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An additive-looking energy expression does not by itself establish two noninteracting subsystems. Their allowed microstates must also form a Cartesian product of independently available states. Here the second label's allowed values depend on the first. The four allowed pairs and their energies areThe canonical partition function, with $\beta =(k_{B}T{)}^{-1}$, is thereforeThe conditional sum over $m_J$ is $2\cosh (\beta ϵJ)$ and depends on $J$; it cannot be pulled out as the partition function of an independent second subsystem. Summing independently over $J\in \{1,2\}$ and all four values $m\in \{-2,-1,1,2\}$ would invent four forbidden pairs and give eight states instead of four.

One can make the state labels independent by writing $s=m_J/J=\pm 1$, but then $E(J,s)=ϵJ(1+s)$ contains a coupling between them. For genuinely additive independent energies, the difference between the $s=+1$ and $s=-1$ states would be independent of $J$; here it is $2ϵJ$. Hence the proposed $J$ and $m_J$ subsystems cannot be treated as noninteracting independent subsystems. The literal two-value reading of the PDF's endpoint notation has been used. If that notation were instead intended as shorthand for full angular-momentum multiplets, including every integer between the endpoints, the eight allowed states would giveThe conclusion about the failure of independent-subsystem factorization is unchanged. The two state-counting conventions should not be mixed.