= Solution
For $n=0$ the proposed <wavefunction> vanishes identically and cannot be normalized. A negative integer gives $\sin(-|n|\pi x/L)=-\sin(|n|\pi x/L)$, which differs from the corresponding positive-integer <wavefunction> only by a physically irrelevant <global phase>. Both have the same positive <energy>, proportional to $n^2$. Thus \b[zero is inadmissible and negative labels are redundant]. A negative integer is not forbidden by a separate negative-energy argument.
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