Quantum Darwinism and spectrum broadcast structure describe the emergence of objectivity in quantum
systems. However, it is unclear whether these two frameworks lead to consistent predictions on the objectivity
of the state of a quantum system in a given scenario. In this paper, we jointly investigate quantum Darwinism
and spectrum broadcasting, as well as the subdivision of quantum Darwinism into accessible information and
quantum discord, in a two-level system interacting with an N-level environment via a random matrix coupling.
We propose a partial trace method to suitably and consistently partition the effective N-level environment and
compare the predictions with those obtained using the partitioning method proposed by Perez [Phys. Rev.
A 81, 052326 (2010)]. We find that quantum Darwinism can apparently emerge under the Perez trace even
when spectrum broadcast structure does not emerge, and the majority of the quantum mutual information
between system and environment fractions is in fact quantum in nature. This work therefore shows there can
be discrepancies between quantum Darwinism and the nature of information and spectrum broadcast structure
