diff --git a/main.tex b/main.tex
index ef7c1dd..11e037a 100644
--- a/main.tex
+++ b/main.tex
@@ -755,7 +755,7 @@ Williams' theoretical result is not merely of academic interest; it describes a
 
 By bridging theory and practice, we provide practitioners with concrete guidance on when and how to apply space-time trade-offs. Our open-source tools and complete experimental data (available at \url{https://github.com/sqrtspace}) democratize these optimizations, making theoretical insights accessible for real-world system design.
 
-The ubiquity of the $\sqrt{n}$ pattern---from database buffers to neural network training---validates Williams' mathematical insight. As data continues to grow exponentially while memory grows linearly, understanding and applying these trade-offs becomes increasingly critical for building efficient systems.
+The ubiquity of the $\sqrt{n}$ pattern, from database buffers to neural network training, validates Williams' mathematical insight. As data continues to grow exponentially while memory grows linearly, understanding and applying these trade-offs becomes increasingly critical for building efficient systems.
 
 \section*{Acknowledgments}
 This work was carried out independently as part of early-stage R\&D at MarketAlly LLC and MarketAlly Pte. Ltd. The author acknowledges the use of large-language models for drafting, code generation, and formatting assistance. The final decisions, content, and interpretations are solely the authors' own.
diff --git a/ubiquity.pdf b/ubiquity.pdf
index 6dca4b1..f59b2e0 100644
Binary files a/ubiquity.pdf and b/ubiquity.pdf differ