Assuming Martin's Conjecture on functions between Turing degrees, is AD + DC consistent with existence of an $f:\mathcal{D}_t → \mathcal{D}_t$ of rank $Θ$ ?
$\mathcal{D}_t$ is the set of Turing degrees. $Θ$ is the supremum of the ordinals that are a surjective image of the reals (under AC, $Θ$ would be $c^+$).
The well-orderings of natural consistency strengths, descriptive complexity levels, and Turing degrees are related to the apparent prewellordering of natural functions $f$ between Turing degrees that are not constant on a cone, where $f ≤ g$ iff for a cone of Turing degrees, $f(x)≤g(x)$. Moreover, natural functions from Turing degrees to many-one degrees, under a certain uniformity restriction, correspond to Wadge ranks (The uniform Martin's conjecture for many-one degrees).
To exclude unnatural sets of reals (and 'pathologies' related to AC), we work ZF + AD + DC. Under a version of Martin's conjecture, functions $\mathcal{D}_t → \mathcal{D}_t$ (that are not constant on a cone) are prewellordered as above, and for sufficiently closed $α$, Turing-invariant functions $ω^ω→ω^ω$ of Wadge rank $<α$ have rank $<α$ in the prewellordering. (Note: Martin's conjecture has been proved for uniformly Turing-invariant functions $ω^ω→ω^ω$. Martin's Conjecture is often (but not here) restricted to Turing-invariant $f:ω^ω→ω^ω$ rather than $f:\mathcal{D}_t → \mathcal{D}_t$.)
This would rule out a function of rank $Θ$, except that an $f:\mathcal{D}_t → \mathcal{D}_t$ only gives a Turing-invariant binary relation on $ω^ω$, and we might not be able to uniformize it. AD gives plenty of failures of choice/uniformization, including countable ordinals ⇒ reals, Turing degrees ⇒ reals, and if $V = HOD(ℝ)$, reals ⇒ reals. However, the unifomization failure here gives something special: A function $f:\mathcal{D}_t → \mathcal{D}_t$ such that for every $g:ω^ω→ω^ω$, for every real $r$ in a cone of Turing degrees, $g(r)$ is computable from every real of degree $f([r]_T)$. Moreover, because $f$ is encodable by a function from reals to countable sets for which uniformization fails, $f$ (or at least some set in $L(f,ℝ)$) is not ∞-Borel, and $\text{AD}^+$ fails (see Uniformization under AD question and Ramsey ultrafilters and countable-to-one uniformization). The consistency of $\text{AD} + ¬\text{AD}^+$ is open, but that would not necessarily rule out an easy negative answer here.
A natural strengthening of the hypotheses is ZF + AD + DC + (optionally) "Θ is regular" + uniformization for upward-closed relations on Turing degrees + (if needed and appropriate) Martin's Conjecture + uniformization fails (for $ℝ^2$). Is it consistent?
The hypotheses may turn out to be inconsistent, or they may turn out to be a curiosity of a secondary importance, or $f$ in the question may provide the sharp for Woodin's Ultimate L and $\text{AD}^+$, and the least (under Wadge rank) natural set of reals that is not universally Baire. Failure of $\text{AD}^+$ under AD + DC implies a failure of uniformization (for $ℝ^2$); we do not know what happens then, but the above presents a possibility that uniformization almost holds, and that its failure (rather being kind of an end) gives a new canonical structure.
