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This paper settles in the negative the following open question: Are \( V_4 \)-magic graphs necessarily \( \mathbb{Z}_4 \)-magic? For an abelian group \( A \), we examine the properties of \( A \)-magic labelings with constant weight \( 0 \), called \emph{zero-sum \( A \)-magic}, and utilize well-known results on edge-colorings in order to construct (from \( 3 \)-regular graphs) infinite families that are \( V_4 \)-magic but not \( \mathbb{Z}_4 \)-magic. Noting that our arguments lead to connected graphs of order \( 2n \) for all \( n \geq 11 \) that are \( V_4 \)-magic and not \( \mathbb{Z}_4 \)-magic, we conclude the paper by investigating the zero-sum integer-magic spectra of graphs, including Cartesian products, and give a conjecture about the zero-sum integer-magic spectra of \( 3 \)-regular graphs.