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Background: A recent inelastic alpha-scattering experiment [Phys. Rev. Lett.129, 102701 (2022)] found $0^+$ resonances in $^{24}$Mg on and above the$^{12}$C+$^{12}$C break-up threshold. It has been conjectured that the stateshave a $^{12}$C+$^{12}$C cluster structure, and play a similar role inaccelerating $^{12}$C+$^{12}$C fusion to the manner in which the Hoyle stateaccelerates production of $^{12}$C in massive stars.  Purpose: We wish to build up a quantitative theoretical basis for theconsiderations of the Hoyle-state paradigm, by calculating the distribution ofthe $0^+$ states in the shell, as well as in the relevant cluster models.  Methods: We determine the spectrum of excited $0^+$ states in $^{24}$Mgnucleus using multiconfigurational dynamical symmetry calculations leading to aunified description of the quartet (or shell), $^{12}$C+$^{12}$C and$^{20}$Ne+$^{4}$He cluster configurations.  Results: The density of $0^+$ states in the quartet spectrum is comparable tothat found in experiment; however, the density of cluster states isconsiderably less.  Conclusions: The recently observed alpha-scattering resonances do not seem tobe simple $^{12}$C+$^{12}$C cluster states, but are more plausibly interpretedas fragmented cluster states due to coupling to quartet excitations, asbackground states.

$^{12}$C and $\alpha$-clusters, $0^+$ spectrum, and Hoyle-state candidates in $^{24}$Mg