Let L[E] be an iterable tame extender model. We analyze to which extent L[E] knows fragments of its own iteration strategy. Specifically, we prove that inside L[E], for every cardinal κ which is not a limit of Woodin cardinals there is some cutpoint t < κ such that Jκ[E] is iterable above t with respect to iteration trees of length less than κ.
As an application we show L[E] to be a model of the following two cardinals versions of the diamond principle. If λ > κ > ω₁ are cardinals, then ◇κ,λ* holds true, and if in addition λ is regular, then ◇κ,λ⁺ holds true.
"The self-iterability of L[E]." J. Symbolic Logic 74 (3) 751 - 779, September 2009. https://doi.org/10.2178/jsl/1245158084