diff --git a/crates/kernel/src/face_source_tests.rs b/crates/kernel/src/face_source_tests.rs
index 8c4e16c7de..c46f0008bf 100644
--- a/crates/kernel/src/face_source_tests.rs
+++ b/crates/kernel/src/face_source_tests.rs
@@ -30,6 +30,9 @@ fn box_certificates_are_six_construction_roles_with_dimension_stable_axes() {
         // A side far shorter than the body (a 0.3 mm label of a metre): its walls' edges lie
         // closer together than the sweep's body-sized nearness, and each keeps its own role.
         (Vec3::new(500., -0.3, 1400.), Vec3::new(1500., 0., 1800.)),
+        // Far thinner than the body's flatness tolerance (0.1 µm in 14 mm): each cap is in its
+        // own plane only.
+        (Vec3::ZERO, Vec3::new(10., 10., 1e-4)),
     ] {
         let body = crate::box_solid(a, b).unwrap().with_box_certificate(42).unwrap();
         let mut roles = std::collections::HashSet::new();
diff --git a/crates/kernel/src/smlib_sweeps.rs b/crates/kernel/src/smlib_sweeps.rs
index 6eb0767668..94cbe3406e 100644
--- a/crates/kernel/src/smlib_sweeps.rs
+++ b/crates/kernel/src/smlib_sweeps.rs
@@ -272,7 +272,12 @@ mod imp {
         let m = body.mesh(&crate::MeshTolerance::chord(mesh_tol))?;
         let faces = face_tris(&m);
         let n = body.face_count();
-        let flat_tol = size * 1e-5 + 1e-9;
+        // A cap's points lie on its plane to the evaluation's precision. Sized to the body, the
+        // flatness tolerance can exceed the gap between parallel start and end planes (a box
+        // 0.1 µm thick in a 14 mm body), which would put each cap in both: it stays under a
+        // quarter of that gap.
+        let gap = end.filter(|e| e.normal().cross(start.normal()).len() < 1e-9).map_or(f64::INFINITY, |e| start.height(e.origin).abs());
+        let flat_tol = (size * 1e-5 + 1e-9).min(0.25 * gap);
         let near = 3.0 * mesh_tol + flat_tol;
         let flat = |f: usize, pl: &Plane| {
             faces.get(f).is_some_and(|ft| !ft.tris.is_empty() && ft.tris.iter().all(|(t, _)| t.iter().all(|q| pl.height(*q).abs() <= flat_tol)))