A property equivalent to the existence of scales

Let ${\text{UNIF}}$ and ${\text{SCALES}}$ be the propositions that every relation on ${\mathbf{R}}$ can be uniformized, and every subset of ${\mathbf{R}}$ admits a scale, respectively. For $A \subset {\mathbf{R}}$, let $w(A)$ denote the Wadge ordinal of $A$, and let $\delta _1^1(A)$ be the supremum of the ordinals realized in the pointclass ${\Delta ^1}_1(A)$.

Theorem ${\text{(AD)}}$. The following are equivalent:

(a) ${\text{SCALES}}$,

(b) ${\text{UNIF}} +$ the set $\{ w(A):\delta _1^1(A) = {(w(A))^ + }\}$ contains an $\omega$-cub subset of $\Theta$.

Using this theorem, Woodin has shown that if the theory ${\text{(ZF}} + {\text{DC}} + {\text{AD}} + {\text{UNIF)}}$ is consistent, then the theory ${\text{(ZF}} + {\text{DC}} + {\text{AD}}_{\mathbf{R}} + {\text{SCALES)}}$ is also consistent. In this paper we give a proof of the above theorem and of a local version of it. We also study the ordinal $\delta _1^1(A)$ and give several characterizations of it.