diff --git a/OpenNest.Core/Diagnostics/OverlapClipping.cs b/OpenNest.Core/Diagnostics/OverlapClipping.cs
new file mode 100644
index 00000000..acc10f2f
--- /dev/null
+++ b/OpenNest.Core/Diagnostics/OverlapClipping.cs
@@ -0,0 +1,124 @@
+using System;
+using System.Collections.Generic;
+using Clipper2Lib;
+using OpenNest.Geometry;
+
+namespace OpenNest.Diagnostics;
+
+/// <summary>
+/// Shared material of two hole-aware outlines as connected islands, through Clipper2.
+/// Callers pass part-local coordinates (rebased near the origin) and add the origin back.
+/// <para>
+/// Coordinates are rounded to a 1e-8 model-unit grid, far below the 0.001 chord tolerance
+/// and the <see cref="OpenNest.Math.Tolerance.Epsilon"/> area threshold. Each operand is
+/// its outline minus the union of its holes (outline positive, holes negative,
+/// <see cref="FillRule.Positive"/>).
+/// Nonfinite or out-of-range coordinates throw <see cref="ArithmeticException"/>, which the
+/// analyzer turns into an incomplete-check issue rather than a clear result.
+/// </para>
+/// </summary>
+internal static class OverlapClipping
+{
+    private const double Scale = 1e8;
+
+    // Scaled coordinates stay several orders of magnitude inside Clipper2's int64 range.
+    private const double MaxCoordinate = 1e9;
+
+    /// <summary>One connected island: a counterclockwise outline and clockwise holes, both closed.</summary>
+    internal sealed record Island(List<Vector> Outline, List<List<Vector>> Holes, PolygonAreaMoments Moments);
+
+    internal static List<Island> Intersect(Polygon outerA, IReadOnlyList<Polygon> holesA,
+        Polygon outerB, IReadOnlyList<Polygon> holesB)
+    {
+        var clipper = new Clipper64();
+        clipper.AddSubject(Region(outerA, holesA));
+        clipper.AddClip(Region(outerB, holesB));
+        var tree = new PolyTree64();
+        if (!clipper.Execute(ClipType.Intersection, FillRule.Positive, tree))
+            throw new InvalidOperationException("Clipper2 could not intersect the overlap material.");
+        var islands = new List<Island>();
+        Collect(tree, islands);
+        return islands;
+    }
+
+    private static void Collect(PolyPath64 node, List<Island> islands)
+    {
+        for (var i = 0; i < node.Count; i++)
+        {
+            var outer = (PolyPath64)node[i];
+            var outline = ToVectors(outer.Polygon, positive: true);
+            var holes = new List<List<Vector>>();
+            for (var h = 0; h < outer.Count; h++)
+            {
+                var hole = (PolyPath64)outer[h];
+                holes.Add(ToVectors(hole.Polygon, positive: false));
+                // An island can sit inside a hole formed by both operands' holes together.
+                Collect(hole, islands);
+            }
+            if (TryMoments(outline, holes, out var moments))
+                islands.Add(new Island(outline, holes, moments));
+            else if (Clipper.Area(outer.Polygon) / (Scale * Scale) >= OpenNest.Math.Tolerance.Epsilon)
+                throw new ArithmeticException("Overlap island area moments are invalid.");
+            // Otherwise a degenerate dust island below the reporting threshold is skipped.
+        }
+    }
+
+    /// <summary>
+    /// Net area and centroid of an outline minus its holes. Fails for degenerate,
+    /// nonfinite or nonpositive results.
+    /// </summary>
+    private static bool TryMoments(List<Vector> outline, List<List<Vector>> holes, out PolygonAreaMoments moments)
+    {
+        moments = default;
+        if (!PolygonAreaMoments.TryCompute(outline, out var outer))
+            return false;
+        var removed = new List<PolygonAreaMoments>(holes.Count);
+        foreach (var hole in holes)
+        {
+            if (!PolygonAreaMoments.TryCompute(hole, out var hollow))
+                return false;
+            removed.Add(hollow);
+        }
+        return PolygonAreaMoments.TrySubtract(outer, removed, out moments);
+    }
+
+    private static Paths64 Region(Polygon outer, IReadOnlyList<Polygon> holes)
+    {
+        var region = new Paths64 { ToPath(outer, positive: true) };
+        if (holes != null)
+            foreach (var hole in holes)
+                region.Add(ToPath(hole, positive: false));
+        return region;
+    }
+
+    private static Path64 ToPath(Polygon polygon, bool positive)
+    {
+        var vertices = polygon.Vertices;
+        var count = vertices.Count;
+        if (count > 1 && vertices[count - 1].X == vertices[0].X && vertices[count - 1].Y == vertices[0].Y)
+            count--;
+        var path = new Path64(count);
+        for (var i = 0; i < count; i++)
+        {
+            var point = vertices[i];
+            if (!double.IsFinite(point.X) || !double.IsFinite(point.Y)
+                || System.Math.Abs(point.X) > MaxCoordinate || System.Math.Abs(point.Y) > MaxCoordinate)
+                throw new ArithmeticException("Overlap material is outside the exact clipping range.");
+            path.Add(new Point64(System.Math.Round(point.X * Scale), System.Math.Round(point.Y * Scale)));
+        }
+        if (Clipper.IsPositive(path) != positive)
+            path.Reverse();
+        return path;
+    }
+
+    private static List<Vector> ToVectors(Path64 path, bool positive)
+    {
+        var points = new List<Vector>(path.Count + 1);
+        foreach (var point in path)
+            points.Add(new Vector(point.X / Scale, point.Y / Scale));
+        if (Clipper.IsPositive(path) != positive)
+            points.Reverse();
+        points.Add(points[0]); // Regions repeat their closing vertex exactly.
+        return points;
+    }
+}
diff --git a/OpenNest.Core/Diagnostics/PlateOverlapAnalyzer.cs b/OpenNest.Core/Diagnostics/PlateOverlapAnalyzer.cs
index 72c8af06..43487a64 100644
--- a/OpenNest.Core/Diagnostics/PlateOverlapAnalyzer.cs
+++ b/OpenNest.Core/Diagnostics/PlateOverlapAnalyzer.cs
@@ -140,7 +140,7 @@ public static class PlateOverlapAnalyzer
         Analyze(Capture(parts, cancellationToken), cancellationToken);
 
     /// <summary>
-    /// Returns deterministic pair reports containing closed world-coordinate overlap fragments.
+    /// Returns deterministic pair reports containing closed world-coordinate overlap islands.
     /// Cancellation throws and publishes no partial report. Check IsComplete before claiming clear.
     /// </summary>
     public static PlateOverlapReport Analyze(PlateOverlapSnapshot snapshot,
@@ -205,32 +205,13 @@ public static class PlateOverlapAnalyzer
                 try
                 {
                     // Keep pair clipping arithmetic near the parts where possible, then
-                    // restore output to world space. Triangulation itself also uses stable
-                    // local-origin winding so tiny holes in a huge part remain correct.
+                    // restore output to world space.
                     var origin = a.Material.Outer.Vertices[0];
                     var localA = a.Material.Transform(0, origin * -1);
                     var localB = b.Material.Transform(0, origin * -1);
-                    var result = Collision.Check(localA.Outer, localB.Outer, localA.Holes, localB.Holes);
-                    if (!result.Overlaps)
+                    if (!TryShare(localA, localB.Outer, localB.Holes, origin, cancellationToken,
+                            out var regions, out var centroid))
                         continue;
-                    // Evaluate every hole-subtracted fragment before restoring world space.
-                    // A failed moment must make this pair incomplete, never an origin marker.
-                    var moments = new List<PolygonAreaMoments>();
-                    var regions = new List<PlateOverlapRegion>();
-                    foreach (var region in result.OverlapRegions)
-                    {
-                        cancellationToken.ThrowIfCancellationRequested();
-                        if (!PolygonAreaMoments.TryCompute(region.Vertices, out var fragment))
-                            throw new ArithmeticException("Overlap fragment area moments are invalid.");
-                        moments.Add(fragment);
-                        regions.Add(new PlateOverlapRegion(region.Vertices.Select(point => point + origin),
-                            fragment.Area));
-                    }
-                    if (!PolygonAreaMoments.TryCombine(moments, out var combined))
-                        throw new ArithmeticException("Combined overlap area moments are invalid.");
-                    var centroid = combined.Centroid + origin;
-                    if (!OverlapMaterial.IsFinite(centroid))
-                        throw new ArithmeticException("Overlap centroid is not finite in world coordinates.");
                     pairs.Add(new PlateOverlapPair(a.Input.Id, b.Input.Id,
                         a.Input.Name, b.Input.Name, regions, centroid));
                 }
@@ -299,25 +280,8 @@ public static class PlateOverlapAnalyzer
                     var origin = part.Material.Outer.Vertices[0];
                     var local = part.Material.Transform(0, origin * -1);
                     var rect = RectanglePolygon(zone, origin * -1);
-                    var result = Collision.Check(local.Outer, rect, local.Holes, null);
-                    if (!result.Overlaps)
+                    if (!TryShare(local, rect, null, origin, cancellationToken, out var regions, out var centroid))
                         continue;
-                    var moments = new List<PolygonAreaMoments>();
-                    var regions = new List<PlateOverlapRegion>();
-                    foreach (var region in result.OverlapRegions)
-                    {
-                        cancellationToken.ThrowIfCancellationRequested();
-                        if (!PolygonAreaMoments.TryCompute(region.Vertices, out var fragment))
-                            throw new ArithmeticException("Clamp overlap fragment area moments are invalid.");
-                        moments.Add(fragment);
-                        regions.Add(new PlateOverlapRegion(
-                            region.Vertices.Select(point => point + origin), fragment.Area));
-                    }
-                    if (!PolygonAreaMoments.TryCombine(moments, out var combined))
-                        throw new ArithmeticException("Combined clamp overlap area moments are invalid.");
-                    var centroid = combined.Centroid + origin;
-                    if (!OverlapMaterial.IsFinite(centroid))
-                        throw new ArithmeticException("Clamp overlap centroid is not finite in world coordinates.");
                     findings.Add(new PlateOverlapPair(part.Input.Id,
                         ClampFindingSlot.IdFor(clamp.Ordinal), part.Input.Name,
                         $"Clamp '{clamp.Id}'", regions, centroid));
@@ -332,6 +296,39 @@ public static class PlateOverlapAnalyzer
         return findings;
     }
 
+    /// <summary>
+    /// Shared material of a rebased operand with another rebased outline/holes, as world
+    /// regions (connected islands) plus the area centroid of them all. Returns false when the
+    /// total shared area does not exceed <see cref="OpenNest.Math.Tolerance.Epsilon"/>: contact and rounding
+    /// dust are not overlap. Invalid moments throw, so the pair becomes incomplete rather
+    /// than clear or marked at the origin.
+    /// </summary>
+    private static bool TryShare(OverlapMaterial localA, Polygon outerB, List<Polygon> holesB, Vector origin,
+        CancellationToken cancellationToken, out List<PlateOverlapRegion> regions, out Vector centroid)
+    {
+        regions = null;
+        centroid = default;
+        var islands = OverlapClipping.Intersect(localA.Outer, localA.Holes, outerB, holesB);
+        var area = islands.Sum(island => island.Moments.Area);
+        if (!double.IsFinite(area))
+            throw new ArithmeticException("Overlap area moments are invalid.");
+        if (area <= OpenNest.Math.Tolerance.Epsilon)
+            return false;
+        regions = new List<PlateOverlapRegion>(islands.Count);
+        foreach (var island in islands)
+        {
+            cancellationToken.ThrowIfCancellationRequested();
+            regions.Add(new PlateOverlapRegion(island.Outline.Select(point => point + origin), island.Moments.Area,
+                island.Holes.Select(hole => hole.Select(point => point + origin))));
+        }
+        if (!PolygonAreaMoments.TryCombine(islands.Select(island => island.Moments), out var combined))
+            throw new ArithmeticException("Combined overlap area moments are invalid.");
+        centroid = combined.Centroid + origin;
+        if (!OverlapMaterial.IsFinite(centroid))
+            throw new ArithmeticException("Overlap centroid is not finite in world coordinates.");
+        return true;
+    }
+
     private static PlateOverlapPair TinyClampFinding(PreparedPart part, CapturedClamp clamp)
     {
         var zone = clamp.Bounds;
diff --git a/OpenNest.Core/Diagnostics/PlateOverlapReport.cs b/OpenNest.Core/Diagnostics/PlateOverlapReport.cs
index a5fd5b47..136c0764 100644
--- a/OpenNest.Core/Diagnostics/PlateOverlapReport.cs
+++ b/OpenNest.Core/Diagnostics/PlateOverlapReport.cs
@@ -80,7 +80,7 @@ public sealed class PlateOverlapPair
     public int PartBId { get; }
     public string PartAName { get; }
     public string PartBName { get; }
-    /// <summary>Convex fragments, not connected islands; no mutable kernel polygons are exposed.</summary>
+    /// <summary>Connected islands of shared material; no mutable kernel polygons are exposed.</summary>
     public IReadOnlyList<PlateOverlapRegion> Regions { get; }
     public double Area { get; }
     /// <summary>
@@ -92,17 +92,34 @@ public sealed class PlateOverlapPair
     public Box Bounds => new(bounds.X, bounds.Y, bounds.Length, bounds.Width);
 }
 
-/// <summary>A positive-area, hole-subtracted convex polygon in world coordinates.</summary>
+/// <summary>
+/// One connected island of shared material in world coordinates, after hole subtraction:
+/// a counterclockwise outline and its clockwise holes.
+/// </summary>
 public sealed class PlateOverlapRegion
 {
-    internal PlateOverlapRegion(IEnumerable<Vector> vertices, double area)
+    private static readonly IReadOnlyList<IReadOnlyList<Vector>> NoHoles = System.Array.AsReadOnly(
+        System.Array.Empty<IReadOnlyList<Vector>>());
+
+    internal PlateOverlapRegion(IEnumerable<Vector> vertices, double area,
+        IEnumerable<IEnumerable<Vector>> holes = null)
     {
         Vertices = System.Array.AsReadOnly(vertices.ToArray());
         Area = area;
+        Holes = holes == null
+            ? NoHoles
+            : System.Array.AsReadOnly(holes
+                .Select(hole => (IReadOnlyList<Vector>)System.Array.AsReadOnly(hole.ToArray()))
+                .ToArray());
     }
 
-    /// <summary>Read-only vertices with an exactly repeated closing vertex.</summary>
+    /// <summary>Read-only counterclockwise outline with an exactly repeated closing vertex.</summary>
     public IReadOnlyList<Vector> Vertices { get; }
+
+    /// <summary>Read-only clockwise hole rings, each with an exactly repeated closing vertex.</summary>
+    public IReadOnlyList<IReadOnlyList<Vector>> Holes { get; }
+
+    /// <summary>Positive net area: the outline's area minus its holes'.</summary>
     public double Area { get; }
 }
 
diff --git a/OpenNest.Core/Geometry/PolygonAreaMoments.cs b/OpenNest.Core/Geometry/PolygonAreaMoments.cs
index 4bb78daa..698ea023 100644
--- a/OpenNest.Core/Geometry/PolygonAreaMoments.cs
+++ b/OpenNest.Core/Geometry/PolygonAreaMoments.cs
@@ -97,5 +97,35 @@ public readonly struct PolygonAreaMoments
         return true;
     }
 
+    /// <summary>
+    /// Net moments of an outline minus nonoverlapping holes that lie inside it, all given as
+    /// positive moments (for example from <see cref="TryCompute"/>). Moments are taken about
+    /// the outline's centroid. Fails for an invalid input or a nonpositive or nonfinite result.
+    /// </summary>
+    public static bool TrySubtract(PolygonAreaMoments outline, IEnumerable<PolygonAreaMoments> holes,
+        out PolygonAreaMoments moments)
+    {
+        moments = default;
+        if (holes == null || !double.IsFinite(outline.Area) || outline.Area <= 0 || !IsFinite(outline.Centroid))
+            return false;
+        var area = outline.Area;
+        var firstMoment = Vector.Zero;
+        foreach (var hole in holes)
+        {
+            if (!double.IsFinite(hole.Area) || hole.Area <= 0 || !IsFinite(hole.Centroid))
+                return false;
+            firstMoment -= (hole.Centroid - outline.Centroid) * hole.Area;
+            area -= hole.Area;
+        }
+
+        if (!double.IsFinite(area) || area <= 0 || !IsFinite(firstMoment))
+            return false;
+        var centroid = outline.Centroid + firstMoment / area;
+        if (!IsFinite(centroid))
+            return false;
+        moments = new PolygonAreaMoments(area, centroid);
+        return true;
+    }
+
     private static bool IsFinite(Vector point) => double.IsFinite(point.X) && double.IsFinite(point.Y);
 }
diff --git a/OpenNest.Tests/Diagnostics/OverlapPairPresentationTests.cs b/OpenNest.Tests/Diagnostics/OverlapPairPresentationTests.cs
index 1d2c3292..f0bb185a 100644
--- a/OpenNest.Tests/Diagnostics/OverlapPairPresentationTests.cs
+++ b/OpenNest.Tests/Diagnostics/OverlapPairPresentationTests.cs
@@ -62,7 +62,7 @@ public class OverlapPairPresentationTests
         var pairs = PlateOverlapAnalyzer.Analyze(new[] { Rectangle("a"), Rectangle("b"), Rectangle("c") }).Pairs;
         var hits = OverlapPairPresentation.HitTest(pairs.Reverse(), p => p, new Vector(2, 2), 96);
         Assert.Equal(new[] { "1/2", "1/3", "2/3" }, hits.Select(OverlapPairPresentation.Label));
-        Assert.All(hits, p => Assert.True(p.Regions.Count > 1));
+        Assert.All(hits, p => Assert.Single(p.Regions)); // One island per coincident pair.
     }
 
     private static Part Rectangle(string name)
diff --git a/OpenNest.Tests/Diagnostics/PlateOverlapAnalyzerTests.cs b/OpenNest.Tests/Diagnostics/PlateOverlapAnalyzerTests.cs
index 390b02bf..ceca6dc0 100644
--- a/OpenNest.Tests/Diagnostics/PlateOverlapAnalyzerTests.cs
+++ b/OpenNest.Tests/Diagnostics/PlateOverlapAnalyzerTests.cs
@@ -487,7 +487,8 @@ public class PlateOverlapAnalyzerTests
         Assert.False(report.IsComplete);
         var issue = Assert.Single(report.Issues);
         Assert.Equal((0, (int?)1), (issue.PartAId, issue.PartBId));
-        Assert.Contains("moment", issue.Message, StringComparison.OrdinalIgnoreCase);
+        // The exact clipping range refuses the pair before any moment could overflow.
+        Assert.Contains("range", issue.Message, StringComparison.OrdinalIgnoreCase);
         var valid = Assert.Single(report.Pairs);
         Assert.Equal((2, 3), (valid.PartAId, valid.PartBId));
         AssertCentroid(valid, -1.5, -1.5);
@@ -501,6 +502,88 @@ public class PlateOverlapAnalyzerTests
         Assert.Equal(y, pair.Centroid.Y, 7);
     }
 
+    // Regions are connected islands of shared material, not triangulation fragments: a
+    // C-shaped overlap (frame half beside its hole) is one region with the C's area.
+    [Fact]
+    public void Analyze_ConcaveOverlapIsOneConnectedRegion()
+    {
+        var frame = WithContours(Square(0, 0, 4), Square(1, 1, 2));
+
+        var report = PlateOverlapAnalyzer.Analyze(new[] { frame, Rectangle(2, -1, 4, 6) });
+
+        Assert.True(report.IsComplete, string.Join("; ", report.Issues));
+        var pair = Assert.Single(report.Pairs);
+        var region = Assert.Single(pair.Regions);
+        Assert.Equal(6, pair.Area, 8);
+        Assert.Equal(6, region.Area, 8);
+        Assert.Equal(region.Vertices[0], region.Vertices[^1]);
+    }
+
+    // A part covering a whole frame shares one island whose hole is the frame's hole.
+    [Fact]
+    public void Analyze_RegionKeepsTheHoleOfSharedMaterial()
+    {
+        var frame = WithContours(Square(0, 0, 4), Square(1, 1, 2));
+
+        var report = PlateOverlapAnalyzer.Analyze(new[] { Rectangle(-1, -1, 6, 6), frame });
+
+        Assert.True(report.IsComplete, string.Join("; ", report.Issues));
+        var pair = Assert.Single(report.Pairs);
+        var region = Assert.Single(pair.Regions);
+        Assert.Equal(12, pair.Area, 8);
+        AssertCentroid(pair, 2, 2);
+        var hole = Assert.Single(region.Holes);
+        Assert.Equal(hole[0], hole[^1]);
+        Assert.Equal(16, SignedArea(region.Vertices), 8); // Outline counterclockwise.
+        Assert.Equal(-4, SignedArea(hole), 8);             // Hole clockwise.
+        Assert.All(hole, point =>
+        {
+            Assert.InRange(point.X, 1, 3);
+            Assert.InRange(point.Y, 1, 3);
+        });
+        Assert.Throws<NotSupportedException>(() => ((IList<IReadOnlyList<Vector>>)region.Holes).Clear());
+        Assert.Throws<NotSupportedException>(() => ((IList<Vector>)hole)[0] = new Vector(9, 9));
+    }
+
+    private static double SignedArea(IReadOnlyList<Vector> ring)
+    {
+        var twice = 0.0;
+        for (var i = 0; i + 1 < ring.Count; i++)
+            twice += ring[i].X * ring[i + 1].Y - ring[i + 1].X * ring[i].Y;
+        return twice / 2;
+    }
+
+    // A's square hole closes the mouth of B's C-shaped hole, so the material enclosed by
+    // both holes is a separate island inside the shared hole and must not be dropped.
+    [Fact]
+    public void Analyze_IslandEnclosedByBothPartsHolesIsKept()
+    {
+        var a = WithContours(Square(0, 0, 10), new[]
+        {
+            new Vector(6.5, 3.5), new Vector(7.5, 3.5), new Vector(7.5, 6.5), new Vector(6.5, 6.5)
+        });
+        var b = WithContours(Square(0, 0, 10), new[]
+        {
+            new Vector(3, 3), new Vector(7, 3), new Vector(7, 4), new Vector(4, 4),
+            new Vector(4, 6), new Vector(7, 6), new Vector(7, 7), new Vector(3, 7)
+        });
+
+        var report = PlateOverlapAnalyzer.Analyze(new[] { a, b });
+
+        Assert.True(report.IsComplete, string.Join("; ", report.Issues));
+        var pair = Assert.Single(report.Pairs);
+        Assert.Equal(87.5, pair.Area, 8); // 100 - |H1 u H2| = 100 - (3 + 10 - 0.5).
+        Assert.Equal(new[] { 5.0, 82.5 }, pair.Regions.Select(r => System.Math.Round(r.Area, 8)).Order());
+        var island = pair.Regions.Single(r => r.Area < 10);
+        Assert.Empty(island.Holes);
+        Assert.All(island.Vertices, point =>
+        {
+            Assert.InRange(point.X, 4, 6.5);
+            Assert.InRange(point.Y, 4, 6);
+        });
+        Assert.Single(pair.Regions.Single(r => r.Area > 10).Holes);
+    }
+
     [Theory]
     [InlineData(Mode.Absolute)]
     [InlineData(Mode.Incremental)]
diff --git a/OpenNest.Tests/Geometry/PolygonAreaMomentsTests.cs b/OpenNest.Tests/Geometry/PolygonAreaMomentsTests.cs
index 5b59de61..16b62623 100644
--- a/OpenNest.Tests/Geometry/PolygonAreaMomentsTests.cs
+++ b/OpenNest.Tests/Geometry/PolygonAreaMomentsTests.cs
@@ -135,6 +135,36 @@ public class PolygonAreaMomentsTests
         Assert.False(PolygonAreaMoments.TryCombine(new[] { default(PolygonAreaMoments), valid }, out _));
     }
 
+    [Theory]
+    [InlineData(0)]
+    [InlineData(1000000000)]
+    [InlineData(-1000000000)]
+    public void TrySubtract_OffCentreHoleShiftsCentroidAway(double offset)
+    {
+        // 4x4 square minus a 1x1 hole centred at (1, 1): area 15, centroid (16*2 - 1)/15 = 2 + 1/15.
+        Assert.True(PolygonAreaMoments.TryCompute(Rectangle(offset, offset, 4, 4), out var outline));
+        Assert.True(PolygonAreaMoments.TryCompute(Rectangle(offset + 0.5, offset + 0.5, 1, 1), out var hole));
+
+        Assert.True(PolygonAreaMoments.TrySubtract(outline, new[] { hole }, out var net));
+
+        Assert.Equal(15, net.Area, 12);
+        Assert.Equal(offset + 2 + 1.0 / 15, net.Centroid.X, 7);
+        Assert.Equal(offset + 2 + 1.0 / 15, net.Centroid.Y, 7);
+        Assert.True(PolygonAreaMoments.TrySubtract(outline, Array.Empty<PolygonAreaMoments>(), out var whole));
+        Assert.Equal((outline.Area, outline.Centroid), (whole.Area, whole.Centroid));
+    }
+
+    [Fact]
+    public void TrySubtract_InvalidOrNonpositiveResultIsRejected()
+    {
+        Assert.True(PolygonAreaMoments.TryCompute(Rectangle(0, 0, 1, 1), out var unit));
+
+        Assert.False(PolygonAreaMoments.TrySubtract(unit, null, out _));
+        Assert.False(PolygonAreaMoments.TrySubtract(default, new[] { unit }, out _));
+        Assert.False(PolygonAreaMoments.TrySubtract(unit, new[] { default(PolygonAreaMoments) }, out _));
+        Assert.False(PolygonAreaMoments.TrySubtract(unit, new[] { unit }, out _));
+    }
+
     private static Vector[] Rectangle(double x, double y, double width, double height) =>
         new[] { new Vector(x, y), new Vector(x + width, y),
             new Vector(x + width, y + height), new Vector(x, y + height) };
diff --git a/OpenNest/Controls/OverlapOverlayController.cs b/OpenNest/Controls/OverlapOverlayController.cs
index d7e01be1..d46c7761 100644
--- a/OpenNest/Controls/OverlapOverlayController.cs
+++ b/OpenNest/Controls/OverlapOverlayController.cs
@@ -571,21 +571,11 @@ internal sealed class OverlapOverlayController : IDisposable
             {
                 foreach (var region in pair.Regions)
                 {
-                    var vertices = region.Vertices;
-                    var points = new PointF[vertices.Count - 1]; // Analyzer repeats the closing vertex.
-                    var signedArea = 0.0;
-                    var start = vertices[0];
-                    for (var i = 0; i < points.Length; i++)
-                    {
-                        var a = vertices[i] - start;
-                        var b = vertices[i + 1] - start;
-                        signedArea += a.X * b.Y - b.X * a.Y;
-                        // The host has already translated by origin: graph, not control coordinates.
-                        points[i] = view.PointWorldToGraph(vertices[i]);
-                    }
-                    if (signedArea < 0)
-                        Array.Reverse(points);
-                    next.AddPolygon(points);
+                    // Outlines wind one way and holes the other, so the winding fill leaves
+                    // holes empty while overlapping pairs' islands still merge.
+                    next.AddPolygon(GraphRing(region.Vertices, positive: true));
+                    foreach (var hole in region.Holes)
+                        next.AddPolygon(GraphRing(hole, positive: false));
                 }
             }
             path = next;
@@ -599,6 +589,28 @@ internal sealed class OverlapOverlayController : IDisposable
         }
     }
 
+    /// <summary>
+    /// Graph points of a closed world ring (the analyzer repeats the closing vertex), wound
+    /// positive for outlines or negative for holes in world space whatever the input order.
+    /// </summary>
+    private PointF[] GraphRing(IReadOnlyList<Vector> vertices, bool positive)
+    {
+        var points = new PointF[vertices.Count - 1];
+        var signedArea = 0.0;
+        var start = vertices[0];
+        for (var i = 0; i < points.Length; i++)
+        {
+            var a = vertices[i] - start;
+            var b = vertices[i + 1] - start;
+            signedArea += a.X * b.Y - b.X * a.Y;
+            // The host has already translated by origin: graph, not control coordinates.
+            points[i] = view.PointWorldToGraph(vertices[i]);
+        }
+        if (signedArea < 0 == positive)
+            Array.Reverse(points);
+        return points;
+    }
+
     private void DrawCutOffClampContacts(Graphics graphics)
     {
         var dpi = view.DeviceDpi / 96f;
diff --git a/OpenNest/Controls/ValidationOverlay.cs b/OpenNest/Controls/ValidationOverlay.cs
index 169902c6..c48bb3f6 100644
--- a/OpenNest/Controls/ValidationOverlay.cs
+++ b/OpenNest/Controls/ValidationOverlay.cs
@@ -71,7 +71,13 @@ internal sealed class ValidationOverlay : IDisposable
                     areas?.Dispose();
                     areas = new GraphicsPath(FillMode.Winding);
                     foreach (var region in overlap.Regions)
+                    {
+                        // The analyzer winds outlines counterclockwise and holes clockwise,
+                        // so the winding fill leaves each island's holes empty.
                         areas.AddPolygon(region.Vertices.Select(view.PointWorldToGraph).ToArray());
+                        foreach (var hole in region.Holes)
+                            areas.AddPolygon(hole.Select(view.PointWorldToGraph).ToArray());
+                    }
                     scale = view.ViewScale;
                 }
                 using var brush = new SolidBrush(Color.FromArgb(100, 255, 0, 80));
diff --git a/docs/geometry/visual-overlap-check.md b/docs/geometry/visual-overlap-check.md
index 4e40e1e4..233408c2 100644
--- a/docs/geometry/visual-overlap-check.md
+++ b/docs/geometry/visual-overlap-check.md
@@ -57,11 +57,10 @@ material), even if the dialog is later canceled. There is no check during export
 ## Analysis API
 
 `OpenNest.Diagnostics.PlateOverlapAnalyzer` in OpenNest.Core checks a group of placed
-parts and returns the shared polygon areas for each overlapping pair. Existing
-`Part.Intersects`, `PartOverlapChecker`, `Plate.HasOverlappingParts`, engine
-validators, and CLI entry points are unchanged. A separate shared-triangulator fix
-uses translation-stable winding, correcting missed clockwise outlines/holes far
-from the origin without changing contact or fragment-area tolerance policies.
+parts and returns the shared material for each overlapping pair as connected
+islands. It clips with Clipper2. Existing `Part.Intersects`, `PartOverlapChecker`,
+`Plate.HasOverlappingParts`, engine validators, and CLI entry points are unchanged
+and still use the hand-written `Collision` kernel (its GPU port is on hold).
 
 ## Synchronous use
 
@@ -77,8 +76,9 @@ foreach (var pair in report.Pairs)
     Console.WriteLine($"{pair.PartAId} / {pair.PartBId}: {pair.Area}, centroid {pair.Centroid}");
     foreach (var region in pair.Regions)
     {
-        // region.Vertices: closed, read-only world-coordinate polygon
-        // region.Area: positive shared material area, in model units squared
+        // region.Vertices: closed, read-only, counterclockwise world-coordinate outline
+        // region.Holes: closed, read-only, clockwise hole rings inside that outline
+        // region.Area: positive net shared material area, in model units squared
     }
 }
 
@@ -116,17 +116,20 @@ conversion cost; callers must not mutate inputs while capture runs.
 
 Analysis clones captured entities before chaining, prepares polygons once per
 source, then prepares each pose. An X-sorted bounds sweep prunes separated pairs.
-It invokes the existing hole-aware `Collision.Check` once per candidate pair,
-without an earlier boolean collision pass. Pairs are rebased near the origin for
-clipping/triangulation and restored to world coordinates; area calculation uses
-translated-origin products to avoid cancellation far from the origin. Snapshots
-can be reused and analyzed concurrently. Callers own freshness checks and must
-not publish results after geometry changes or after a newer request supersedes them.
+Each candidate pair is intersected once with Clipper2: each operand is its outline
+minus the union of its holes (outline positive, holes negative, positive fill rule).
+Pairs are rebased near the origin and rounded to a 1e-8 model-unit grid for
+clipping, then restored to world coordinates. A coordinate more than 1e9 units from
+the pair's origin, or a nonfinite one, makes that pair incomplete rather than clear.
+Area calculation uses translated-origin products to avoid cancellation far from the
+origin. Snapshots can be reused and analyzed concurrently. Callers own freshness
+checks and must not publish results after geometry changes or after a newer request
+supersedes them.
 
 Cancellation throws `OperationCanceledException`; it never returns a partial
 all-clear. Checks occur between source/pose preparation, validation loops, and
 candidate pairs, and before return. An individual conversion, polygonization,
-triangulation, or `Collision.Check` call is not internally interruptible.
+or Clipper2 intersection is not internally interruptible.
 
 ## Material contract
 
@@ -171,13 +174,14 @@ quadrant change moves the zones); posting runs its own fresh check with per-atte
 
 ## Interpretation and limits
 
-Regions are the kernel's convex, hole-subtracted fragments, not merged connected
-islands. Fill the fragments for a visual overlay; do not outline triangulation
-seams as physical boundaries. Areas within one pair may be summed. Areas across
+Regions are connected islands of shared material. Each has a counterclockwise
+outline and clockwise `Holes`; an island can also sit inside a hole that both parts'
+holes enclose together. Fill outlines and holes in one nonzero-winding path so holes
+stay empty. Areas within one pair may be summed. Areas across
 pairs are not a union: three coincident parts produce three overlapping pairs,
 so summing all pair areas double-counts shared plate locations.
 
-`pair.Centroid` is the true area-weighted center of every shared-material fragment
+`pair.Centroid` is the true area-weighted center of every shared-material island
 for that pair, after hole subtraction. It is not an average of crossings or
 vertices. For a disconnected or concave overlap, the mathematical centroid can
 lie outside the red material (for example, between two separate patches). This
@@ -185,9 +189,10 @@ is intentional: shaded Areas are authoritative for actual overlap locations;
 use Both for detailed inspection rather than interpreting a centroid as an
 interior collision point. There is one centroid per pair, not per connected island.
 
-`PolygonAreaMoments` evaluates signed moments about local origins and combines
-fragments with positive area weights independent of winding. The analyzer does
-this on rebased clipping fragments before adding the world origin back. Invalid,
+`PolygonAreaMoments` evaluates signed moments about local origins. The analyzer
+subtracts each island's hole moments from its outline's (`TrySubtract`), then
+combines the islands with positive area weights (`TryCombine`), on rebased
+coordinates before adding the world origin back. Invalid,
 degenerate, or nonfinite moments produce an incomplete pair issue, not a marker
 at zero. Curved-outline centroids inherit the polygonization approximation.
 
@@ -198,11 +203,12 @@ repairs, export blocks, or machining-validity guarantees.
 
 Arc/circle flattening uses a chord tolerance of `0.001` model units (also exposed
 as `report.ChordTolerance`). Curved overlaps and topology are therefore polygonal
-approximations. The unchanged collision kernel applies dimensional bounds and
-fragment-area thresholds using `Tolerance.Epsilon` (`0.00001`); sufficiently small
-slivers are below its reporting policy. Floating-point coordinates still have
-finite resolution. Contact and fragment thresholds are unchanged; only the shared
-triangulator's winding arithmetic was stabilized in the prerequisite fix.
+approximations. A pair is reported when its total shared area exceeds
+`Tolerance.Epsilon` (`0.00001` square units). Contact and rounding dust below that
+are not overlap. The engine's `Collision` kernel instead drops each convex fragment
+at or below that area, so a sliver it splits into small fragments can read clear
+there while this diagnostic reports it. Floating-point coordinates still have finite
+resolution.
 
 ## Desktop lifecycle and rendering
 
@@ -247,21 +253,28 @@ whose ordered input slots changed; this display-only path does not renumber them
 Explicit geometry invalidation, plate switch, handle loss, cancellation and
 failure clear all retained highlights.
 
-Measured on 501 real PEP-converted plates with 2 to 384 parts, a from-scratch check
-takes median 1 ms, p99 368 ms and max 6.5 s (a 299-part plate); an incremental
-recheck after moving one part takes median 0.1 ms, p99 20 ms and max 35 ms. A
-comparison against uncached full analysis over 2505 edits (nudge, drag onto
+Measured on 389 real PEP-converted plates (252 nests), a from-scratch check takes
+median 0.8 ms, p99 97 ms and max 3.2 s. Nearly all of the slow cases are first-time
+material preparation for a hole-heavy drawing. With material prepared, clipping every
+candidate pair takes median 0.05 ms, p99 27 ms and max 52 ms. The triangle-pair kernel
+took up to 2.6 s on the same plates. On those plates Clipper2 and the previous kernel
+found identical pair, clamp and issue sets. Areas differed by at most 0.0002 square
+units, always smaller from the kernel, consistent with its dropping fragments below
+its area threshold. With the previous kernel, an
+incremental recheck after moving one part took median 0.1 ms, p99 20 ms and max
+35 ms. A comparison against uncached full analysis over 2505 edits (nudge, drag onto
 another part, move three, delete first, swap order) matched exactly. It checks freshness before
 publication and painting. Handle destruction/disposal cancels work and releases
 paths; old completions cannot replace a newer report. Snapshot conversion and
 clipping never run in paint or mouse-move handlers.
 
 PlateView draws the controller overlay after work-area/debug-remnant drawing and
-before action paint subscribers and hover tooltips. One consistently wound path
-is filled once, avoiding fragment outlines, internal triangulation seams, and
-darker triple coverage. World-to-graph conversion excludes pan, because PlateView
-already applies origin translation. Paths are rebuilt for displayed-pair/scale changes,
-not ordinary repaints or panning. The state label saves/restores graphics state.
+before action paint subscribers and hover tooltips. Island outlines and reversed
+holes go into one consistently wound path that is filled once, so holes stay empty
+and triple coverage does not darken. World-to-graph conversion excludes pan,
+because PlateView already applies origin translation. Paths are rebuilt for
+displayed-pair/scale changes, not ordinary repaints or panning. The state label
+saves/restores graphics state.
 Centroid hit tests use only cached report coordinates and DPI-scaled screen
 radii. Hover clears on edits, mode/request/view changes, leave, and teardown.
 Diagnostic details draw above action adorners and take precedence over the normal
@@ -269,14 +282,14 @@ part-name tooltip only while visible.
 
 Next hardening: the first check of a plate with a hole-heavy drawing can still
 take seconds, and cancellation cannot interrupt the interior of material
-preparation or one kernel operation. Prepared per-part triangulations are not yet
-cached across requests.
+preparation or one Clipper2 intersection.
 
 ## Verification
 
 `OpenNest.Tests/Diagnostics/PlateOverlapAnalyzerTests.cs` exercises analytical
 rectangle regions/areas, containment and contact, both operands' holes, concave
-and disconnected intersections, curves, baseline rotation, large translations,
+and disconnected intersections, connected islands with holes (including an island
+enclosed by both parts' holes), curves, baseline rotation, large translations,
 deterministic pair ordering against an exhaustive rectangle oracle, invalid
 inputs, snapshot isolation, read-only output, cutting-program independence, and
 cancellation. Run:
@@ -294,8 +307,8 @@ waits, and the no-retry rule for canceled or failed layouts.
 freshness, per-pair display retention through repeated edits and pending checks,
 hard invalidation, cancellation, and incomplete-versus-clear messaging.
 `PolygonAreaMomentsTests` covers analytic
-centers, unequal/disconnected fragments, winding, closure, large translations,
-and invalid/overflow cases. `OverlapPairPresentationTests` checks adaptive unit
+centers, unequal/disconnected fragments, hole subtraction, winding, closure, large
+translations, and invalid/overflow cases. `OverlapPairPresentationTests` checks adaptive unit
 formatting, sequence labels, coincident ordering, and zoom-independent DPI hit
 radii. `OverlapHoverPagesTests` proves bounded continuation pages retain every
 pair and long/Unicode name, with navigation bounds and explicit tiny-view failure.