  We examine properties of the mean-field wave function of the one-dimensional
Kitaev model supporting Majorana Zero Modes (MZMs) \emph{when restricted} to a
fixed number of particles. Such wave functions can in fact be realized as exact
ground states of interacting number-conserving Hamiltonians and amount to a
more realistic description of the finite isolated superconductors. Akin to
their mean-field parent, the fixed-number wave functions encode a single
electron spectral function at zero energy that decays exponentially away from
the edges, with a localization length that agrees with the mean-field value.
Based purely on the structure of the number-projected ground states, we
construct the fixed particle number generalization of the MZM operators. They
can be used to compute the edge tunneling conductance; however, notably the
value of the zero-bias conductance remains the same as in the mean-field case,
quantized to $2e^2/h$. We also compute the topological entanglement entropy for
the number-projected wave functions and find that it contains a `robust'
$\log(2)$ component as well as a logarithmic correction to the mean field
result, which depends on the precise partitioning used to compute it. The
presence of the logarithmic term in the entanglement entropy indicates the
absence of a spectral gap above the ground state; as one introduces
fluctuations in the number of particles, the correction vanishes smoothly.
