= Solution
Inverting the detector rotation at this time gives
$$
|\Psi\rangle=\frac1{\sqrt2}\left[
|d_\nearrow\rangle\big(c|\uparrow E_\leftarrow\rangle+s|\downarrow E_\rightarrow\rangle\big)
+|d_\swarrow\rangle\big(s|\uparrow E_\leftarrow\rangle-c|\downarrow E_\rightarrow\rangle\big)
\right].
$$
For a representation with one system factor and one environment factor per displayed detector state, each parenthesized system–environment vector would have to be a <product state>. But the first has coefficient matrix $\operatorname{diag}(c,s)$ in the indicated orthonormal bases; the second has matrix $\operatorname{diag}(s,-c)$. Their <matrix ranks> are two whenever $cs\ne0$. Equivalently, their two nonzero <Schmidt coefficients> are $|c|,|s|$, which no independent basis changes can remove. Thus \b[the proposed factorization is impossible when $\sin\theta\ne0$].
The exception matters: at $\theta=m\pi$, the detector basis is the original basis up to phases or interchange, and a factorization does exist. In particular, the literal condition $\theta\ne0$ is insufficient if it includes $\theta=\pi$. The intended obstruction concerns genuinely mixed detector bases.
The environment has selected robust detector records: its interaction commutes with the original projectors, and orthogonal environmental records suppress their interference in the <reduced density matrix>. A rotated detector branch instead leaves the system and environment entangled. This is <environment-induced pointer-basis selection>, the issue analyzed by <W. H. Zurek> in his apparatus-environment model.<Pointer basis of quantum apparatus: Into what mixture does the wave packet collapse? by W. H. Zurek>[1981 abstract: the apparatus-environment interaction selects recorded observables through a commutation condition]{ref}
This addresses the basis ambiguity but does not by itself select one realized outcome from the global superposition. Moreover, a two-state environment has recurrences: $\cos2A$ returns to unit modulus, so the toy model does not establish irreversible macroscopic collapse. A large environment can make records effectively persistent over experimental times, but that requires further dynamical assumptions. The distinction is part of the <measurement problem>, not an algebraic error cured by a partial trace.
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