Projectively well-ordered inner model

An inner model $M$ is projectively well-ordered when there is a projective relation that well-orders the real numbers belonging to $M$.

For an inner model $M$, the ordinal ${\omega }_1^M$ is the least ordinal that $M$ regards as uncountable. Equivalently, it is the supremum of the order types of the well-order codes belonging to $M$. It can be countable in the ambient universe.

If $M$ is projectively well-ordered and projective determinacy holds, then ${\omega }_1^M$ is countable in the ambient universe. Otherwise, selecting with the projective well-order the least $M$-code for each countable ordinal produces an uncountable projective set of unique well-order codes. It has no perfect subset by the boundedness theorem for well-order codes, contradicting the perfect set property implied by projective determinacy.