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Joint work with Mario Kummer, Andreas Thom.
We develop new methods for exhibiting convex semialgebraic sets that are not spectrahedral shadows, and prove that the cone of copositive matrices of size $n\geq 5$ is not a spectrahedral shadow, answering a question of Scheiderer. Our arguments are based on the model theoretic observation that any formula defining a spectrahedral shadow must be preserved by every unital $\mathbb R$-linear completely positive map $R\to R$ on a real closed field extension $R$ of $\mathbb R$.