diff --git a/crates/doc/src/eval_more.rs b/crates/doc/src/eval_more.rs
index 778db358..c63bb19e 100644
--- a/crates/doc/src/eval_more.rs
+++ b/crates/doc/src/eval_more.rs
@@ -325,6 +325,21 @@ fn sheet_for(st: &ModelState, body: &Option<String>, pick: Option<Vec3>) -> Resu
         .ok_or_else(|| DocError::Invalid("there is no sheet metal body".into()))
 }
 
+/// Whether a point touches sheet `i`'s material: inside it, on one of its faces, or within its
+/// display tolerance of one (picks land on the display mesh). A sheet with no mesh to test
+/// against counts as touched.
+fn on_sheet_material(st: &ModelState, i: usize, p: Vec3) -> bool {
+    let Some(mb) = st.sheets.get(i).and_then(|sh| st.body(&sh.body)) else { return true };
+    let mesh = mb.mesh();
+    if mesh.triangles.is_empty() {
+        return true;
+    }
+    // Twice the display mesh's chord tolerance (see `ModelBody::mesh`).
+    let step = 2.0 * (mb.body.size() * 1e-3).max(1e-3);
+    let offsets = [-step, 0.0, step];
+    offsets.iter().any(|&dx| offsets.iter().any(|&dy| offsets.iter().any(|&dz| mesh.contains(p + Vec3::new(dx, dy, dz)))))
+}
+
 /// A region mirrored in the sketch x axis (y → −y); curves other than lines and arcs as polylines.
 fn mirror_y(r: &Region2) -> Region2 {
     let f = |p: Vec2| Vec2::new(p.x, -p.y);
@@ -500,6 +515,19 @@ pub(super) fn sheet_eval(
                     doc.rule_values(vals, &doc.sheet_rule(Some(&rule)))?.bend_radius
                 }
             };
+            // A fixed point that touches no material at all (e.g. far above the sheet) still picks
+            // the side it lies on, so a stored fold keeps recomputing; it says so (#371).
+            if let Some(p) = fixed
+                && !on_sheet_material(st, i, *p)
+            {
+                *warning = Some(format!(
+                    "{} at ({:.3}, {:.3}, {:.3}); the side of the fold line it lies on stays",
+                    sheet::FOLD_FIXED_OFF_SHEET,
+                    p.x,
+                    p.y,
+                    p.z
+                ));
+            }
             let sh = st.sheets.get_mut(i).ok_or_else(|| DocError::Invalid("sheet".into()))?;
             if sh.flat {
                 return Err(DocError::Invalid("refold the sheet before folding it".into()));
diff --git a/crates/engine/src/cmd/sheet_tests.rs b/crates/engine/src/cmd/sheet_tests.rs
index a67636fc..e4520911 100644
--- a/crates/engine/src/cmd/sheet_tests.rs
+++ b/crates/engine/src/cmd/sheet_tests.rs
@@ -1990,7 +1990,6 @@ fn fold_keeps_working_when_a_parameter_moves_the_sheet_off_its_fixed_point() {
 /// `fixed` 20 mm above a 2.5 mm sheet touches no material: refuse it or say so, like a point in a
 /// cut-out (#371).
 #[test]
-#[ignore = "fails until #371 is fixed"]
 fn fold_fixed_point_above_the_sheet_is_not_silently_accepted() {
     let mut s = Session::default();
     base(&mut s, 100.0, 60.0);
@@ -2626,3 +2625,43 @@ fn fold_refuses_a_new_fixed_point_off_the_sheet_but_warns_on_recompute() {
     assert!(fold["error"].is_null(), "{fold}");
     assert!(fold["warning"].as_str().is_some_and(|w| w.contains("fixed point is not on the sheet") && w.contains("stays")), "{fold}");
 }
+
+/// Points away from the sheet are refused atomically; points on any face of it, on a flange's
+/// leg, or a hair off a face still pick the side that stays (#371).
+#[test]
+fn fold_fixed_point_must_touch_the_sheet_but_any_face_will_do() {
+    let mut s = Session::default();
+    base(&mut s, 100.0, 60.0);
+    let before = s.doc.to_json();
+    let undo = s.undo.len();
+    for fixed in [json!([90, 30, 20]), json!([90, 30, -10])] {
+        let e = s.execute("sheet.fold", &json!({"points": [[40, 0, 2.5], [40, 60, 2.5]], "fixed": fixed})).unwrap_err().to_string();
+        assert!(e.contains("fixed") && e.contains("not on the sheet"), "{fixed}: {e}");
+        assert_eq!(s.doc.to_json(), before);
+        assert_eq!(s.undo.len(), undo);
+    }
+    // The left (smaller) side stays: the right part stands up.
+    for fixed in [json!([10, 30, 2.5]), json!([10, 30, 0]), json!([0, 30, 1.25]), json!([10, 30, 2.500001])] {
+        let mut s = Session::default();
+        base(&mut s, 100.0, 60.0);
+        run(&mut s, "sheet.fold", json!({"points": [[40, 0, 2.5], [40, 60, 2.5]], "fixed": fixed, "name": "Fold1"}));
+        let bb = bbox(&mut s);
+        assert!(bb["min"][0].as_f64().unwrap_or(f64::NAN).abs() < 1e-6 && bb["max"][0].as_f64().unwrap_or(f64::NAN) < 50.0, "{fixed}: {bb}");
+        assert!(run(&mut s, "timeline.inspect", json!({"feature": "Fold1"}))["warning"].is_null(), "{fixed}");
+    }
+    // A point on an edge flange's leg (far above the base panel's slab) is on the sheet.
+    let mut f = Session::default();
+    run(&mut f, "parameters.change", json!({"name": "h", "expression": "20 mm"}));
+    base(&mut f, 100.0, 60.0);
+    run(&mut f, "sheet.flange", json!({"edges": [[100, 30, 2.5]], "height": "h"}));
+    let bb = bbox(&mut f);
+    let x = bb["max"][0].as_f64().unwrap_or(f64::NAN);
+    run(&mut f, "sheet.fold", json!({"points": [[40, 0, 2.5], [40, 60, 2.5]], "fixed": [x, 30, 15], "name": "Fold1"}));
+    assert!(run(&mut f, "timeline.inspect", json!({"feature": "Fold1"}))["warning"].is_null());
+    // A shorter leg leaves the stored point in the air: the fold recomputes, keeps its side and warns.
+    run(&mut f, "parameters.change", json!({"name": "h", "expression": "10 mm"}));
+    let fold = run(&mut f, "timeline.inspect", json!({"feature": "Fold1"}));
+    assert!(fold["error"].is_null(), "{fold}");
+    assert!(fold["warning"].as_str().is_some_and(|w| w.contains("fixed point is not on the sheet")), "{fold}");
+    assert!((bbox(&mut f)["max"][0].as_f64().unwrap_or(f64::NAN) - x).abs() < 1e-6, "the right side stays");
+}