TASK 22: global audio sync offset (positive delay at the mixer out)

Adds AudioSyncOffsetMs (0..500ms, default 0) that delays the whole
interleaved-stereo mix so audio lands on the video when it runs ahead —
OBS's documented lip-sync fix. Positive-only: advancing audio needs a
video-side delay and is out of the audio layer's scope.

- Services/Audio/AudioSyncDelay.cs: pure delay line, flushed on Configure
- AudioMixer: Func<int> syncOffsetMs seam + delay applied post-limiter
- LayoutStore.Settings + MainViewModel.Audio/VM: load/save + binding
- PreviewPane mic bar: SYNC slider + status dot (IntToSyncBrushConverter)
- AudioSyncDelayTests: identity, negative/beyond-500 clamps, 10ms→960 samples

Reference (external scan): https://obs-versions.com/blog/how-to-fix-audio-delay-on-obs
(audio ahead => positive delay). Verified: build 0 warnings; 3/3 delay tests pass.

ViewModels/MainViewModel.Audio.cs

@@ -29,6 +29,7 @@ public partial class MainViewModel
     private double _gameVolume = 1.0;
     private bool _gameMuted;
     private double? _gameVolumeBeforeMute;
+    private int _audioSyncOffsetMs;
     private AudioGainProvider? _gainProvider;
     private readonly IMicrophoneEnumerator _microphoneEnumerator;
     private readonly AudioMixer _audioMixer;
@@ -154,6 +155,21 @@ public partial class MainViewModel
         private set => SetProperty(ref _micSourceName, value);
     }
 
+    /// <summary>Global audio-sync delay in milliseconds (0..500, default 0),
+    /// applied to the whole mix so audio lands on the video when it runs ahead
+    /// — the OBS fix for lip-sync. Consumed live by the mixer's delay line via
+    /// a Func seam; persisted with the layout.</summary>
+    public int AudioSyncOffsetMs
+    {
+        get => _audioSyncOffsetMs;
+        set
+        {
+            var clamped = Math.Clamp(value, 0, 500);
+            if (SetProperty(ref _audioSyncOffsetMs, clamped))
+                ScheduleSave();
+        }
+    }
+
     private void ToggleMicMute()
     {
         if (MicMuted)

ViewModels/MainViewModel.cs

@@ -233,6 +233,7 @@ public partial class MainViewModel : ViewModelBase
         // the same, already-vetted device (green dot) or reports it missing
         // (yellow) instead of silently falling back to the default.
         MicSourceName = _layoutStore.LoadMicSourceName();
+        AudioSyncOffsetMs = _layoutStore.LoadAudioSyncOffsetMs();
 
         _cameraEnumerator = new MediaCaptureCameraEnumerator();
         _cameraManager = new CameraManager(
@@ -258,7 +259,8 @@ public partial class MainViewModel : ViewModelBase
             new WasapiLoopbackAudioSource(),
             message => AppLog.Write(message),
             micGain: _gainProvider.MicGain,
-            loopbackGain: _gainProvider.LoopbackGain);
+            loopbackGain: _gainProvider.LoopbackGain,
+            syncOffsetMs: () => AudioSyncOffsetMs);
         _audioMixer.MicLevelChanged += OnMicLevelChanged;
         _audioMixer.LoopbackLevelChanged += OnLoopbackLevelChanged;
         _audioMixer.MicConnected += OnMicConnected;
@@ -409,6 +411,7 @@ public partial class MainViewModel : ViewModelBase
             _layoutStore.Save(Scenes, _webcam, _socials, _music);
             _layoutStore.SaveTransitionSettings(
                 SelectedTransitionType.ToString(), TransitionDurationMs);
+            _layoutStore.SaveAudioSyncOffsetMs(AudioSyncOffsetMs);
         }
         catch (Exception ex)
         {