= Solution
The standard normal <distribution function> $\Phi$ gives
$$
P(Z_i=1\mid x_i)=P(\epsilon_i>\log c-\mu-x_i^T\beta)
=\Phi\!\left(\frac{\mu-\log c}{\sigma}+x_i^T\frac\beta\sigma\right).
$$
The observations are independent <Bernoulli random variables>, because the original errors are independent. Thus the <probit regression> has intercept $\alpha=(\mu-\log c)/\sigma$ and coefficient vector $\gamma=\beta/\sigma$. With the specified known $\sigma$ and $c$, recovery is
$$
\boxed{\mu=\log c+\sigma\alpha,\qquad\beta=\sigma\gamma}.
$$
The scale would not be recoverable from threshold data alone if $\sigma$ were unknown.
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