Kurepa equivalence

Write $\mathsf{S}\mathsf{T}$, $\mathsf{S}\mathsf{L}$, and $\mathsf{S}\mathsf{A}$ for the respective existence assertions in clauses 1-3. These are genuine existence statements under the fixed nonempty, nontrivial conventions in F1 and the cited tree interfaces.

If $\mathsf{S}\mathsf{T}$ holds, F2 constructs a Suslin line, so $\mathsf{S}\mathsf{T}\Rightarrow \mathsf{S}\mathsf{L}$. Conversely, if $\mathsf{S}\mathsf{L}$ holds, F3 constructs a Suslin tree, so $\mathsf{S}\mathsf{L}\Rightarrow \mathsf{S}\mathsf{T}$. Hence $\mathsf{S}\mathsf{T}⟺\mathsf{S}\mathsf{L}$.

If $\mathsf{S}\mathsf{T}$ holds, first apply F4 to obtain a normal splitting Suslin tree, then apply F5 to its reverse-order regular-open completion. The output is a Suslin algebra, so $\mathsf{S}\mathsf{T}\Rightarrow \mathsf{S}\mathsf{A}$.

Conversely, F6 sends any Suslin algebra to a normal splitting Suslin tree, so $\mathsf{S}\mathsf{A}\Rightarrow \mathsf{S}\mathsf{T}$. Together with step 2.2 this gives $\mathsf{S}\mathsf{T}⟺\mathsf{S}\mathsf{A}$.

Steps 2.1, 2.2, and 3.1 prove the three-way equivalence. By F1, SH is $\mathrm{\neg }\mathsf{S}\mathsf{L}$; negating either proved biconditional gives $\mathrm{\neg }\mathsf{S}\mathsf{L}⟺\mathrm{\neg }\mathsf{S}\mathsf{T}⟺\mathrm{\neg }\mathsf{S}\mathsf{A}$. Thus SH is equivalent to either stated nonexistence assertion. This proof uses only the five fully authored construction suppliers and not the earlier Recorded Kurepa remark.