Liouville number stratification conjecture

Let $\mathbb{R}_{S_n}$ be the reals at definability level $S_n$, let $\mathbb{R}_{S_\omega}=\bigcup_{n\in\mathbb{N}}\mathbb{R}_{S_n}$, and let $\mathbb{A}_{S_n}$ and $\mathbb{T}_{S_n}$ denote the algebraic and transcendental numbers at level $S_n$, respectively. A Liouville number is a real number admitting exceptionally strong rational approximations: for every positive integer $m$, there exist integers $p$ and $q>1$ such that $\left|L-\frac pq\right|<\frac{1}{q^m}$. Liouville number stratification conjecture. All Liouville numbers belong to $\mathbb{R}_{S_\omega}$; for every $n$, there exists a Liouville number $L\in\mathbb{T}_{S_{n+1}}\setminus\mathbb{T}_{S_n}$; and no Liouville number belongs to any $\mathbb{A}_{S_n}$. This proposes that Liouville numbers are distributed across the stratified constructive continuum while remaining transcendental at every definability level. The source supplies no resolution evidence, and the conjecture is therefore recorded as open.