Update app.py

app.py

@@ -89,15 +89,17 @@ training.
     """
 
 ## Introduction
-We propose **Elastic Reasoning**, a novel framework for scalable chain of thoughts
-that explicitly separates reasoning into two phases—`thinking and solution`—with
-independently allocated budgets. At test time, Elastic Reasoning prioritize that
-completeness of solution segments, significantly improving reliability under tight
-resource constraints. To train models that are robust to truncated thinking, we
-introduce a lightweight `budget-constrained rollout` strategy, integrated into GRPO,
-which teaches the model to reason adaptively when the thinking process is cut
-short and generalizes effectively to unseen budget constraints without additional
-training.
+Building upon the same core insight as **Elastic Reasoning**—that correct answers can often be derived without waiting for a full chain-of-thought (CoT)—**Fractured Sampling** shifts focus to the **sampling strategy** of reasoning.
+
+Instead of relying on complete, uninterrupted reasoning sequences, Fractured Sampling **breaks the CoT along the temporal dimension**, exploring whether it's possible to "get the right answer without thinking all the way through."
+
+This method introduces sampling control along three key dimensions:
+
+- **Solution Diversity (m) — sampling multiple final outputs from a single reasoning trace.
+- **Trajectory Diversity (n) — sampling multiple independent reasoning traces with different seeds (vanilla CoT sampling).
+- **Reasoning Depth Diversity (H) — sampling at different intermediate stages of a single reasoning trace.
+
+Among these, the novel **reasoning depth `H`** plays a critical role: by sampling outputs at different depths of partially completed reasoning chains, the model creates multiple sets of "fragmented thoughts + solutions," which are then jointly evaluated to select the most trustworthy outcome.
     """)
     gr.Image("figs/frac-frame.png", label="Framework", show_label=False, elem_id="my-img")
     gr.Image("figs/single.png", label="Framework", show_label=False, elem_id="my-img")