TPO Mathematical Supplement: Detailed Formulations and Proofs

**Definition 1.1** (Conversation Graph): A conversation graph $G = (V, E, \mathbf{C}, \mathbf{M})$ where: - $V = \{v_1, v_2, ..., v_n\}$ is the set of message nodes - $E \subseteq V \times V$ is the set of directed edges representing reply relationships - $\mathbf{C}: V \rightarrow \mathbb{R}^5$ maps each node to its DLM coordinates - $\mathbf{M}: V \rightarrow \Sigma^*$ maps each node to its message content **Definition 1.2** (Path): A path $P = \langle v_{i_1}, v_{i_2}, ..., v_{i_k} \rangle$ is a sequence of nodes where $(v_{i_j}, v_{i_{j+1}}) \in E$ for all $j \in [1, k-1]$. **Definition 1.3** (Root-to-Leaf Path): A path $P$ is root-to-leaf if $\text{in-degree}(v_{i_1}) = 0$ and $\text{out-degree}(v_{i_k}) = 0$. For each node $v_i$, the DLM coordinates are: $$\mathbf{c}_i = (x_i, y_i, z_i, t_i, n_i) \in \mathbb{R}^5$$ **Depth Coordinate**: $x_i \in \mathbb{R}_{\geq 0}$ represents hierarchical depth $$x_i = \text{depth}(v_i) + \text{fractional\_offset}(v_i)$$