Consequences of $\oplus \mathbf{P} \subseteq \mathbf{NP}$ I have part of a proof attempt of $\oplus \mathbf{P} \subseteq \mathbf{NP}$. The proof attempt consists of a Karp reduction from the $\oplus \mathbf{P}$-complete problem $\oplus$3-REGULAR VERTEX COVER to SAT.

Given a cubic graph $G$, the reduction outputs a CNF formula $F$ having both the following properties:


$F$ has at most $1$ satisfying assignment.
$F$ is satisfiable if and only if the number of vertex covers of $G$ is odd.



  Questions
  
  
  Which would be the consequences of $\oplus \mathbf{P} \subseteq \mathbf{NP}$ ? A consequence which I'm already aware of is the following: $\mathbf{PH}$ would be reducible to $\mathbf{NP}$ via two-sided randomized reduction. In other words we would have $\mathbf{PH}\subseteq\mathbf{BPP}^{\mathbf{NP}}$ (using Toda's Theorem, which states that $\mathbf{PH}\subseteq\mathbf{BPP}^{\oplus\mathbf{P}}$, by merely replacing $\oplus\mathbf{P}$ with $\mathbf{NP}$). I do not know if $\mathbf{BPP}^{\mathbf{NP}}$ has been shown to be contained in some level $i$ of the Polynomial Hierarchy: if yes, a further consequence would be that $\mathbf{PH}$ collapses to such level $i$. Moreover, under widely accepted derandomization assumptions ($\mathbf{BPP} = \mathbf{P}$), the Polynomial Hierarchy would collapse between the first and the second level, as we would have $\mathbf{PH} = \mathbf{P}^\mathbf{NP} = \Delta_2^\mathbf{P}$ (I'm told this is not true, however I will not erase this line until I fully understand why).
  If I'm not mistaken, the aforementioned reduction would actually prove more than $\oplus \mathbf{P} \subseteq \mathbf{NP}$. It would prove $\oplus \mathbf{P} \subseteq \mathbf{UP}$. Which would be the consequences of $\oplus \mathbf{P} \subseteq \mathbf{UP}$, in addition to those already implied by $\oplus \mathbf{P} \subseteq \mathbf{NP}$ ? I do not know exactly whether $\oplus \mathbf{P} \subseteq \mathbf{UP}$ would add more surprise to the already surprising consequences of $\oplus \mathbf{P} \subseteq \mathbf{NP}$, nor to what extent. Intuitively I presume it would, and to a pretty wide extent.
There are two oracle sets defined in T88 such that ${\bf NP^A \not \subseteq \oplus P ^A}$ and ${\bf \oplus P^B \not \subseteq NP^B}$.