Left panel: overlap of the CMB and 21 cm kernels that generate the cross-spectrum, shown for two multipoles as a function of the primordial wavenumber ﹩{k}^{{\prime} }﹩. The blue and orange curves are the CMB transfer function and the 21 cm transfer function. The black curve is the integrand of the cross-spectrum, and the red dashed curve is its running integral ﹩{\int }^{{k}^{{\prime} }}﹩, whose endpoint (thin horizontal line) is the full integral ﹩{C}_{\ell }^{\times }({k}_{{\rm{mid}}})﹩. The dotted vertical line marks ﹩{k}^{{\prime} }=\ell /{r}_{* }﹩, the spherical Bessel function turn-on. Right panel: colored curves show squared cross-correlations (21 cm and CMB) for shells at redshift z = 50, 100, and 200 (orange, green, and blue), while the black curve shows the coefficient R_ℓ when conditioned on the full stack of shells, using the simulated maps. Purple dashed and gray dashed lines show the analytical coefficient R_ℓ when the survey range is extended to z = 600 for the Planck primordial power spectrum and for our own survey using 21cmFAST. Dotted orange and blue lines show reference potential–potential squared correlations at z = 50 and 200, respectively.