diff --git a/three_manifold_no_conjugate_v2.txt b/three_manifold_no_conjugate_v2.txt
index 6ec7d4f..39717b3 100644
--- a/three_manifold_no_conjugate_v2.txt
+++ b/three_manifold_no_conjugate_v2.txt
@@ -16,7 +16,7 @@ NOTATION CONVENTION
   T, gamma': tangent to a geodesic gamma
   p^T      : transpose of matrix/vector p
   v'       : derivative of a function v
-  Inline math uses $...$ like LaTeX.  The file is plain text.
+  Inline math uses ... like LaTeX.  The file is plain text.
 
 
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@@ -116,29 +116,6 @@ from (b) and the Ivanov-Kapovitch equivalence for closed 3-manifolds.
 
 
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-PART 3.  PROOF ROADMAP
-================================================================
-
-The proof splits into two logically independent halves.
-
-HALF A (topological).  Nonpositive curvature is impossible.
-  Part 4: define M as a specific gluing of two copies of Sigma x S^1.
-  Part 5: prove pi_1(M) cannot act on any CAT(0) space as above.
-  Conclusion: parts (b), (c), (d) of the theorem.
-
-HALF B (geometric).  A metric without conjugate points exists.
-  Part 6: build a specific smooth metric g on M.  Outside a small
-          "neck" (0,1) x T^2, g has sec <= 0.  Inside the neck, g is
-          a cleverly warped family of flat tori.
-  Part 7: classify the possible geodesic passages through the neck.
-  Parts 8-9: compute the second variation and bound its index form
-          below, uniformly in every passage.
-  Part 10: sum over all passages, deduce no conjugate points, and
-          combine with Part 5 to finish.
-
-Part 11 is an appendix: an explicit hyperbolic punctured torus, used
-as the "core" of each product piece in Part 6.
-
 PART 4.  THE MANIFOLD M
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