diff --git a/CHANGELOG.md b/CHANGELOG.md
index ec4c721d..bc48c1e9 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -1,5 +1,13 @@
 ## Unreleased
 
+### Fixed
+
+- GJK no longer reports a stalled simplex as "touching" when the penetration bound is merely within
+  `sqrt(eps)` of the largest CSO support magnitude. With a large shape (e.g. a 200x200 ground cuboid)
+  that tolerance exceeded the size of the other shape, and the stale search direction was returned as
+  the contact normal. The tolerance is now `10 * eps` of that magnitude, so such stalls are
+  perturbed and, failing that, resolved by EPA.
+
 ### Modified
 
 - Deserializing a `ContactManifoldsWorkspace` that was serialized by a custom contact-manifolds
diff --git a/crates/parry3d/tests/geometry/cylinder_large_cuboid_normals.rs b/crates/parry3d/tests/geometry/cylinder_large_cuboid_normals.rs
new file mode 100644
index 00000000..c3dc2fa0
--- /dev/null
+++ b/crates/parry3d/tests/geometry/cylinder_large_cuboid_normals.rs
@@ -0,0 +1,140 @@
+use parry3d::math::{Pose, Rotation, Vector};
+use parry3d::query::details::contact_manifold_pfm_pfm_shapes;
+use parry3d::query::ContactManifold;
+use parry3d::shape::{Cuboid, Cylinder};
+
+// A wheel (cylinder, radius 0.1, half-height 0.05) spinning at ~43 rad/s and rolling on a
+// 200x200x0.5 ground cuboid. The poses are consecutive simulation steps, so the manifold's
+// cached normal is carried from one step to the next, as in a physics engine.
+//
+// The wheel hovers ~1e-4 above the ground. With the ground's far corners, the CSO support
+// points reach ~144, and GJK stalled with a lower bound of -3.3e-2 on the penetration. That
+// was accepted as "touching" because the tolerance scaled with `sqrt(eps) * support_scale`
+// (~0.15), more than the whole wheel, and the stale `old_dir` was returned as the contact
+// normal: (-0.51, 0.69, 0.51) instead of (0, 1, 0).
+#[test]
+fn cylinder_rolling_on_large_cuboid_keeps_a_vertical_normal() {
+    // (body translation, body rotation as xyzw) for 9 consecutive steps.
+    const POSES: [([f32; 3], [f32; 4]); 9] = [
+        (
+            [2.1498301029205322, 0.12448107451200485, 0.3996926546096802],
+            [
+                0.2489599585533142,
+                -0.6622065305709839,
+                0.24911077320575714,
+                0.6613963842391968,
+            ],
+        ),
+        (
+            [2.1495745182037354, 0.12121988832950592, 0.3993242681026459],
+            [
+                0.46536311507225037,
+                -0.5327503085136414,
+                0.46630358695983887,
+                0.5312016010284424,
+            ],
+        ),
+        (
+            [2.149324655532837, 0.11523353308439255, 0.39895594120025635],
+            [
+                0.6219001412391663,
+                -0.335325688123703,
+                0.6241053342819214,
+                0.333600789308548,
+            ],
+        ),
+        (
+            [2.149080514907837, 0.10652196407318115, 0.398587703704834],
+            [
+                0.698938250541687,
+                -0.0951799601316452,
+                0.7025531530380249,
+                0.09405004233121872,
+            ],
+        ),
+        (
+            [2.1422312259674072, 0.10064999759197235, 0.3972579836845398],
+            [
+                0.6860884428024292,
+                0.16269077360630035,
+                0.6905015707015991,
+                -0.16131402552127838,
+            ],
+        ),
+        (
+            [2.1318469047546387, 0.10105381906032562, 0.3973941504955292],
+            [
+                0.5856216549873352,
+                0.39615002274513245,
+                0.5881026387214661,
+                -0.39274439215660095,
+            ],
+        ),
+        (
+            [2.120213747024536, 0.10016228258609772, 0.3976336121559143],
+            [
+                0.409687876701355,
+                0.5779147148132324,
+                0.4103825092315674,
+                -0.5742447972297668,
+            ],
+        ),
+        (
+            [2.105894088745117, 0.1001090407371521, 0.3976670503616333],
+            [
+                0.1809343695640564,
+                0.6852918267250061,
+                0.1802733689546585,
+                -0.6820118427276611,
+            ],
+        ),
+        (
+            [2.0811259746551514, 0.11391204595565796, 0.40611711144447327],
+            [
+                -0.03625503554940224,
+                0.7045886516571045,
+                -0.047442466020584106,
+                -0.7070997357368469,
+            ],
+        ),
+    ];
+    const PREDICTION: f32 = 0.02;
+
+    let ground = Cuboid::new(Vector::new(100.0, 0.25, 100.0));
+    let ground_pose = Pose::from_translation(Vector::new(0.0, -0.25, 0.0));
+    let wheel = Cylinder::new(0.05, 0.1);
+    // The cylinder's axis (local Y) is laid along the body's local X.
+    let axle_to_body_x = Rotation::from_xyzw(
+        0.0,
+        0.0,
+        core::f32::consts::FRAC_1_SQRT_2,
+        core::f32::consts::FRAC_1_SQRT_2,
+    );
+
+    let mut manifold = ContactManifold::<(), ()>::new();
+    for (step, (t, q)) in POSES.iter().enumerate() {
+        let body = Pose::from_parts(
+            Vector::new(t[0], t[1], t[2]),
+            Rotation::from_xyzw(q[0], q[1], q[2], q[3]),
+        );
+        let wheel_pose = body * Pose::from_parts(Vector::ZERO, axle_to_body_x);
+        let pos12 = ground_pose.inv_mul(&wheel_pose);
+
+        contact_manifold_pfm_pfm_shapes(
+            &pos12,
+            &ground,
+            &wheel,
+            None,
+            None,
+            PREDICTION,
+            &mut manifold,
+        );
+
+        assert!(!manifold.points.is_empty(), "step {step}: no contact");
+        assert!(
+            manifold.local_n1.y > 0.99,
+            "step {step}: contact normal {:?} is not vertical",
+            manifold.local_n1
+        );
+    }
+}
diff --git a/crates/parry3d/tests/geometry/mod.rs b/crates/parry3d/tests/geometry/mod.rs
index 7eb925aa..5eacd54e 100644
--- a/crates/parry3d/tests/geometry/mod.rs
+++ b/crates/parry3d/tests/geometry/mod.rs
@@ -4,6 +4,7 @@ mod ball_triangle_toi;
 mod convex_hull;
 mod cuboid_ray_cast;
 mod cylinder_cuboid_contact;
+mod cylinder_large_cuboid_normals;
 mod epa3;
 mod still_objects_toi;
 mod time_of_impact3;
diff --git a/src/query/gjk/gjk.rs b/src/query/gjk/gjk.rs
index 56f4caea..9cb64cb3 100644
--- a/src/query/gjk/gjk.rs
+++ b/src/query/gjk/gjk.rs
@@ -412,7 +412,7 @@ where
                 } else {
                     return GJKResult::Proximity(old_dir);
                 }
-            } else if exact_dist && best_min_bound >= -_eps_rel * support_scale {
+            } else if exact_dist && best_min_bound >= -_eps_tol * support_scale {
                 // Origin on the CSO boundary: the pair is touching, so the witnesses
                 // describe the contact.
                 let (p1, p2) = result(simplex, true);
@@ -457,7 +457,7 @@ where
                 } else {
                     return GJKResult::Proximity(dir);
                 }
-            } else if exact_dist && best_min_bound >= -_eps_rel * support_scale {
+            } else if exact_dist && best_min_bound >= -_eps_tol * support_scale {
                 let (p1, p2) = result(simplex, false);
                 return GJKResult::ClosestPoints(p1, p2, dir);
             } else if nb_perturbs < MAX_PERTURBATIONS {