= Solution
The characteristic halo mass inferred from the stellar mapping is
$$
M_{h,\rm ch}=\frac{10^{10}M_\odot}{0.01013}\simeq9.87\times10^{11}M_\odot.
$$
The <peak-height calibration from an exponential mass-function cutoff> and $\beta=1/4$ give
$$
\sigma(M,3)=\frac{\delta_{\rm sc}}{\sqrt2}\left(\frac{M}{M_{h,\rm ch}}\right)^{-1/4}.
$$
Using $\delta_{\rm sc}\simeq1.686$, appropriate to the nearly matter-dominated collapse at this redshift,
$$
\boxed{\sigma(10^{14}M_\odot,3)\simeq\frac{1.686}{\sqrt2}\left(\frac{10^{14}}{9.87\times10^{11}}\right)^{-1/4}\simeq0.376.}
$$
The barrier in the original complementary-error-function expression is extrapolated to the present, but this answer is the variance at $z=3$. Rewriting the ratio with $\delta_{\rm sc}$ and $\sigma(M,3)$ consistently cancels the growth normalization; using the present-extrapolated barrier with the redshift-three variance would double-count growth.
The <baryon conversion efficiency of a halo> is not observationally constant. The <stellar-to-halo mass ratio> rises from low-mass haloes to a broad maximum near galactic halo masses, then decreases toward groups and clusters; it also varies with redshift and has intrinsic scatter. Photoheating and stellar-feedback-driven outflows suppress baryon retention and <star formation> in small haloes. Long cooling times, hot atmospheres and feedback from active nuclei limit efficient conversion in massive haloes. The peak commonly occurs around halo masses of order $10^{12}M_\odot$, with its position evolving with epoch.
Therefore a mass-independent \b[$f_*\simeq0.05$ is a simplified average, not a realistic stellar-to-halo relation]. Observed ratios must be compared with $\epsilon_*=f_bf_*$, not with $f_*$ alone. Scatter, satellite populations and the mass-dependent efficiency change the stellar mass-function shape and invalidate an exact constant-rescaling of the halo mass function.
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