diff --git a/crates/kernel/src/body.rs b/crates/kernel/src/body.rs
index e6e771c9..b089e58e 100644
--- a/crates/kernel/src/body.rs
+++ b/crates/kernel/src/body.rs
@@ -255,8 +255,20 @@ impl Body {
             return Err(KernelError::Failed("empty result".into()));
         }
         let mut solid = rephase_revolved(solid);
-        // Orient outward: a closed solid must have positive volume.
-        let v = guard("orient", || Ok(mesh_shells(&solid, 1.0).iter().map(|s| s.to_polygon().volume()).sum::<f64>()))?;
+        // Orient outward: a closed solid must have positive volume (meshed finely enough that
+        // every face comes out: a face missing from a coarse mesh can flip the sign).
+        let span = {
+            let mut lo = Vec3::new(f64::INFINITY, f64::INFINITY, f64::INFINITY);
+            let mut hi = -lo;
+            for v in solid.vertex_iter() {
+                let p = from_p3(v.point());
+                lo = Vec3::new(lo.x.min(p.x), lo.y.min(p.y), lo.z.min(p.z));
+                hi = Vec3::new(hi.x.max(p.x), hi.y.max(p.y), hi.z.max(p.z));
+            }
+            if lo.x.is_finite() { (hi - lo).len() } else { 1.0 }
+        };
+        let tol = (span * 0.01).clamp(1e-3, 1.0);
+        let v = guard("orient", || Ok(mesh_shells(&solid, tol).iter().map(|s| s.to_polygon().volume()).sum::<f64>()))?;
         if v < 0.0 {
             solid.not();
         }
diff --git a/crates/kernel/src/ops.rs b/crates/kernel/src/ops.rs
index 97684525..3e34a144 100644
--- a/crates/kernel/src/ops.rs
+++ b/crates/kernel/src/ops.rs
@@ -4,7 +4,7 @@ use serde::{Deserialize, Serialize};
 use solvecraft_geom::Vec3;
 use truck_modeling as mt;
 
-use crate::body::{Body, p3, v3};
+use crate::body::{Body, from_p3, p3, v3};
 use crate::{KernelError, Result, guard};
 
 #[derive(Clone, Copy, Debug, PartialEq, Eq, Serialize, Deserialize)]
@@ -110,10 +110,46 @@ fn trivial(a: &Body, b: &Body, op: BoolOp, ma: &solvecraft_geom::Mesh, mb: &solv
         BoolOp::Intersect if a_in_b > ALL => Some(Some(a.clone())),
         BoolOp::Union if b_in_a > ALL => Some(Some(a.clone())),
         BoolOp::Union if a_in_b > ALL => Some(Some(b.clone())),
+        // The tool wholly inside, clear of the outside: a void.
+        BoolOp::Cut if b_in_a > ALL && clear_inside(ma, mb, a.size().max(b.size()) * 1e-6) => void(a, b).ok(),
         _ => None,
     }
 }
 
+/// Does every vertex of `inner` lie inside `outer` and farther than `gap` from its surface?
+fn clear_inside(outer: &solvecraft_geom::Mesh, inner: &solvecraft_geom::Mesh, gap: f64) -> bool {
+    if outer.triangles.len() * inner.positions.len() > 40_000_000 {
+        return false;
+    }
+    let tris: Vec<[Vec3; 3]> = outer.triangles.iter().filter_map(|t| outer.tri(t)).collect();
+    inner.positions.iter().all(|p| {
+        outer.contains(*p)
+            && tris.iter().all(|[a, b, c]| {
+                // Distance to the triangle's plane bounds the distance to the triangle from below.
+                let n = (*b - *a).cross(*c - *a);
+                let Some(nn) = n.normalized() else { return true };
+                let h = (*p - *a).dot(nn).abs();
+                h > gap || {
+                    let q = *p - nn * (*p - *a).dot(nn);
+                    let inside = [(*a, *b), (*b, *c), (*c, *a)].iter().all(|(u, v)| (*v - *u).cross(q - *u).dot(n) >= 0.0);
+                    !inside && [(*a, *b), (*b, *c), (*c, *a)].iter().all(|(u, v)| p.dist_to_segment(*u, *v) > gap)
+                }
+            })
+    })
+}
+
+/// `a` with the inside of `b` (wholly within it) taken out: a second, inward-facing shell.
+fn void(a: &Body, b: &Body) -> Result<Option<Body>> {
+    guard("void", || {
+        let mut inner = b.deep_copy();
+        inner.not();
+        let mut shells = a.solid.boundaries().clone();
+        shells.extend(inner.boundaries().iter().cloned());
+        let s = crate::body::Solid::try_new(shells).map_err(|e| KernelError::Failed(format!("void: {e}")))?;
+        Ok(Some(Body::new(s)?))
+    })
+}
+
 thread_local! {
     /// Set while a boolean runs on bodies whose coincident faces were pushed apart (no
     /// second round of pushing).
@@ -267,6 +303,26 @@ pub fn boolean(a: &Body, b: &Body, op: BoolOp) -> Result<Option<Body>> {
         Ok(None) => return Ok(None),
         Err(e) => last = format!("{last}; {e}"),
     }
+    // Last resort: nudge the tool by a micron-scale screw motion (turned 2e-6 rad about a
+    // skew axis and moved 2e-6 of the size), which breaks tangencies and seams lying exactly on
+    // the other body. The result moves by no more than that.
+    if !APART.with(|c| c.get()) {
+        APART.with(|c| c.set(true));
+        let c = b.solid.vertex_iter().fold(Vec3::ZERO, |acc, v| acc + from_p3(v.point())) * (1.0 / b.solid.vertex_iter().count().max(1) as f64);
+        let r = transform(b, Vec3::new(0.61, -0.37, 0.71) * (size * 2e-6), c, Vec3::new(0.27, 0.83, -0.49), 2e-6).and_then(|nb| boolean(a, &nb, op));
+        APART.with(|c| c.set(false));
+        match r {
+            Ok(Some(body)) => {
+                let v = volume(&body);
+                if v > 0.0 && plausible(v, &body) {
+                    return Ok(Some(body));
+                }
+                last = format!("{last}; the nudged tool gave an implausible volume {v:.4}");
+            }
+            Ok(None) => return Ok(None),
+            Err(e) => last = format!("{last}; nudged: {e}"),
+        }
+    }
     Err(KernelError::Failed(format!("boolean {op:?}: {last}")))
 }
 
diff --git a/crates/kernel/src/tests.rs b/crates/kernel/src/tests.rs
index 9b98ce61..a4e9ae59 100644
--- a/crates/kernel/src/tests.rs
+++ b/crates/kernel/src/tests.rs
@@ -1102,3 +1102,76 @@ fn fillet_an_open_curved_edge() {
     let want = r * r * (1.0 - PI / 4.0) * (10.0 - r * (10.0 - 3.0 * PI) / (3.0 * (4.0 - PI))) * PI;
     assert!(rel(removed, want) < 1e-3, "{removed} vs {want}");
 }
+
+/// Boolean survey: random boxes, cylinders and spheres combined every way; prints what fails.
+#[test]
+#[ignore]
+fn boolean_survey() {
+    let mut seed: u64 = 0x2545_f491_4f6c_dd1d;
+    let mut rnd = || {
+        seed ^= seed << 13;
+        seed ^= seed >> 7;
+        seed ^= seed << 17;
+        (seed % 1_000_000) as f64 / 1_000_000.0
+    };
+    let mut shapes: Vec<(String, Body)> = Vec::new();
+    for i in 0..40 {
+        let kind = i % 4;
+        let (x, y, z) = ((rnd() - 0.5) * 20.0, (rnd() - 0.5) * 20.0, (rnd() - 0.5) * 20.0);
+        // Snap some coordinates to a 5 mm grid: coincident faces and tangencies.
+        let snap = |v: f64, s: bool| if s { (v / 5.0).round() * 5.0 } else { v };
+        let s = rnd() < 0.5;
+        let (x, y, z) = (snap(x, s), snap(y, s), snap(z, s));
+        let a = 5.0 + snap(rnd() * 20.0, s);
+        let ax = (rnd() * 3.0) as usize % 3;
+        let b = match kind {
+            0 => box_solid(Vec3::new(x, y, z), Vec3::new(x + a, y + a * 0.8, z + a * 0.6)),
+            1 => cylinder(Vec3::new(x, y, z), [Vec3::X, Vec3::Y, Vec3::Z][ax], a * 0.4, a),
+            2 => sphere(Vec3::new(x, y, z), a * 0.5),
+            _ => box_solid(Vec3::new(x, y, z), Vec3::new(x + a * 0.5, y + a, z + a * 0.3)),
+        };
+        if let Ok(b) = b {
+            shapes.push((format!("{}{i}@({x:.3},{y:.3},{z:.3}) a={a:.3} ax={ax}", ["box", "cyl", "sph", "slab"][kind]), b));
+        }
+    }
+    let (mut ok, mut empty, mut fail) = (0, 0, Vec::new());
+    for i in 0..shapes.len() {
+        for j in i + 1..(i + 4).min(shapes.len()) {
+            for op in [BoolOp::Union, BoolOp::Cut, BoolOp::Intersect] {
+                let (na, a) = &shapes[i];
+                let (nb, b) = &shapes[j];
+                match boolean(a, b, op) {
+                    Ok(Some(_)) => ok += 1,
+                    Ok(None) => empty += 1,
+                    Err(e) => fail.push(format!("{na} {op:?} {nb}: {}", e.to_string().chars().take(140).collect::<String>())),
+                }
+            }
+        }
+    }
+    println!("ok {ok} empty {empty} failed {}", fail.len());
+    for f in &fail {
+        println!("  {f}");
+    }
+}
+
+#[test]
+fn booleans_found_by_the_survey() {
+    // A tool wholly inside: a void (the solid keeps an inner shell).
+    let sp = sphere(Vec3::new(-2.894, 7.008, 8.081), 8.869).unwrap();
+    let slab = box_solid(Vec3::new(-5.0, 0.0, 5.0), Vec3::new(-2.5, 5.0, 6.5)).unwrap();
+    let c = boolean(&sp, &slab, BoolOp::Cut).unwrap().unwrap();
+    let want = 4.0 / 3.0 * PI * 8.869f64.powi(3) - 18.75;
+    assert!(rel(measure(&c).unwrap().volume, want) < 1e-3, "{}", measure(&c).unwrap().volume);
+    // A sphere clipping a box's corner: the result comes out outward-facing.
+    let sp = sphere(Vec3::new(-8.226, 9.539, -1.251), 11.8435).unwrap();
+    let bx = box_solid(Vec3::new(0.0, -5.0, -5.0), Vec3::new(15.0, 7.0, 4.0)).unwrap();
+    let c = boolean(&bx, &sp, BoolOp::Cut).unwrap().unwrap();
+    let v = measure(&c).unwrap().volume;
+    assert!(v > 1400.0 && v < 1620.0, "{v}");
+    // A cylinder and a box sharing an end plane, the box's side near-tangent to the cylinder.
+    let cy = cylinder(Vec3::new(-5.0, 0.0, 0.0), Vec3::X, 8.0, 20.0).unwrap();
+    let bx = box_solid(Vec3::new(0.0, -5.0, -5.0), Vec3::new(15.0, 7.0, 4.0)).unwrap();
+    let u = boolean(&cy, &bx, BoolOp::Union).unwrap().unwrap();
+    let vu = measure(&u).unwrap().volume;
+    assert!(vu > PI * 64.0 * 20.0 && vu < PI * 64.0 * 20.0 + 1620.0, "{vu}");
+}