Thin equivalence relations and inner models

We describe the inner models with representatives in all equivalence classes of thin equivalence relations in a given projective pointclass of even level assuming projective determinacy. These models are characterized by their correctness and the property that they correctly compute the tree from the appropriate scale. The main lemma shows that the tree from a scale can be reconstructed in a generic extension of an iterate of a mouse. We construct models with this property as generic extensions of iterates of mice if the corresponding projective ordinal is below ω2. On the way we consider several related problems, including the question when forcing does not add equivalence classes to thin projective equivalence relations. For example, we show that if every set has a sharp, then reasonable forcing does not add equivalence classes to thin provably ∆3 equivelence relations, and generalize this to all projective levels.