diff --git a/four_dim_kakeya_hausdorff_v2.txt b/four_dim_kakeya_hausdorff_v2.txt
index 43d0da3..1156fcf 100644
--- a/four_dim_kakeya_hausdorff_v2.txt
+++ b/four_dim_kakeya_hausdorff_v2.txt
@@ -1698,7 +1698,7 @@ higher-order jet estimate at a smaller scale. Recurse to build up
 polynomial coefficients Taylor-step by Taylor-step. The entropy
 control from Proposition 5.5 ensures the recursion stays tight.
 
-For high school readers, the essential content is: "you get a
+The essential content: "you get a
 degree-M polynomial fit to the row on a definite chunk of time,
 from a purely combinatorial support hypothesis." That is the
 engine that will run in every narrowness regime.
@@ -2305,7 +2305,7 @@ the row-fit condition produces a rotated structure, and the 90-
 degree rotation (-Z, t) appears from the specific relation between
 Z and t that forces "constant row + rotated row" structure.
 
-For high-school readers, the essence is: you cannot have a
+The essence: you cannot have a
 row that evaluates to the same value on lots of very different
 lines. The lines must all be squeezed into two parallel strips
 specific to the row.
@@ -2989,7 +2989,7 @@ cases are the ones that Theorem 1.1 answers.
 That closes the mathematical content of the paper.
 
 
-APPENDIX A. A MATH TOOLKIT FOR HIGH SCHOOL READERS
+APPENDIX A. MATH TOOLKIT
 
 
 A.0 Purpose