The indecomposable objects in the center of Deligne's category Rep St

We classify the indecomposable objects in the monoidal center of Deligne's interpolation category Rep S-t by viewing Rep S-t as a model-theoretic limit in rank and characteristic. We further prove that the center of Rep S-t is semisimple if and only if t is not a non-negative integer. In addition, we identify the associated graded Grothendieck ring of this monoidal center with that of the graded sum of the centers of representation categories of finite symmetric groups with an induction product.