  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 novel 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.
