  We investigate quantum objectivity in the boson-spin model, where a central
harmonic oscillator interacts with a thermal bath of spin-1/2 systems. We
analyze how information about a continuous position variable can be encoded
into discrete, finite-dimensional environments. More precisely, we study
conditions under which the so-called Spectrum Broadcast Structures (SBS) can be
formed in the model. These are multipartite quantum state structures,
representing a mode-refined form of decoherence. Working in the recoil-less
limit, we use the Floquet theory to show that despite its apparent simplicity,
the model has a rich structure with different regimes, depending on the motion
of the central system. In one of them, the faithful encoding of the position
and hence objectivity are impossible irrespectively of the resources used. In
other, large enough collections of spins will faithfully encode the position
information. We derive the characteristic length scales, corresponding to
decoherence and precision of the encoding.
