= Solution
Assume standard radiation-dominated expansion, negligible chemical potential, one neutrino helicity and one antineutrino helicity, and a stable species $X$ with no later entropy injection. Use <natural units> and unreduced <Planck mass> $m_{\rm pl}\simeq1.22\times10^{19}\,\mathrm{GeV}$. The relativistic energy density is $\rho_R=\pi^2g_*T^4/30$. The <Friedmann equation> then gives
$$
H=\sqrt{\frac{4\pi^3g_*}{45}}\frac{T^2}{m_{\rm pl}}\simeq1.66\sqrt{g_*}\frac{T^2}{m_{\rm pl}}.
$$
For <thermal decoupling in cosmology>, equate the interaction rate $G_A^2T_D^5$ to $H(T_D)$:
$$
\boxed{T_D\simeq\left(\frac{1.66\sqrt{g_*}}{G_A^2m_{\rm pl}}\right)^{1/3}
\simeq1.1\times10^2\,\mathrm{GeV}.}
$$
This order-of-magnitude estimate assumes the supplied rate law remains applicable there.
A decoupled relativistic species cools as $T_X\propto a^{-1}$, while annihilations heat the coupled bath. Treat the supplied high-temperature count $100$ as the remaining bath's entropy count, approximately equal to its energy count. <Cosmological entropy conservation> down to ordinary <neutrino decoupling> gives
$$
\frac{T_X}{T_\nu}=\left(\frac{10.75}{100}\right)^{1/3}.
$$
Subsequent <electron-positron annihilation in cosmology> heats photons but neither neutrino population. Therefore the <two-stage entropy dilution of an early-decoupled neutrino> gives
$$
\boxed{\frac{T_X}{T_\gamma}=\left(\frac{10.75}{100}\frac4{11}\right)^{1/3}
=\left(\frac{43}{1100}\right)^{1/3}\simeq0.339.}
$$
The ordinary <Cosmic neutrino background> has $T_\nu/T_\gamma=(4/11)^{1/3}$. If the rounded count $100$ instead includes the new two helicities, the remaining bath count is $98.25$; this changes the answer only slightly, to about $0.341$. Keeping the extra species' entropy separate is essential in either convention.
For the two relativistic fermion states, $g_{*s,X}=(7/8)2=7/4$. Photon and three standard neutrino species have present entropy degrees of freedom, referred to $T_\gamma$,
$$
g_{*s,\mathrm{std}}=2+\frac78\,6\left(\frac4{11}\right)=\frac{43}{11}.
$$
Writing $\theta=T_X/T_\gamma$, the <fractional entropy of a relativistically decoupled species> is
$$
\boxed{\frac{s_X}{s_{\rm total}}=
\frac{(7/4)\theta^3}{43/11+(7/4)\theta^3}
=\frac{7/4}{100+7/4}\simeq0.0172.}
$$
Thus approximately \b[1.7% of the thermal particle entropy is in the new neutrinos and antineutrinos]. If $100$ counts all states before the new decoupling, the fraction is $1.75\%$. These are the same rounded estimate. The question concerns the radiation/relic thermal entropy budget; it does not supply a model for entropy in later black holes or other gravitational structures.
If $X$ has a small mass but decoupled relativistically, its comoving particle number is still conserved. The <relativistic-neutrino number-density ratio> gives
$$
\frac{n_X}{n_\gamma}=\frac34\theta^3\simeq0.0293,\qquad
\frac{n_X}{n_{\nu,\mathrm{one\ standard\ species}}}=\frac{10.75}{100}=0.1075.
$$
Once nonrelativistic, $\rho_X\simeq m_Xn_X$. The <density bound on a relativistically decoupled massive relic> is therefore
$$
\boxed{m_X\leq\frac{\Omega_{X,\max}\rho_c}{n_X}.}
$$
Using the standard present photon abundance and critical-density normalization, this is approximately $\Omega_Xh^2\simeq m_X/(9\times10^2\,\mathrm{eV})$, where $h=H_0/(100\,\mathrm{km\,s^{-1}\,Mpc^{-1}})$ is the present Hubble normalization, not a magnetic field. A stable relic must not exceed the permitted matter density; for example, if $\Omega_Mh^2=0.15$, even assigning all matter to it gives only an order-$10^2\,\mathrm{eV}$ mass ceiling.
A stronger restriction can follow from <neutrino free streaming> and the observed formation of <large-scale structure of the universe>: a light thermal population acts as hot rather than cold <dark matter>, erasing structure below its streaming scale. If at most a fraction $f_{\max}$ of the matter may be in this population, replace the overclosure ceiling by $m_X\lesssim9\times10^2 f_{\max}\Omega_Mh^2\,\mathrm{eV}$. The allowed fraction is observational input and is not supplied, so the paper does not determine a unique tighter mass limit. A species still relativistic today is radiation rather than matter; its contribution to the <effective number of neutrino species> is $\Delta N_{\rm eff}=(T_X/T_\nu)^4\simeq0.051$, and must instead satisfy the radiation-density constraints. These conclusions assume stable thermally populated states; decay or nonstandard entropy histories change the bounds.
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