.A.

aq Department of Physics, The Ohio State University,
191 West Woodruff Avenue, Columbus, OH 43210, U.S.A.

ar The Ohio State University, Columbus, 43210 OH, U.S.A.

as Kavli Institute for Particle Astrophysics and Cosmology, Stanford University,
Menlo Park, CA 94305, U.S.A.

at SLAC National Accelerator Laboratory, Menlo Park, CA 94305, U.S.A.

au Instituto de Astrofísica de Andalucía (CSIC), Glorieta de la Astronomía, s/n, E-18008 Granada, Spain

av Institute of Physics, Laboratory of Astrophysics, École Polytechnique Fédérale de Lausanne (EPFL),
Observatoire de Sauverny, CH-1290 Versoix, Switzerland

aw Departamento de Física, Universidad de los Andes,
Cra. 1 No. 18A-10, Edificio Ip, CP 111711, Bogotá, Colombia

ax Observatorio Astronómico, Universidad de los Andes,
Cra. 1 No. 18A-10, Edificio H, CP 111711 Bogotá, Colombia

ay Department of Physics, The University of Texas at Dallas, Richardson, TX 75080, U.S.A.

az Center for Computational Astrophysics, Flatiron Institute,
162 5th Avenue, New York, NY 10010, U.S.A.

ba Scientific Computing Core, Flatiron Institute, 162 5th Avenue, New York, NY 10010, U.S.A.

bb Institut d'Estudis Espacials de Catalunya (IEEC), 08034 Barcelona, Spain

bc Institute of Space Sciences, ICE-CSIC, Campus UAB,
Carrer de Can Magrans s/n, 08913 Bellaterra, Barcelona, Spain

bd Departamento de Física Quàntica i Astrofísica, Universitat de Barcelona,
Martí i Franquès 1, E08028 Barcelona, Spain

be Institut de Ciències del Cosmos (ICCUB), Universitat de Barcelona (UB),
c. Martí i Franquès, 1, 08028 Barcelona, Spain

bf Consejo Nacional de Ciencia y Tecnología, Av. Insurgentes Sur 1582,
Colonia Crédito Constructor, Del. Benito Juárez C.P. 03940, México D.F., México

bg Departamento de Física, Universidad de Guanajuato - DCI, C.P. 37150, Leon, Guanajuato, México

bh Centro de Investigación Avanzada en Física Fundamental (CIAFF), Facultad de Ciencias, Universidad
Autónoma de Madrid, ES-28049 Madrid, Spain

bi Excellence Cluster ORIGINS, Boltzmannstrasse 2, D-85748 Garching, Germany

bj University Observatory, Faculty of Physics, Ludwig-Maximilians-Universität,
Scheinerstr. 1, 81677 München, Germany

bk Department of Astrophysical Sciences, Princeton University, Princeton NJ 08544, U.S.A.

bl Institut d'Astrophysique de Paris, 98 bis boulevard Arago. 75014 Paris, France

bm Institute for Fundamental Physics of the Universe, via Beirut 2, 34151 Trieste, Italy

bn International School for Advanced Studies, Via Bonomea 265, 34136 Trieste, Italy

bo Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge CB3 0HA, U.K.

bp Department of Astronomy, The Ohio State University,
4055 McPherson Laboratory, 140 W 18th Avenue, Columbus, OH 43210, U.S.A.

bq Department of Physics, Southern Methodist University, 3215 Daniel Avenue, Dallas, TX 75275, U.S.A.

br The Ohio State University, Columbus, 43210 OH, U.S.A.

bs Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, U.K.

bt Department of Physics and Astronomy, University of Waterloo,
200 University Ave W, Waterloo, ON N2L 3G1, Canada

bu Perimeter Institute for Theoretical Physics, 31 Caroline St. North, Waterloo, ON N2L 2Y5, Canada

bv Waterloo Centre for Astrophysics, University of Waterloo,
200 University Ave W, Waterloo, ON N2L 3G1, Canada

bw Graduate Institute of Astrophysics and Department of Physics, National Taiwan University,
No. 1, Sec. 4, Roosevelt Rd., Taipei 10617, Taiwan

bx Schmidt Sciences, 155 W 23rd St, New York, NY 10011, U.S.A.

by Sorbonne Université, CNRS/IN2P3, Laboratoire de Physique Nucléaire et de Hautes Energies (LPNHE),
FR-75005 Paris, France
bz Department of Astronomy and Astrophysics, UCO/Lick Observatory, University of California,
1156 High Street, Santa Cruz, CA 95064, U.S.A.

ca Department of Astronomy and Astrophysics, University of California, Santa Cruz,
1156 High Street, Santa Cruz, CA 95065, U.S.A.

cb Department of Astronomy & Astrophysics, University of Toronto, Toronto, ON M5S 3H4, Canada

cc University of Science and Technology, 217 Gajeong-ro, Yuseong-gu, Daejeon 34113, Republic of Korea

cd Departament de Física, Serra Húnter, Universitat Autònoma de Barcelona,
08193 Bellaterra (Barcelona), Spain

ce Laboratoire de Physique Subatomique et de Cosmologie, 53 Avenue des Martyrs, 38000 Grenoble, France

cf Institució Catalana de Recerca i Estudis Avançats,
Passeig de Lluís Companys, 23, 08010 Barcelona, Spain

cg Max Planck Institute for Extraterrestrial Physics, Gießenbachstraße 1, 85748 Garching, Germany

ch Department of Physics and Astronomy, Siena College, 515 Loudon Road, Loudonville, NY 12211, U.S.A.

ci Department of Physics & Astronomy, University of Wyoming,
1000 E. University, Dept. 3905, Laramie, WY 82071, U.S.A.

cj National Astronomical Observatories, Chinese Academy of Sciences,
A20 Datun Rd., Chaoyang District, Beijing, 100012, P.R. China

ck Steward Observatory, University of Arizona, 933 N, Cherry Ave, Tucson, AZ 85721, U.S.A.

cl Aix Marseille Univ, CNRS, CNES, LAM, Marseille, France

cm Departament de Física, EEBE, Universitat Politècnica de Catalunya,
c/Eduard Maristany 10, 08930 Barcelona, Spain

cn Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France

co University of California Observatories, 1156 High Street, Sana Cruz, CA 95065, U.S.A.

cp Department of Physics & Astronomy, Ohio University, Athens, OH 45701, U.S.A.

cq Department of Physics and Astronomy, Sejong University, Seoul, 143-747, Korea

cr Abastumani Astrophysical Observatory, Tbilisi, GE-0179, Georgia

cs Faculty of Natural Sciences and Medicine, Ilia State University, 0194 Tbilisi, Georgia

ct Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, U.S.A.

cu Centre for Advanced Instrumentation, Department of Physics, Durham University,
South Road, Durham DH1 3LE, U.K.

cv Physics Department, Brookhaven National Laboratory, Upton, NY 11973, U.S.A.

cw Beihang University, Beijing 100191, China

cx Department of Astronomy, Tsinghua University,
30 Shuangqing Road, Haidian District, Beijing, China, 100190

cy Physics Department, Stanford University, Stanford, CA 93405, U.S.A.

cz Department of Physics, University of California, Berkeley,
366 LeConte Hall MC 7300, Berkeley, CA 94720-7300, U.S.A.