= Solution
The given cooling law yields $T_{c,i}=[0.056L_\odot/0.09]^{2/7}\simeq3.4\times10^7$ K. Combine it with the <ion> <thermal energy> to obtain
$$
\boxed{\Delta t=\frac{k_BM_\odot}{8m_u(0.09)}\frac25\left(T_{c,f}^{-5/2}-T_{c,i}^{-5/2}\right).}
$$
For $T_{c,f}=0.1T_{c,i}$ this is about $1.32\times10^{10}$ years; the exact value is retained in the independent numerical check. At approximately fixed <radius>, $T_{\mathrm{eff}}\propto L^{1/4}\propto T_c^{7/8}$, giving $\boxed{T_{\mathrm{eff},f}\simeq3.3\times10^3\ \mathrm K}$. The required cooling time is comparable to the cosmic age and leaves little time for progenitor evolution. Such objects would also be very faint. This explains scarcity in the toy model, rather than a universal prohibition on observing ultracool <White dwarf>[white dwarfs]. The source's categorical absence claim is too strong; detailed cooling histories can differ. The ideal-ion <heat capacity> and <luminosity> law are being extrapolated into regimes where crystallization, atmosphere physics and other corrections matter, so this is not a precision cooling age.
A concrete observational counterexample to a categorical absence in this <temperature> regime is the observed ultracool population including LHS 1402, LHS 3250 and SDSS J1337+00, with published atmosphere fits spanning about 3240–3480 K. https://doi.org/10.1088/0067-0049/190/1/77[Section 3.3.4, Ultracool WDs: observed objects and <temperature> fits]{ref}
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