diff --git a/crates/specialized/rtx-cfd/src/mesh/patch_gen.rs b/crates/specialized/rtx-cfd/src/mesh/patch_gen.rs
index c06fcd9..c0b3873 100644
--- a/crates/specialized/rtx-cfd/src/mesh/patch_gen.rs
+++ b/crates/specialized/rtx-cfd/src/mesh/patch_gen.rs
@@ -952,9 +952,16 @@ pub fn cylinder_flag_outline_deformed_tip(
     let mut pts: Vec<[f64; 2]> = Vec::new();
     // 1. Top edge: from the arc's top end toward the root, along the top
     //    polyline (given tip → root) from arclength `radius` to x = x_f.
-    // The corner radius: `radius` (the semicircle) or the flat tip's.
-    let rc = tip_corner.min(radius).max(0.0);
-    let flat = rc < radius - 1e-12;
+    // The corner radius: `radius` (the semicircle) or the flat tip's. The
+    // branch is chosen against the UNDEFORMED half-thickness `t`, never the
+    // deformed tip's `radius` (which breathes by nanometres under the FEA's
+    // deformation and flipped the topology at the semicircle setting).
+    let flat = tip_corner < t - 1e-9;
+    let rc = if flat {
+        tip_corner.min(radius).max(0.0)
+    } else {
+        radius
+    };
     // Corner arc centres and the tangent points on the faces.
     let c_top = [
         tc[0] - rc * axis[0] - rc * normal[0],
diff --git a/crates/specialized/rtx-cfd/tests/patch_cylinder_flag_flat_tip.rs b/crates/specialized/rtx-cfd/tests/patch_cylinder_flag_flat_tip.rs
index 619cc84..2988f85 100644
--- a/crates/specialized/rtx-cfd/tests/patch_cylinder_flag_flat_tip.rs
+++ b/crates/specialized/rtx-cfd/tests/patch_cylinder_flag_flat_tip.rs
@@ -25,7 +25,14 @@ const FILLET: f64 = 0.5 * 0.41 / 41.0;
 const CORNER: f64 = 0.0025;
 
 fn edges(a: f64, nx: usize, ny: usize) -> FlagEdges {
+    edges_breathing(a, nx, ny, 0.0)
+}
+
+/// The flag's edges with the thickness scaled by `1 + eps` (the FEA's
+/// deformed tip is never exactly `t` thick).
+fn edges_breathing(a: f64, nx: usize, ny: usize, eps: f64) -> FlagEdges {
     let len = X1 - X0;
+    let th = T * (1.0 + eps);
     let centre = |x: f64| {
         let xi = (x - X0) / len;
         let y = a * xi * xi * (3.0 - xi) / 2.0;
@@ -35,7 +42,7 @@ fn edges(a: f64, nx: usize, ny: usize) -> FlagEdges {
     let edge = |x: f64, side: f64| -> [f64; 2] {
         let (y, dy) = centre(x);
         let n = (1.0 + dy * dy).sqrt();
-        [x - side * T * dy / n, CENTRE[1] + y + side * T / n]
+        [x - side * th * dy / n, CENTRE[1] + y + side * th / n]
     };
     let m = 2 * nx;
     let bottom: Vec<[f64; 2]> = (0..=m)
@@ -234,3 +241,47 @@ fn flat_tip_patch_builds_straight_and_bent_cold_and_warm() -> CfdResult<()> {
     }
     Ok(())
 }
+
+/// The semicircle setting (`tip_corner = t`) must keep its topology when
+/// the deformed tip is a few nanometres thicker or thinner than `t` (the
+/// P6-b ny 41 cap-12 marches died at every α on "topology changed
+/// 143x12 -> 144x12" within the first steps of the release — the flat /
+/// semicircle branch was chosen against the DEFORMED half-thickness).
+#[test]
+fn semicircle_setting_keeps_its_topology_under_thickness_breathing() {
+    for &h in &[0.01, 0.41 / 41.0, 0.41 / 62.0] {
+        let n0 = outline_breathing(0.0, h, T, 0.0).len();
+        for &eps in &[-1e-6, -1e-9, 1e-12, 1e-9, 1e-6, 1e-4] {
+            for &a in &[0.0, 1e-5, -1e-5, 0.08] {
+                let n = outline_breathing(a, h, T, eps).len();
+                assert_eq!(n, n0, "h = {h}, eps = {eps:e}, a = {a}: {n} points vs {n0}");
+            }
+        }
+        // The flat tip is insensitive too.
+        let f0 = outline_breathing(0.0, h, CORNER, 0.0).len();
+        for &eps in &[-1e-6, 1e-6, 1e-4] {
+            assert_eq!(
+                outline_breathing(0.0, h, CORNER, eps).len(),
+                f0,
+                "flat, h = {h}, eps = {eps:e}"
+            );
+        }
+    }
+}
+
+fn outline_breathing(a: f64, h: f64, corner: f64, eps: f64) -> Vec<[f64; 2]> {
+    cylinder_flag_outline_deformed_tip(
+        CENTRE,
+        R,
+        T,
+        &edges_breathing(a, 35, 2, eps),
+        X1,
+        FILLET,
+        3,
+        16,
+        h,
+        1.15,
+        corner,
+    )
+    .0
+}