diff --git a/crates/geom/src/brep/eval.rs b/crates/geom/src/brep/eval.rs
index e4cf59bdd2..9b34a8d1fd 100644
--- a/crates/geom/src/brep/eval.rs
+++ b/crates/geom/src/brep/eval.rs
@@ -294,20 +294,8 @@ impl SurfaceEval {
     /// 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
+        let Some((_, su, sv)) = self.derivs(uv[0], uv[1]) else { return [false; 2] };
+        uv::collapsed([su.len(), sv.len()], &self.domain(), reach)
     }
 }
 
diff --git a/crates/geom/src/tess/uv.rs b/crates/geom/src/tess/uv.rs
index 0ec7e28bb1..65040c9425 100644
--- a/crates/geom/src/tess/uv.rs
+++ b/crates/geom/src/tess/uv.rs
@@ -320,6 +320,32 @@ pub fn degenerate(s: &EvalSurface, uv: [f64; 2], d: usize) -> bool {
     this.len() <= 1e-9 * other.len().max(1e-12)
 }
 
+/// [`degenerate`], or within `reach` mm of a pole (see [`collapsed`]): a boundary point on a
+/// patch side collapsed to within a tolerance, not to a point, where its parameter along `d` is
+/// as free as at a pole.
+pub fn degenerate_within(s: &EvalSurface, dom: &Domain, uv: [f64; 2], d: usize, reach: f64) -> bool {
+    let Some((_, su, sv)) = s.derivs(uv[0], uv[1]) else { return false };
+    let speed = [su.len(), sv.len()];
+    collapsed(speed, dom, reach).get(d).copied().unwrap_or(false)
+}
+
+/// Along each direction, whether a point whose parameter speeds (mm per unit) are `speed` is a
+/// pole, or within `reach` mm of one: moving along the direction there not at all (to 1e-9 of
+/// the other's speed), or across half its period (or its whole range) by no more than `reach`.
+pub fn collapsed(speed: [f64; 2], dom: &Domain, reach: f64) -> [bool; 2] {
+    let [a, b] = speed;
+    let mut out = [a <= 1e-9 * b.max(1e-12), b <= 1e-9 * a.max(1e-12)];
+    for (d, flag) in out.iter_mut().enumerate() {
+        let span = match (dom.period[d], dom.range[d]) {
+            (Some(period), _) => 0.5 * period,
+            (None, Some((lo, hi))) => (hi - lo).abs(),
+            (None, None) => continue,
+        };
+        *flag |= speed[d] * span <= reach;
+    }
+    out
+}
+
 #[cfg(test)]
 mod tests {
     use super::*;
diff --git a/crates/kernel/src/body.rs b/crates/kernel/src/body.rs
index 97b46c9412..6bb9fcccd6 100644
--- a/crates/kernel/src/body.rs
+++ b/crates/kernel/src/body.rs
@@ -878,7 +878,8 @@ impl Body {
     }
 
     /// The body's volume from a mesh at `tol`, measured once per (record, tolerance): clones of
-    /// a body share the measurement. `None` when the body cannot be meshed.
+    /// a body share the measurement. `None` when the body cannot be meshed, or a face of it was
+    /// left out of the mesh (an open mesh has no volume).
     pub fn volume_at(&self, tol: f64) -> Option<f64> {
         let key = if tol.is_finite() && tol > 0.0 { tol } else { 0.05 };
         let key = key.to_bits();
@@ -889,7 +890,11 @@ impl Body {
         {
             return Some(volume);
         }
-        let volume = self.mesh(&MeshTolerance::chord(f64::from_bits(key))).ok()?.measure().volume;
+        let mesh = self.mesh(&MeshTolerance::chord(f64::from_bits(key))).ok()?;
+        if self.faces_without_triangles(&mesh) > 0 {
+            return None;
+        }
+        let volume = mesh.measure().volume;
         if let Some(cell) = &self.volume
             && let Ok(mut slot) = cell.lock()
         {
@@ -913,7 +918,8 @@ impl Body {
     /// joins, protected creases). Triangles carry their face index and the edges their polylines,
     /// in the record's order; neighbouring faces share every boundary node, so the mesh is
     /// watertight wherever the B-rep is. A mesh body gives its own triangles. A face the planner
-    /// cannot mesh is an error that names it and why.
+    /// cannot mesh is left out (logged with why; [`Body::unmeshed_faces`] counts it) and the rest
+    /// meshed; a body none of whose faces mesh is an error that names the first and why.
     pub fn mesh(&self, t: &MeshTolerance) -> Result<Mesh> {
         crate::profile::timed(crate::profile::Kind::Tessellate, "mesh", || self.mesh_untimed(t))
     }
@@ -956,6 +962,10 @@ impl Body {
         for f in &stats.lenient {
             log::warn!("display plan: face {} filled leniently in (u, v), a defect in its pcurves or seams: {}", f.face, f.reason);
         }
+        // A face no method plans is left out (no triangles; `unmeshed_faces` counts it) and named.
+        for f in &stats.unmeshed {
+            log::warn!("display plan: face {} left out of the mesh: {}", f.face, f.reason);
+        }
         Ok(plan)
     }
 
@@ -1047,6 +1057,14 @@ impl Body {
             return 0;
         }
         let Ok(m) = self.mesh(&MeshTolerance::chord(coarse_tolerance(self.size()))) else { return self.face_count() };
+        self.faces_without_triangles(&m)
+    }
+
+    /// Faces of the body with no triangles in its mesh `m` (left out by the mesher).
+    fn faces_without_triangles(&self, m: &Mesh) -> usize {
+        if self.is_mesh() {
+            return 0;
+        }
         let mut has = vec![false; self.face_count()];
         for f in &m.tri_face {
             if let Some(h) = has.get_mut(*f as usize) {
diff --git a/crates/kernel/src/tess/mod.rs b/crates/kernel/src/tess/mod.rs
index 08d9b192f8..d956b2adb2 100644
--- a/crates/kernel/src/tess/mod.rs
+++ b/crates/kernel/src/tess/mod.rs
@@ -18,9 +18,10 @@
 //!    named in the plan's stats ([`PlanStats::lenient`], with why it was refused) so it is
 //!    reported and fixed there; it is filled leniently in (u, v) (the sides crossing one already
 //!    kept are left out, see [`Lenient`]) so the display has no hole. A face that refuses even
-//!    that (its loops do not close in (u, v), or wind round a seam or pole), or whose pcurves do
-//!    not lie on its edges, is an error naming the face (and the edge) and why; there is no
-//!    other mesher to fall back to.
+//!    that (its loops do not close in (u, v), or wind round a seam or pole) is left out of the
+//!    plan with no triangles and named in [`PlanStats::unmeshed`], and the rest of the body is
+//!    meshed; there is no other mesher to fall back to. A face whose pcurves do not lie on its edges is an error naming the face and
+//!    the edge, as is a body none of whose faces can be planned.
 
 pub(crate) mod cdt;
 pub(crate) mod collar;
@@ -64,12 +65,15 @@ pub struct PlanStats {
     pub delaunay: usize,
     /// Faces every method refused, filled leniently in (u, v) (see [`Lenient`]) so the display
     /// has no hole: each still a defect upstream, in its pcurves or seams.
-    pub lenient: Vec<LenientFace>,
+    pub lenient: Vec<RefusedFace>,
+    /// Faces even the lenient fill refused: left out of the plan (no triangles), so the rest of
+    /// the body is still meshed, each with why it could not be planned.
+    pub unmeshed: Vec<RefusedFace>,
 }
 
-/// A face filled leniently: its index in the record and why the strict plan refused it.
+/// A face a method refused: its index in the record and why.
 #[derive(Clone, Debug, PartialEq, Eq)]
-pub struct LenientFace {
+pub struct RefusedFace {
     pub face: usize,
     pub reason: String,
 }
@@ -198,7 +202,7 @@ pub(crate) fn plan(b: &brep::Brep, tol: f64, turn: Option<size::Turn>) -> Result
             return Err(unsupported(format!("face {fi}: missing")));
         };
         let refused = tangled.iter().find(|(f, _)| *f == fi).map(|(_, why)| why.as_str());
-        let (fp, stats) = plan_or_fill(fi, kind(face.parameters()), eval, &fb.loops, &rules, *size, refused)?;
+        let (fp, stats) = plan_or_fill(fi, kind(face.parameters()), eval, &fb.loops, &rules, *size, fb.reach, refused)?;
         let fwd = forward.get(fi).copied().unwrap_or(true);
         let fp =
             FacePlan { flip: !fwd, triangles: if fwd { fp.triangles } else { fp.triangles.into_iter().map(|[a, b, c]| [a, c, b]).collect() }, ..fp };
@@ -206,16 +210,31 @@ pub(crate) fn plan(b: &brep::Brep, tol: f64, turn: Option<size::Turn>) -> Result
     });
     let mut plan = TessPlan::default();
     let mut stats = PlanStats::default();
-    for (r, eval) in planned.into_iter().zip(evals) {
-        let (fp, s) = r?;
+    // A face no method plans is left out, named in the stats, and the rest of the body is
+    // meshed; a body none of whose faces plan is the first face's error.
+    let mut first_error: Option<KernelError> = None;
+    for (fi, (r, eval)) in planned.into_iter().zip(evals).enumerate() {
         let surface = u32::try_from(plan.surfaces.len()).map_err(|_| unsupported("too many faces"))?;
         plan.surfaces.push(eval);
+        let (fp, s) = match r {
+            Ok(done) => done,
+            Err(e) => {
+                stats.unmeshed.push(RefusedFace { face: fi, reason: e.to_string() });
+                first_error.get_or_insert(e);
+                (FacePlan { face: u32::try_from(fi).unwrap_or(u32::MAX), ..FacePlan::default() }, PlanStats::default())
+            }
+        };
         plan.faces.push(FacePlan { surface: Some(surface), ..fp });
         stats.planar += s.planar;
         stats.structured += s.structured;
         stats.delaunay += s.delaunay;
         stats.lenient.extend(s.lenient);
     }
+    if let Some(e) = first_error
+        && stats.unmeshed.len() == faces.len()
+    {
+        return Err(e);
+    }
     plan.edges = nodes.into_iter().map(|n| n.points).collect();
     plan.edge_faces = edge_faces.into_iter().map(|fs| fs.into_iter().map(|f| u32::try_from(f).unwrap_or(u32::MAX)).collect()).collect();
     Ok((plan, stats))
@@ -226,7 +245,8 @@ pub(crate) fn plan(b: &brep::Brep, tol: f64, turn: Option<size::Turn>) -> Result
 /// at the node's curve parameter. A seam edge's two uses on one face have their own pcurves, so
 /// its two sides are told apart. A pcurve runs on continuously; where one edge meets the next, a
 /// pcurve that starts a whole period along from where the previous one ended is moved back by
-/// it, except in a direction the meeting point is a pole of: there the parameter is free, and
+/// it, except in a direction the meeting point is a pole of (or lies on a patch side collapsed
+/// to within its vertex's tolerance, see [`Boundary::reach`]): there the parameter is free, and
 /// the two pcurves' own values are where the face's boundary runs along the pole from and to
 /// (a meridian revolved half a turn meets the other at the pole half a period away; unwrapped,
 /// its loop would wind round the surface). Each loop then starts at a node off every pole whose
@@ -251,7 +271,21 @@ fn boundary(
 ) -> Result<Boundary> {
     let dom = uv::domain(s);
     let fail = |e: u32, why: String| unsupported(format!("face {fi} ({}): edge {e}: {why}", kind(&face.surface)));
-    let mut out = Boundary { loops: Vec::with_capacity(face.loops.len()), segments: Vec::with_capacity(face.loops.len()) };
+    // How near a pole a node is one: the tolerance of a vertex of the face that lies on a patch
+    // side collapsed to within it, not to a point, as the record check takes a loop's pole; at
+    // most the chord tolerance, finer than which the mesh resolves nothing (a vertex tolerance
+    // as large as the surface's features would make a pole of every node near one).
+    let mut reach: f64 = 0.0;
+    for oe in face_edges(face) {
+        let (Some(edge), Some([p0, p1])) = (b.edges.get(oe.edge as usize), oe.pcurve.interval()) else { continue };
+        let tol = edge.vertices.get(usize::from(!oe.forward)).and_then(|v| b.vertices.get(*v as usize)).map_or(0.0, |v| v.tolerance.min(rules.tol));
+        let Some(q) = oe.pcurve.eval(if oe.forward { p0 } else { p1 }).map(|q| map.to_eval(q)) else { continue };
+        if tol.is_finite() && tol > reach && (0..2).any(|d| uv::degenerate_within(s, &dom, q, d, tol)) {
+            reach = tol;
+        }
+    }
+    let pole_along = |q: [f64; 2], d: usize| uv::degenerate_within(s, &dom, q, d, reach);
+    let mut out = Boundary { loops: Vec::with_capacity(face.loops.len()), segments: Vec::with_capacity(face.loops.len()), reach };
     // Each vertex's parameters, from the first edge use of the face to start there.
     let mut at_vertex: std::collections::HashMap<u32, [f64; 2]> = std::collections::HashMap::new();
     for l in &face.loops {
@@ -290,7 +324,7 @@ fn boundary(
             if let (Some(e), Some(&start)) = (end, eq.first()) {
                 for d in 0..2 {
                     let Some(per) = dom.period[d] else { continue };
-                    if !(uv::degenerate(s, e, d) || uv::degenerate(s, start, d)) {
+                    if !(pole_along(e, d) || pole_along(start, d)) {
                         shift[d] = per * ((e[d] - start[d]) / per).round();
                     }
                 }
@@ -321,7 +355,7 @@ fn boundary(
             uvs.extend(eq);
         }
         let n = uvs.len();
-        let pole = |q: &[f64; 2]| (0..2).any(|d| uv::degenerate(s, *q, d));
+        let pole = |q: &[f64; 2]| (0..2).any(|d| pole_along(*q, d));
         if let Some(k) = (0..n).find(|&k| uvs.get(k).is_some_and(|q| !pole(q)) && uvs.get((k + n - 1) % n).is_some_and(|q| !pole(q))) {
             uvs.rotate_left(k);
             pts.rotate_left(k);
@@ -336,10 +370,13 @@ fn boundary(
 }
 
 /// A face's boundary loops, and for each loop node the edge segment that starts there: the
-/// edge, and the segment's place along it from its start vertex.
+/// edge, and the segment's place along it from its start vertex. `reach`: how near a pole a
+/// point of the face is taken as one (see [`uv::degenerate_within`]): the tolerance of a vertex
+/// of it on a patch side collapsed to within that tolerance, else 0.
 struct Boundary {
     loops: Vec<seam::Loop>,
     segments: Vec<Vec<(u32, u32)>>,
+    reach: f64,
 }
 
 /// Edges are sampled to the chord tolerance each on its own, so where two curves of one face
@@ -360,7 +397,7 @@ fn untangle(
 ) -> Result<Vec<(usize, String)>> {
     let crossing = |fi: usize, fb: &Boundary| -> Vec<(u32, u32)> {
         let Some(s) = evals.get(fi) else { return Vec::new() };
-        tangle::crossing(s, &fb.loops).into_iter().filter_map(|(l, k)| fb.segments.get(l)?.get(k).copied()).collect()
+        tangle::crossing(s, fb.reach, &fb.loops).into_iter().filter_map(|(l, k)| fb.segments.get(l)?.get(k).copied()).collect()
     };
     let indexed: Vec<usize> = (0..boundaries.len()).collect();
     let curved = evals.iter().filter(|e| !matches!(e, EvalSurface::Plane { .. })).count();
@@ -499,9 +536,9 @@ fn area2(uv: &[[f64; 2]]) -> f64 {
 /// unchanged across the pole. Across a pole it may jump by any amount, so the run from the
 /// second pole back to the first is moved one period along, whichever way keeps the loop
 /// within one period with a positive area: the seam's two uses then lie a period apart.
-fn open_pole_to_pole(s: &EvalSurface, dom: &uv::Domain, l: &mut seam::Loop) {
+fn open_pole_to_pole(s: &EvalSurface, dom: &uv::Domain, reach: f64, l: &mut seam::Loop) {
     let n = l.uv.len();
-    let poles: Vec<usize> = (0..n).filter(|&k| l.uv.get(k).is_some_and(|q| (0..2).any(|d| uv::degenerate(s, *q, d)))).collect();
+    let poles: Vec<usize> = (0..n).filter(|&k| l.uv.get(k).is_some_and(|q| (0..2).any(|d| uv::degenerate_within(s, dom, *q, d, reach)))).collect();
     let [p0, p1] = poles.as_slice() else { return };
     for d in 0..2 {
         let Some(per) = dom.period[d] else { continue };
@@ -547,11 +584,11 @@ fn open_pole_to_pole(s: &EvalSurface, dom: &uv::Domain, l: &mut seam::Loop) {
 
 /// The poles of a loop where it turns straight back along a seam: the nodes either side have
 /// the same parameters.
-fn turning_poles(s: &EvalSurface, per: f64, uv: &[[f64; 2]]) -> Vec<usize> {
+fn turning_poles(s: &EvalSurface, dom: &uv::Domain, reach: f64, per: f64, uv: &[[f64; 2]]) -> Vec<usize> {
     let n = uv.len();
     let near = |a: Option<&[f64; 2]>, b: Option<&[f64; 2]>| a.zip(b).is_some_and(|(a, b)| (a[0] - b[0]).abs().max((a[1] - b[1]).abs()) <= 1e-9 * per);
     (0..n)
-        .filter(|&k| uv.get(k).is_some_and(|q| (0..2).any(|d| uv::degenerate(s, *q, d))))
+        .filter(|&k| uv.get(k).is_some_and(|q| (0..2).any(|d| uv::degenerate_within(s, dom, *q, d, reach))))
         .filter(|&k| near(uv.get((k + n - 1) % n), uv.get((k + 1) % n)))
         .collect()
 }
@@ -564,7 +601,7 @@ fn turning_poles(s: &EvalSurface, per: f64, uv: &[[f64; 2]]) -> Vec<usize> {
 /// to the pole, and starts at the node before it: the seam's uses then bound the face's parameter box on either side, and share
 /// their nodes as every other edge does. Kept as it was unless the opened loop spans at most a
 /// period with a positive area.
-fn open_at_pole(s: &EvalSurface, dom: &uv::Domain, l: &mut seam::Loop) {
+fn open_at_pole(s: &EvalSurface, dom: &uv::Domain, reach: f64, l: &mut seam::Loop) {
     let n = l.uv.len();
     if n < 4 || l.pts.len() != n {
         return;
@@ -574,7 +611,7 @@ fn open_at_pole(s: &EvalSurface, dom: &uv::Domain, l: &mut seam::Loop) {
         if (seam::drift(l, d, per).abs() - per).abs() > per * 1e-6 {
             continue;
         }
-        let [pole] = turning_poles(s, per, &l.uv)[..] else { continue };
+        let [pole] = turning_poles(s, dom, reach, per, &l.uv)[..] else { continue };
         // From the node after the pole round to the pole, unwrapped along the way.
         let order: Vec<usize> = (1..=n).map(|i| (pole + i) % n).collect();
         let mut uv: Vec<[f64; 2]> = Vec::with_capacity(n);
@@ -633,9 +670,10 @@ fn holes_beside_outer(dom: &uv::Domain, loops: &mut [seam::Loop]) {
 /// One face, planned by [`plan_face`]; a face it refuses because its loops cross in (u, v) is
 /// planned once more leniently (see [`Lenient`]) and named in the stats with the first refusal,
 /// so the defect upstream is still reported. When the lenient fill refuses too, the first
-/// refusal is the error. `refused`: a face whose boundary still crosses itself after
-/// [`untangle`] (two edges meeting tangentially at a cusp, whose chords cross however often
-/// they are halved), filled leniently straight away with that reason.
+/// refusal is the error (and [`plan`] leaves the face out, named). `refused`: a face whose
+/// boundary still crosses itself after [`untangle`] (two edges meeting tangentially at a cusp,
+/// whose chords cross however often they are halved), filled leniently straight away with that
+/// reason.
 fn plan_or_fill(
     fi: usize,
     kind: &str,
@@ -643,20 +681,21 @@ fn plan_or_fill(
     loops: &[seam::Loop],
     rules: &size::Rules,
     size: (f64, [f64; 2]),
+    reach: f64,
     refused: Option<&str>,
 ) -> Result<(FacePlan, PlanStats)> {
     let mut stats = PlanStats::default();
     let strict = match refused {
         Some(why) => Err(unsupported(why.to_string())),
-        None => plan_face(fi, kind, s, loops, rules, size, &mut stats, Lenient::No),
+        None => plan_face(fi, kind, s, loops, rules, size, reach, &mut stats, Lenient::No),
     };
     match strict {
         Ok(fp) => Ok((fp, stats)),
         Err(first) => {
             let mut stats = PlanStats::default();
-            match plan_face(fi, kind, s, loops, rules, size, &mut stats, Lenient::Yes) {
+            match plan_face(fi, kind, s, loops, rules, size, reach, &mut stats, Lenient::Yes) {
                 Ok(fp) => {
-                    stats.lenient.push(LenientFace { face: fi, reason: first.to_string() });
+                    stats.lenient.push(RefusedFace { face: fi, reason: first.to_string() });
                     Ok((fp, stats))
                 }
                 Err(_) => Err(first),
@@ -667,7 +706,8 @@ fn plan_or_fill(
 
 /// One face: boundary loops in 3D (shared edge nodes) to parameters and triangles.
 /// `size`: the face's target side length (mm) and the surface's mean speed along u and v.
-/// `lenient`: the last resort for a face no method could plan (see [`Lenient`]).
+/// `reach`: how near a pole a point is taken as one ([`Boundary::reach`]). `lenient`: the last resort
+/// for a face no method could plan (see [`Lenient`]).
 #[allow(clippy::too_many_arguments)]
 fn plan_face(
     fi: usize,
@@ -676,6 +716,7 @@ fn plan_face(
     loops: &[seam::Loop],
     rules: &size::Rules,
     size: (f64, [f64; 2]),
+    reach: f64,
     stats: &mut PlanStats,
     lenient: Lenient,
 ) -> Result<FacePlan> {
@@ -689,7 +730,7 @@ fn plan_face(
     // A loop that runs up a seam from a pole and straight back down it (a half sphere bounded
     // by its rim and its seam used both ways) winds once as projected: it is opened at the pole.
     for l in projected.iter_mut() {
-        open_at_pole(s, &dom, l);
+        open_at_pole(s, &dom, reach, l);
     }
     let wound: Vec<usize> = (0..2).filter(|&d| dom.period[d].is_some_and(|per| seam::winds(&projected, d, per))).collect();
     // A face cut along a seam is a long winding strip whose sides are the cut lines and the
@@ -708,7 +749,7 @@ fn plan_face(
     // kernel writes it) projects onto one parameter line: its second run is a period along.
     if !seamed {
         for l in projected.iter_mut() {
-            open_pole_to_pole(s, &dom, l);
+            open_pole_to_pole(s, &dom, reach, l);
         }
         holes_beside_outer(&dom, &mut projected);
     }
@@ -723,7 +764,7 @@ fn plan_face(
             // At a pole the parameter along the degenerate direction is free: one copy with
             // the previous node's value, one with the next node's.
             let (prev, next) = (luv.get((k + n - 1) % n).copied(), luv.get((k + 1) % n).copied());
-            let pole = (0..2).find(|&d| uv::degenerate(s, *q, d));
+            let pole = (0..2).find(|&d| uv::degenerate_within(s, &dom, *q, d, reach));
             match (pole, prev, next) {
                 (Some(d), Some(a), Some(b)) => {
                     let mut qa = *q;
@@ -875,7 +916,7 @@ fn plan_face(
     all.extend(extra);
     // Whatever the layout, no triangle leaves the surface by more than the tolerance.
     if !planar {
-        sag::refine(s, tol, &fixed, &mut all, &mut triangles, MAX_REFINE);
+        sag::refine(s, tol, reach, &fixed, &mut all, &mut triangles, MAX_REFINE);
     }
     // Both evaluators take a closed spline's parameters inside its active interval.
     for q in &mut all {
diff --git a/crates/kernel/src/tess/sag.rs b/crates/kernel/src/tess/sag.rs
index 1d96e79948..02ebb35ef5 100644
--- a/crates/kernel/src/tess/sag.rs
+++ b/crates/kernel/src/tess/sag.rs
@@ -14,12 +14,15 @@ use solidcraft_geom::tess::EvalSurface;
 
 /// Split the triangles `tris` of a face (indices into `uv`; the first `fixed.len()` points sit
 /// at `fixed`, the rest on the surface at their parameters) until none of them lies farther
-/// than `tol` from the surface `s`, adding at most `max_points` points to `uv`.
-pub(crate) fn refine(s: &EvalSurface, tol: f64, fixed: &[Vec3], uv: &mut Vec<[f64; 2]>, tris: &mut Vec<[u32; 3]>, max_points: usize) {
+/// than `tol` from the surface `s`, adding at most `max_points` points to `uv`. `reach`: how near
+/// a pole a point is taken as one (see [`super::uv::degenerate_within`]).
+pub(crate) fn refine(s: &EvalSurface, tol: f64, reach: f64, fixed: &[Vec3], uv: &mut Vec<[f64; 2]>, tris: &mut Vec<[u32; 3]>, max_points: usize) {
     if !(tol.is_finite() && tol > 0.0) || tris.is_empty() {
         return;
     }
     let at = |q: [f64; 2]| s.eval(q[0], q[1]).0;
+    let dom = super::uv::domain(s);
+    let mean = |qs: &[[f64; 2]]| mean(s, &dom, reach, qs);
     // How far the flat point `c` lies from the surface, measured along the normal at the surface
     // point of the same parameters: those parameters need not be the nearest (a straight line in
     // the parameters is no geodesic near a pole), and only the departure across the surface
@@ -64,7 +67,7 @@ pub(crate) fn refine(s: &EvalSurface, tol: f64, fixed: &[Vec3], uv: &mut Vec<[f6
             if len <= shortest {
                 continue;
             }
-            let sag = *sag_of.entry(e).or_insert_with(|| off(mean(s, &[ua, ub]), (xa + xb) * 0.5));
+            let sag = *sag_of.entry(e).or_insert_with(|| off(mean(&[ua, ub]), (xa + xb) * 0.5));
             inner.push((sag, len, e));
         }
         // The surface's departure from a flat triangle is about quadratic over it: at the
@@ -72,7 +75,7 @@ pub(crate) fn refine(s: &EvalSurface, tol: f64, fixed: &[Vec3], uv: &mut Vec<[f6
         // inside the face divides by four. A triangle whose middle is off splits its side inside
         // the face that sags most, while that side still contributes; what is left comes from
         // its boundary sides, which the edges' own sampling keeps within the tolerance.
-        let centroid = mean(s, &[qa, qb, qc]);
+        let centroid = mean(&[qa, qb, qc]);
         let worst = inner.iter().max_by(|x, y| x.0.total_cmp(&y.0)).copied();
         let split = match worst {
             Some((sag, _, e)) if sag > tol => e,
@@ -80,7 +83,7 @@ pub(crate) fn refine(s: &EvalSurface, tol: f64, fixed: &[Vec3], uv: &mut Vec<[f6
             _ => continue,
         };
         let (Some(ua), Some(ub)) = (q(split.0), q(split.1)) else { continue };
-        let m = mean(s, &[ua, ub]);
+        let m = mean(&[ua, ub]);
         let mi = uv.len() as u32;
         uv.push(m);
         pos.push(at(m));
@@ -92,10 +95,10 @@ pub(crate) fn refine(s: &EvalSurface, tol: f64, fixed: &[Vec3], uv: &mut Vec<[f6
 /// The mean of parameter points, as the surface point between them: at a pole the parameter
 /// along the degenerate direction is free (the pole copy carries a neighbour's), so there the
 /// mean takes it from the other points alone.
-fn mean(s: &EvalSurface, qs: &[[f64; 2]]) -> [f64; 2] {
+fn mean(s: &EvalSurface, dom: &super::uv::Domain, reach: f64, qs: &[[f64; 2]]) -> [f64; 2] {
     let mut out = [0.0; 2];
     for (d, slot) in out.iter_mut().enumerate() {
-        let free: Vec<f64> = qs.iter().filter(|q| !super::uv::degenerate(s, **q, d)).map(|q| q[d]).collect();
+        let free: Vec<f64> = qs.iter().filter(|q| !super::uv::degenerate_within(s, dom, **q, d, reach)).map(|q| q[d]).collect();
         let pick: Vec<f64> = if free.is_empty() { qs.iter().map(|q| q[d]).collect() } else { free };
         *slot = pick.iter().sum::<f64>() / pick.len().max(1) as f64;
     }
diff --git a/crates/kernel/src/tess/tangle.rs b/crates/kernel/src/tess/tangle.rs
index 68db1a9760..c53a982cef 100644
--- a/crates/kernel/src/tess/tangle.rs
+++ b/crates/kernel/src/tess/tangle.rs
@@ -33,10 +33,10 @@ impl Chord {
 /// another, each once and in order. On a closed surface the boundary crosses itself where any
 /// copies of two chords, whole periods apart, cross; chords ending on a pole, where the
 /// parameter along the pole is free, are left out. Adjacent chords of a loop share a node and
-/// never cross.
-pub(crate) fn crossing(s: &EvalSurface, loops: &[seam::Loop]) -> Vec<(usize, usize)> {
+/// never cross. `reach`: how near a pole a node is taken as one (see [`uv::degenerate_within`]).
+pub(crate) fn crossing(s: &EvalSurface, reach: f64, loops: &[seam::Loop]) -> Vec<(usize, usize)> {
     let dom = uv::domain(s);
-    let pole = |q: [f64; 2]| (0..2).any(|d| uv::degenerate(s, q, d));
+    let pole = |q: [f64; 2]| (0..2).any(|d| uv::degenerate_within(s, &dom, q, d, reach));
     let mut chords: Vec<Chord> = Vec::new();
     for (li, l) in loops.iter().enumerate() {
         let n = l.uv.len();
diff --git a/crates/kernel/src/tess/tests.rs b/crates/kernel/src/tess/tests.rs
index 5ee223aaa9..3efbef471d 100644
--- a/crates/kernel/src/tess/tests.rs
+++ b/crates/kernel/src/tess/tests.rs
@@ -99,7 +99,7 @@ fn a_box_is_a_regular_grid_on_every_face() {
 fn primitives_are_watertight_with_the_right_volume() {
     for tol in [0.05, 0.005] {
         let s = check("box", &ok(crate::box_solid(Vec3::ZERO, Vec3::new(10.0, 6.0, 4.0))), tol, 240.0);
-        assert_eq!(s, PlanStats { planar: 6, structured: 0, delaunay: 0, lenient: Vec::new() });
+        assert_eq!(s, PlanStats { planar: 6, ..PlanStats::default() });
         let s = check("cylinder", &ok(crate::cylinder(Vec3::new(1.0, 2.0, 3.0), Vec3::new(0.0, 1.0, 1.0), 2.5, 7.0)), tol, PI * 2.5 * 2.5 * 7.0);
         assert_eq!(s.delaunay, 0, "cylinder walls are structured");
         let s = check("torus", &ok(crate::torus(Vec3::ZERO, 6.0, 1.5)), tol, 2.0 * PI * PI * 6.0 * 1.5 * 1.5);
@@ -725,8 +725,8 @@ fn a_curved_face_crossing_itself_is_filled_on_its_surface() {
     let size = size::face_size(&s, &uv::domain(&s), &rules);
     let loops = std::slice::from_ref(&l);
     let mut stats = PlanStats::default();
-    assert!(plan_face(0, "spline", &s, loops, &rules, size, &mut stats, Lenient::No).is_err());
-    let Ok(fp) = plan_face(0, "spline", &s, loops, &rules, size, &mut stats, Lenient::Yes) else { panic!("a lenient fill") };
+    assert!(plan_face(0, "spline", &s, loops, &rules, size, 0.0, &mut stats, Lenient::No).is_err());
+    let Ok(fp) = plan_face(0, "spline", &s, loops, &rules, size, 0.0, &mut stats, Lenient::Yes) else { panic!("a lenient fill") };
     assert!(!fp.triangles.is_empty());
     assert!(fp.uv.len() > fp.fixed.len(), "{} points, {} on the boundary", fp.uv.len(), fp.fixed.len());
     // Each point inside the face on the surface where its parameters put it.
@@ -743,7 +743,7 @@ fn a_curved_face_crossing_itself_is_filled_on_its_surface() {
 fn a_face_filled_leniently_is_named_in_the_stats() {
     let (s, l, rules) = a_face_crossing_itself();
     let size = size::face_size(&s, &uv::domain(&s), &rules);
-    let Ok((fp, stats)) = plan_or_fill(7, "spline", &s, std::slice::from_ref(&l), &rules, size, None) else { panic!("filled leniently") };
+    let Ok((fp, stats)) = plan_or_fill(7, "spline", &s, std::slice::from_ref(&l), &rules, size, 0.0, None) else { panic!("filled leniently") };
     assert!(!fp.triangles.is_empty());
     assert_eq!((stats.planar, stats.structured, stats.delaunay), (0, 0, 0));
     let [lenient] = stats.lenient.as_slice() else { panic!("one lenient face: {:?}", stats.lenient) };
@@ -760,7 +760,7 @@ fn a_face_untangling_cannot_resolve_is_filled_leniently_and_named() {
     let (s, l, rules) = a_face_crossing_itself();
     let size = size::face_size(&s, &uv::domain(&s), &rules);
     let why = "face 7 (spline): boundary crosses itself: edges [1, 2] run closer together than their chords can follow";
-    let Ok((fp, stats)) = plan_or_fill(7, "spline", &s, std::slice::from_ref(&l), &rules, size, Some(why)) else { panic!("filled leniently") };
+    let Ok((fp, stats)) = plan_or_fill(7, "spline", &s, std::slice::from_ref(&l), &rules, size, 0.0, Some(why)) else { panic!("filled leniently") };
     assert!(!fp.triangles.is_empty());
     let [lenient] = stats.lenient.as_slice() else { panic!("one lenient face: {:?}", stats.lenient) };
     assert_eq!(lenient.face, 7);
@@ -807,3 +807,59 @@ fn a_face_bounded_by_two_short_arcs_keeps_an_area_to_mesh() {
     let lens = made.faces.first().and_then(|&f| plan.faces.get(f)).expect("the lens's plan");
     assert!(!lens.triangles.is_empty(), "the lens has triangles");
 }
+
+/// Half of a square pyramid's sides as one spline face: degree 1, closed round the square
+/// (u in [0, 4], a corner at each whole u) and from the base (v = 0) up to a top side collapsed
+/// to a square 1e-7 mm across, not to a point, as the patch's side in Poppy Humanoid's connector
+/// a quarter of a sphere cap's corner sits on. Its loop runs down the side at u = 0, along the
+/// base to u = 2 and up again, then across the top back to the corner it started at, half a
+/// period along. Taken for no pole, that step is taken as the nearest copy, and the loop winds
+/// once round the surface with no partner to cut it along.
+fn pyramid_past_a_near_pole(pole_tolerance: f64) -> solidcraft_geom::brep::Brep {
+    use solidcraft_geom::brep::{self as b, Curve, Curve2, Surface};
+    let top = 1e-7;
+    let corner = |u: usize, r: f64, z: f64| {
+        let [x, y] = [[1.0, 0.0], [0.0, 1.0], [-1.0, 0.0], [0.0, -1.0]][u % 4];
+        [r * x, r * y, z]
+    };
+    let points: Vec<Vec<[f64; 3]>> = (0..5).map(|u| vec![corner(u, 1.0, 0.0), corner(u, top, 1.0)]).collect();
+    let surface = Surface::Spline { u_knots: vec![0.0, 0.0, 1.0, 2.0, 3.0, 4.0, 4.0], v_knots: vec![0.0, 0.0, 1.0, 1.0], points };
+    let edge = |curve: Curve, interval: [f64; 2], vertices: [u32; 2]| b::Edge { nurbs: None, curve, interval, vertices, tolerance: 1e-6, uses: Vec::new() };
+    let edges = vec![
+        edge(Curve::Line { start: corner(0, 1.0, 0.0), end: corner(0, top, 1.0) }, [0.0, 1.0], [1, 0]),
+        edge(Curve::Polyline { points: vec![corner(0, 1.0, 0.0), corner(1, 1.0, 0.0), corner(2, 1.0, 0.0)] }, [0.0, 2.0], [1, 2]),
+        edge(Curve::Line { start: corner(2, 1.0, 0.0), end: corner(2, top, 1.0) }, [0.0, 1.0], [2, 0]),
+    ];
+    let line = |t: [f64; 2], a: [f64; 2], z: [f64; 2]| Curve2::Spline { knots: vec![t[0], t[0], t[1], t[1]], points: vec![a, z] };
+    let uses = vec![
+        b::OrientedEdge { edge: 0, forward: false, pcurve: line([0.0, 1.0], [0.0, 0.0], [0.0, 1.0]) },
+        b::OrientedEdge { edge: 1, forward: true, pcurve: line([0.0, 2.0], [0.0, 0.0], [2.0, 0.0]) },
+        b::OrientedEdge { edge: 2, forward: true, pcurve: line([0.0, 1.0], [2.0, 0.0], [2.0, 1.0]) },
+    ];
+    let face = b::Face { nurbs: None, surface, domain: [[0.0, 2.0], [0.0, 1.0]], loops: vec![b::Loop::Edges { edges: uses }] };
+    let vertex = |point: [f64; 3], tolerance: f64| b::Vertex { point, tolerance };
+    let loose = b::Brep {
+        version: b::VERSION,
+        tolerance: 1e-7,
+        regions: vec![b::Region { shells: vec![0], void: false }],
+        shells: vec![b::Shell::Faces { faces: vec![b::OrientedFace { face: 0, forward: true }], wires: Vec::new() }],
+        faces: vec![face],
+        edges,
+        vertices: vec![vertex([0.0, 0.0, 1.0], pole_tolerance), vertex(corner(0, 1.0, 0.0), 1e-7), vertex(corner(2, 1.0, 0.0), 1e-7)],
+    };
+    ok(loose.canonical().map_err(|e| crate::KernelError::Invalid(format!("{e:?}")))).brep
+}
+
+/// The corner of [`pyramid_past_a_near_pole`] within its vertex's tolerance of the collapsed top
+/// side is a pole, and the face plans strictly over both its sides; with a tolerance short of
+/// the top's size it is no pole, and the loop winds.
+#[test]
+fn a_loop_corner_within_its_tolerance_of_a_pole_is_a_pole() {
+    let (made, stats) = ok(plan(&pyramid_past_a_near_pole(1e-6), 0.01, None));
+    assert!(stats.lenient.is_empty() && stats.unmeshed.is_empty(), "{stats:?}");
+    // Two of the pyramid's sides, each a triangle of area √3 / 2.
+    let area = made.evaluate().measure().area;
+    assert!((area - 3f64.sqrt()).abs() < 0.01 * 3f64.sqrt(), "area {area}");
+    let Err(e) = plan(&pyramid_past_a_near_pole(1e-8), 0.01, None) else { panic!("no pole without the tolerance") };
+    assert!(e.to_string().contains("winds around the surface"), "{e}");
+}
diff --git a/crates/kernel/src/tess/uv.rs b/crates/kernel/src/tess/uv.rs
index b4cb790ec9..7ee1b0f237 100644
--- a/crates/kernel/src/tess/uv.rs
+++ b/crates/kernel/src/tess/uv.rs
@@ -3,7 +3,7 @@
 //! face's parameter box.
 
 use solidcraft_geom::tess::EvalSurface;
-pub(crate) use solidcraft_geom::tess::uv::{Domain, degenerate, domain, unwrap};
+pub(crate) use solidcraft_geom::tess::uv::{Domain, degenerate_within, domain, unwrap};
 
 use super::size::Rules;