The angle-partition field conjecture for algebraic numbers in $\mathcal{T}_1$

Let $\mathcal{P}$ be the field created by allowing arbitrary partition of angles, and let $\mathcal{T}_1$ be the relevant construction field. The algebraic numbers in $\mathcal{T}_1$ are the algebraic numbers contained in $\mathcal{P}$. Angle-partition field conjecture. The field $\mathcal{P}$ completely encapsulates the algebraic numbers in $\mathcal{T}_1$. The question is presented as unresolved; the proposed strategy constructs algebraic-to-algebraic functions from the permitted operations, but it does not establish that no other constructions of algebraic numbers are possible.