diff --git a/crates/geom/src/brep/eval.rs b/crates/geom/src/brep/eval.rs
index 420579fac5..73b6260ccc 100644
--- a/crates/geom/src/brep/eval.rs
+++ b/crates/geom/src/brep/eval.rs
@@ -241,6 +241,26 @@ impl SurfaceEval {
         let (a, b) = (su.len(), sv.len());
         [a <= 1e-9 * b.max(1e-12), b <= 1e-9 * a.max(1e-12)]
     }
+
+    /// [`Self::pole`], or within `reach` mm of one: moving along a direction across half its
+    /// period (or its whole range) moves the point no more than `reach` (a boundary point that
+    /// lies at a pole to within a tolerance, where its periodic parameter is free).
+    pub fn pole_within(&self, uv: [f64; 2], reach: f64) -> [bool; 2] {
+        let mut out = self.pole(uv);
+        let Some((_, su, sv)) = self.derivs(uv[0], uv[1]) else { return out };
+        let dom = self.domain();
+        for (d, speed) in [su.len(), sv.len()].into_iter().enumerate() {
+            let span = match (dom.period.get(d).copied().flatten(), dom.range.get(d).copied().flatten()) {
+                (Some(period), _) => 0.5 * period,
+                (None, Some((a, b))) => (b - a).abs(),
+                (None, None) => continue,
+            };
+            if let Some(flag) = out.get_mut(d) {
+                *flag |= speed * span <= reach;
+            }
+        }
+        out
+    }
 }
 
 /// A tensor spline's rows (`rows[i][j]`, i along u) as the evaluator's u-fastest array.
diff --git a/crates/geom/src/brep/pcurve.rs b/crates/geom/src/brep/pcurve.rs
index 8460e5dff3..5c31d60ae3 100644
--- a/crates/geom/src/brep/pcurve.rs
+++ b/crates/geom/src/brep/pcurve.rs
@@ -529,19 +529,7 @@ impl On {
     /// parameter by half its period (or across its range) moves the point by no more than `reach`
     /// at its speed there. There the parameter is as free as at the pole itself.
     fn pole(&self, uv: [f64; 2], reach: f64) -> [bool; 2] {
-        let mut out = self.surface.pole(uv);
-        let Some((_, su, sv)) = self.surface.derivs(uv[0], uv[1]) else { return out };
-        for (d, speed) in [su.len(), sv.len()].into_iter().enumerate() {
-            let span = match (self.dom.period.get(d).copied().flatten(), self.dom.range.get(d).copied().flatten()) {
-                (Some(period), _) => 0.5 * period,
-                (None, Some((a, b))) => (b - a).abs(),
-                (None, None) => continue,
-            };
-            if let Some(flag) = out.get_mut(d) {
-                *flag |= speed * span <= reach;
-            }
-        }
-        out
+        self.surface.pole_within(uv, reach)
     }
 
     /// The loop's uses moved by whole periods so each starts where the one before it ends (in
diff --git a/crates/geom/src/brep/tests.rs b/crates/geom/src/brep/tests.rs
index 02fd84fe1f..48fa9e14f6 100644
--- a/crates/geom/src/brep/tests.rs
+++ b/crates/geom/src/brep/tests.rs
@@ -844,3 +844,16 @@ fn a_face_box_reaches_its_boundary_extremes() {
     let [_, v] = form::extent(&cylinder, &[vec![(&cut, [0.1, 0.1 + TAU], true)]]).expect("an extent");
     assert!((v[0] - 10.0).abs() < 1e-9 && (v[1] - 50.0).abs() < 1e-9, "{v:?}");
 }
+
+/// A point next to a sphere's pole is a pole to within the reach it is given: crossing half a
+/// turn of longitude there moves it about π times its distance from the axis, and no further.
+#[test]
+fn a_point_beside_a_pole_is_one_within_its_reach() {
+    let sphere = eval::SurfaceEval::new(&Surface::Sphere { frame: a_frame(), radius: 2. }).unwrap();
+    let near = [0.3, FRAC_PI_2 - 1e-7];
+    // 2e-7 mm from the axis: half a turn moves it 2π·1e-7.
+    assert_eq!(sphere.pole(near), [false, false]);
+    assert_eq!(sphere.pole_within(near, 1e-6), [true, false]);
+    assert_eq!(sphere.pole_within(near, 1e-7), [false, false]);
+    assert_eq!(sphere.pole_within([0.3, 0.2], 1e-3), [false, false]);
+}
diff --git a/crates/step/src/topology.rs b/crates/step/src/topology.rs
index 5ce1bd5668..e4d5ba0ca4 100644
--- a/crates/step/src/topology.rs
+++ b/crates/step/src/topology.rs
@@ -1104,10 +1104,11 @@ impl<'b, 'a> Builder<'b, 'a> {
         // does not wrap: a loop around a closed spline surface without a seam edge jumps across
         // its domain.
         let natural = surface.natural();
-        // At a pole (where the surface collapses to a point across a direction) a loop may jump
-        // in that direction: the seam's two uses meet there a period apart.
+        // At a pole (where the surface collapses to a point across a direction, to within the
+        // tolerance, as the pcurves closed the loop there) a loop may jump in that direction: the
+        // seam's two uses meet there a period apart.
         let poles = SurfaceEval::new(&surface).ok();
-        let at_pole = |uv: [f64; 2]| poles.as_ref().is_some_and(|e| e.pole(uv).iter().any(|p| *p));
+        let at_pole = |uv: [f64; 2]| poles.as_ref().is_some_and(|e| e.pole_within(uv, tol).iter().any(|p| *p));
         for (l, pc) in edge_loops.iter().zip(&pcs) {
             let mut ends = Vec::with_capacity(l.len());
             for (&(_, forward), c) in l.iter().zip(pc) {