diff --git a/OpenNest.Core/Diagnostics/PostVerificationGeometry.cs b/OpenNest.Core/Diagnostics/PostVerificationGeometry.cs
index cfff678..aefaff5 100644
--- a/OpenNest.Core/Diagnostics/PostVerificationGeometry.cs
+++ b/OpenNest.Core/Diagnostics/PostVerificationGeometry.cs
@@ -9,6 +9,10 @@ namespace OpenNest.Diagnostics;
 internal static class PostVerificationGeometry
 {
     internal const double Epsilon = 1e-8;
+    // Matches NominalContourClosure.Tolerance: the largest gap capture admits as a closed ring.
+    // Ray parity snaps vertex Y comparisons within this band so an admitted seam cannot flip an
+    // exterior point to interior (see Curve.CrossesRay).
+    internal const double RaySeamTolerance = 1e-6;
     private const double TwoPi = 2 * System.Math.PI;
 
     /// <summary>An axis-aligned extent used only to skip queries that cannot meet.</summary>
@@ -385,8 +389,18 @@ internal static class PostVerificationGeometry
 
         internal bool CrossesRay(Vector point)
         {
+            // Ray parity needs both edges meeting at a shared vertex to agree which side of the ray
+            // that vertex is on. Rings are only nominally closed: capture admits closure seams and
+            // PEP-converted or rotated geometry carries sub-micro-inch vertex jogs, so an exact
+            // comparison can split one vertex across the ray and report far-away exterior points as
+            // interior — which made LeadPathValidator refuse leads that could touch nothing.
+            // Snapping within the admitted seam tolerance keeps the decision consistent on both
+            // edges; real features (notches, tabs, arc spans) are far larger than the tolerance and
+            // still count. Containment of a point that itself lies within the tolerance of a
+            // boundary edge stays ambiguous by design, exactly as it was before.
+            var rayY = point.Y + RaySeamTolerance;
             if (Center is not { } center)
-                return (Start.Y > point.Y) != (End.Y > point.Y)
+                return (Start.Y > rayY) != (End.Y > rayY)
                     && Start.X + (point.Y - Start.Y) * (End.X - Start.X) / (End.Y - Start.Y) > point.X;
 
             // Split arcs at vertical extrema, giving monotone-Y pieces. Apply the same
@@ -407,7 +421,7 @@ internal static class PostVerificationGeometry
                 var b = StartAngle + System.Math.Sign(Sweep) * breaks[i];
                 var ya = i == 1 ? Start.Y : center.Y + Radius * System.Math.Sin(a);
                 var yb = i == breaks.Count - 1 ? End.Y : center.Y + Radius * System.Math.Sin(b);
-                if ((ya > point.Y) == (yb > point.Y))
+                if ((ya > rayY) == (yb > rayY))
                     continue;
                 var dy = point.Y - center.Y;
                 var dx = System.Math.Sqrt(System.Math.Max(0, Radius * Radius - dy * dy));
diff --git a/OpenNest.Tests/CuttingPlanning/MaterialContainmentTests.cs b/OpenNest.Tests/CuttingPlanning/MaterialContainmentTests.cs
new file mode 100644
index 0000000..35ed8cc
--- /dev/null
+++ b/OpenNest.Tests/CuttingPlanning/MaterialContainmentTests.cs
@@ -0,0 +1,121 @@
+using System;
+using System.Collections.Generic;
+using System.Threading;
+using OpenNest.CNC.CuttingPlanning;
+using OpenNest.Diagnostics;
+using OpenNest.Geometry;
+using Xunit;
+using Curve = OpenNest.Diagnostics.PostVerificationGeometry.Curve;
+
+namespace OpenNest.Tests.CuttingPlanning;
+
+/// <summary>
+/// Captured material outlines are only nominally closed: capture admits closure slack up to
+/// NominalContourClosure.Tolerance, and PEP-converted or rotated geometry carries sub-micro-inch
+/// vertex jogs where two segments meet. The point-in-material ray test must stay consistent across
+/// such a jog. Without that, a jog lying exactly on the test ray's Y reports far-away exterior
+/// points as interior, so LeadPathValidator refuses leads that cannot touch anything: plans get
+/// needlessly expensive routes or give up inside the search budget.
+/// </summary>
+public class MaterialContainmentTests
+{
+    [Fact]
+    public void SubMicroinchJogAtRayHeight_KeepsDistantExteriorPointExterior()
+    {
+        var rings = Ring(
+            Line((21.2, 0.6), (31.2, 0.6)),
+            Line((31.2, 0.6000001), (41.2, 0.6000001)),   // admitted seam jog on the ray line
+            Line((41.2, 0.6000001), (41.2, 46.16)),
+            Line((41.2, 46.16), (21.2, 46.16)),
+            Line((21.2, 46.16), (21.2, 0.6)));
+
+        Assert.False(LeadMaterialSnapshot.Inside(new Vector(20.75, 0.6), rings[0], CancellationToken.None));
+        Assert.False(LeadMaterialSnapshot.Inside(new Vector(20.75, 0.6000001), rings[0], CancellationToken.None));
+        Assert.False(LeadMaterialSnapshot.Inside(new Vector(20.75, 0.60000005), rings[0], CancellationToken.None));
+        Assert.True(LeadMaterialSnapshot.Inside(new Vector(30, 20), rings[0], CancellationToken.None));
+    }
+
+    [Fact]
+    public void ClosedWithoutJog_ReportsTheSameAnswersAsTheJoggedCopy()
+    {
+        var clean = Ring(
+            Line((21.2, 0.6), (41.2, 0.6)),
+            Line((41.2, 0.6), (41.2, 46.16)),
+            Line((41.2, 46.16), (21.2, 46.16)),
+            Line((21.2, 46.16), (21.2, 0.6)));
+        var jogged = Ring(
+            Line((21.2, 0.6), (31.2, 0.6)),
+            Line((31.2, 0.6000001), (41.2, 0.6000001)),
+            Line((41.2, 0.6000001), (41.2, 46.16)),
+            Line((41.2, 46.16), (21.2, 46.16)),
+            Line((21.2, 46.16), (21.2, 0.6)));
+
+        foreach (var y in new[] { 0.6, 0.60000005, 0.6000001 })
+        {
+            var expected = LeadMaterialSnapshot.Inside(new Vector(20.75, y), clean[0], CancellationToken.None);
+            Assert.Equal(expected, LeadMaterialSnapshot.Inside(new Vector(20.75, y), jogged[0], CancellationToken.None));
+        }
+    }
+
+    [Fact]
+    public void CircleTestedExactlyAtAnExtremum_KeepsExteriorExteriorAndInteriorInterior()
+    {
+        // Full circle r=5 about (10,10) as four quarter arcs. Testing the ray exactly along both
+        // vertical extrema is where the half-open rule and arc splitting have to agree.
+        var center = new Vector(10, 10);
+        var rings = Ring(
+            Curve.Create(new Vector(15, 10), new Vector(10, 15), center, false),
+            Curve.Create(new Vector(10, 15), new Vector(5, 10), center, false),
+            Curve.Create(new Vector(5, 10), new Vector(10, 5), center, false),
+            Curve.Create(new Vector(10, 5), new Vector(15, 10), center, false));
+
+        Assert.False(LeadMaterialSnapshot.Inside(new Vector(-4, 10), rings[0], CancellationToken.None));
+        Assert.False(LeadMaterialSnapshot.Inside(new Vector(-4, 5), rings[0], CancellationToken.None));
+        Assert.False(LeadMaterialSnapshot.Inside(new Vector(-4, 15), rings[0], CancellationToken.None));
+        Assert.True(LeadMaterialSnapshot.Inside(new Vector(10, 10), rings[0], CancellationToken.None));
+        Assert.True(LeadMaterialSnapshot.Inside(new Vector(7.5, 10), rings[0], CancellationToken.None));
+    }
+
+    [Fact]
+    public void SeamJogBetweenLineAndArc_KeepsDistantExteriorPointExterior()
+    {
+        // Rounded-left-edge material whose semicircular edge meets the bottom line 1e-7 apart:
+        // an admitted closure seam sitting exactly on a datum Y shared with neighbours.
+        var center = new Vector(1, 23.38000005);
+        var rings = Ring(
+            Line((1, 0.6), (41.2, 0.6)),
+            Line((41.2, 0.6), (41.2, 46.16)),
+            Line((41.2, 46.16), (1, 46.16)),
+            Curve.Create(new Vector(1, 46.16), new Vector(1, 0.6000001), center, false));
+
+        Assert.False(LeadMaterialSnapshot.Inside(new Vector(0.5, 0.6), rings[0], CancellationToken.None));
+        Assert.False(LeadMaterialSnapshot.Inside(new Vector(-4, 0.6), rings[0], CancellationToken.None));
+        Assert.False(LeadMaterialSnapshot.Inside(new Vector(-4, 0.6000001), rings[0], CancellationToken.None));
+        Assert.True(LeadMaterialSnapshot.Inside(new Vector(30, 20), rings[0], CancellationToken.None));
+    }
+
+    [Fact]
+    public void RealConcavityIsNotErasedByTheRayTolerance()
+    {
+        // A 0.01 in deep notch is far above any admitted seam tolerance: it must still count.
+        var rings = Ring(
+            Line((0, 0), (1, 0)),
+            Line((1, 0), (1, 1)),
+            Line((1, 1), (0.5, 1)),
+            Line((0.5, 1), (0.5, 0.99)),      // notch floor
+            Line((0.5, 0.99), (0, 0.99)),
+            Line((0, 0.99), (0, 0)));
+
+        // The 0.01 in notch sits above the left part's top edge; the right tower rises through it.
+        Assert.False(LeadMaterialSnapshot.Inside(new Vector(0.25, 0.995), rings[0], CancellationToken.None));
+        Assert.True(LeadMaterialSnapshot.Inside(new Vector(0.75, 0.995), rings[0], CancellationToken.None));
+        Assert.True(LeadMaterialSnapshot.Inside(new Vector(0.25, 0.5), rings[0], CancellationToken.None));
+        Assert.False(LeadMaterialSnapshot.Inside(new Vector(-1, 0.5), rings[0], CancellationToken.None));
+    }
+
+    private static IReadOnlyList<IReadOnlyList<Curve>> Ring(params Curve[] curves) =>
+        new List<IReadOnlyList<Curve>> { new List<Curve>(curves) };
+
+    private static Curve Line((double X, double Y) start, (double X, double Y) end) =>
+        Curve.Create(new Vector(start.X, start.Y), new Vector(end.X, end.Y), null, false);
+}