Update app.py

app.py

@@ -102,7 +102,23 @@ This method introduces sampling control along three key dimensions:
 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.Markdown(
+    """
+### 🔍 Scaling Analysis of *n*, *m*, and *H* in DeepSeek-R1 Models
+
+A detailed test-time scaling analysis on the DeepSeek-R1 series reveals the individual impact of the three sampling dimensions: `n` (number of reasoning paths), `m` (number of answers per path), and `H` (depth-wise reasoning samples).
+
+Across multiple reasoning benchmarks, the results show that increasing **`H` — sampling across reasoning depths — yields the highest cost-effectiveness**. That is, sampling more intermediate answers along the depth of a single reasoning path leads to **greater accuracy improvements with fewer additional tokens**, compared to simply increasing the number of paths (`n`) or answers (`m`).
+    """)
     gr.Image("figs/single.png", label="Framework", show_label=False, elem_id="my-img")
+    gr.Markdown(
+    """
+### 🔄 Joint Sampling of *n*, *m*, and *H* for Enhanced Accuracy
+
+In practical scenarios, the sampling dimensions `n`, `m`, and `H` can be **jointly optimized** rather than tuned in isolation. By **dynamically allocating the sampling budget across these dimensions**, the model can significantly enhance its reasoning accuracy.
+
+This joint sampling strategy leverages the complementary strengths of each dimension—diversity (`n`), redundancy (`m`), and depth-awareness (`H`)—to achieve robust performance under a fixed token budget.
+    """)
     gr.Image("figs/combine.png", label="Framework", show_label=False, elem_id="my-img")
     gr.Markdown(
     """