TASK 31: SceneGraph component + baked-crust compositor optimization

- Add ElementKind (Static/Dynamic) to SceneElement base; Source/WebcamSceneConfig classify
- Services/SceneGraph.cs: owns the Scenes collection (ViewModel's Scenes delegates to it),
  the element mutation surface (Add/Insert/Remove/Move, each invalidating the bake), and the
  queries that were scattered LINQ (GetBackground/GetWebcam/GetChatBoxes/GetSplitPoint/IsStatic)
- SceneCompositor: split-aware BakeStaticBase + CompositeLayers + Render(.., staticBase, split);
  builds/caches the static base below the split point in source-rect space
- FramePump: optional SceneGraph -> optimized bake+composite path; falls back to full render
- MainViewModel: routes element mutations through the graph; invalidates the bake on static
  layout/opacity/visibility/useDefaultBackground changes and after background heal/Ensure
- Integration test SceneGraphTests.BakedStaticBase_WithDynamicLayer_CompositesCorrectly

Derivative survey (mandated): tried before writing — OBS does per-source opacity/visibility
caching and static-scene baking; this mirrors OBS's 'cached static source' optimization.
3 documented defensive deviations from the spec (ChatOverlayLayer stays decoupled;
background helpers stay VM-static for direct testability; full facade peel deferred post-1.0)
in TASKS.md + ai.md. Scope check passed; clean build 0 warnings.

Services/Encoder/FramePump.cs

@@ -31,6 +31,7 @@ public sealed class FramePump : IDisposable
     private readonly Func<(VideoFrame? Frame, SocialBarPosition Position)>? _socialBar;
     private readonly TransitionService? _transition;
     private readonly Func<Scene?>? _fromSceneProvider;
+    private readonly SceneGraph? _sceneGraph;
     private readonly SceneCompositor _compositor = new();
 
     private readonly object _gate = new();
@@ -55,7 +56,8 @@ public sealed class FramePump : IDisposable
         Func<TimeSpan, CancellationToken, Task>? pacingDelay = null,
         Func<(VideoFrame? Frame, SocialBarPosition Position)>? socialBar = null,
         TransitionService? transition = null,
-        Func<Scene?>? fromSceneProvider = null)
+        Func<Scene?>? fromSceneProvider = null,
+        SceneGraph? sceneGraph = null)
     {
         _sceneProvider = sceneProvider ?? throw new ArgumentNullException(nameof(sceneProvider));
         _frameResolver = frameResolver ?? throw new ArgumentNullException(nameof(frameResolver));
@@ -67,6 +69,7 @@ public sealed class FramePump : IDisposable
         _socialBar = socialBar;
         _transition = transition;
         _fromSceneProvider = fromSceneProvider;
+        _sceneGraph = sceneGraph;
     }
 
     public bool IsRunning { get; private set; }
@@ -240,16 +243,16 @@ public sealed class FramePump : IDisposable
                     {
                         var fromScene = _fromSceneProvider();
                         var fromFrame = fromScene != null
-                            ? _compositor.Render(fromScene, _frameResolver, null, compositorOptions, socialBarFrame, socialBarTop)
-                            : _compositor.Render(scene, _frameResolver, null, compositorOptions, socialBarFrame, socialBarTop);
-                        var toFrame = _compositor.Render(scene, _frameResolver, null, compositorOptions, socialBarFrame, socialBarTop);
+                            ? RenderScene(fromScene, compositorOptions, socialBarFrame, socialBarTop)
+                            : RenderScene(scene, compositorOptions, socialBarFrame, socialBarTop);
+                        var toFrame = RenderScene(scene, compositorOptions, socialBarFrame, socialBarTop);
                         frame = transition.BlendFrame(toFrame);
                         transition.Tick(lastTick.Elapsed.TotalMilliseconds);
                         lastTick.Restart();
                     }
                     else
                     {
-                        frame = _compositor.Render(scene, _frameResolver, null, compositorOptions, socialBarFrame, socialBarTop);
+                        frame = RenderScene(scene, compositorOptions, socialBarFrame, socialBarTop);
                         lastTick.Restart();
                     }
 
@@ -287,6 +290,39 @@ public sealed class FramePump : IDisposable
 
     private void OnHealthUpdated(object? sender, StreamHealth health) => HealthUpdated?.Invoke(this, health);
 
+    /// <summary>Render a scene, using the baked-crust optimization when a <see cref="SceneGraph"/>
+    /// is wired in: bake/cache the static layers below the split point, then composite the
+    /// dynamic/above-split layers per frame. Without a SceneGraph, falls back to a full render
+    /// (identical output — see SceneCompositorTests).</summary>
+    private VideoFrame RenderScene(
+        Scene scene,
+        CompositorOptions options,
+        VideoFrame? socialBarFrame,
+        int socialBarTop)
+    {
+        if (_sceneGraph == null)
+            return _compositor.Render(scene, _frameResolver, null, options, socialBarFrame, socialBarTop);
+
+        var split = _sceneGraph.GetSplitPoint(scene);
+        if (split == scene.Elements.Count)
+        {
+            // Fully static scene: bake once, reuse.
+            var baked = _sceneGraph.GetBakedBase(scene, _frameResolver, _compositorOptions);
+            if (baked != null)
+                return StretchMath.BilinearScale(baked, options.OutputWidth, options.OutputHeight);
+        }
+
+        var baseFrame = _sceneGraph.GetBakedBase(scene, _frameResolver, _compositorOptions);
+        if (baseFrame != null)
+        {
+            return SceneCompositor.CompositeLayers(
+                baseFrame, scene, split, _frameResolver, options, socialBarFrame, socialBarTop);
+        }
+
+        // No static base (first layer is dynamic or empty scene) — full render.
+        return _compositor.Render(scene, _frameResolver, null, options, socialBarFrame, socialBarTop);
+    }
+
     private void OnProcessFailed(object? sender, string message)
     {
         _log?.Invoke($"FramePump: encoder process failed: {message}");