feat(9): real broadcast close-out — transition(complete) in StopStream after RTMP EOF

The specced 'End stream -> transition(complete)' call never existed: stopping relied
entirely on enableAutoStop (viewers sat on a frozen stream-offline for ~a minute, VOD
finalized late). Found during the 2026-09-01 recording-verification pass while the
creator asked 'if there's proper close-out info yt needs, we'll provide it?'

EndBroadcastAsync POSTs liveBroadcasts/transition?broadcastStatus=complete&id=..&part=status,
called after the pump stops (RTMP EOF first) and only when a live session had a broadcast —
record-only stops stay offline. invalidTransition/410 (autoStop already ended it) is logged
and returned as an error string, never thrown: a stop must never fail over close-out.
ONE integration test (URL shape + never-throws on 403). ai.md/TASKS.md design lines marked
SHIPPED with the map-lie note.

Ref: https://developers.google.com/youtube/v3/live/docs/liveBroadcasts/transition

ViewModels/MainViewModel.Streaming.Operations.cs

@@ -238,6 +238,19 @@ public partial class MainViewModel : ViewModelBase
         // monitoring); only the live/report pipe, the pump, and the session stop here.
         _audioMixer.StopLive();
         await _framePump.StopAsync();
+
+        // Proper close-out (TASK 9 decision 6, built 2026-09-01): transition(complete)
+        // AFTER the RTMP push is closed so no frames post-date the end — YouTube then
+        // finalizes the VOD immediately instead of leaving viewers on a frozen
+        // "stream offline" until enableAutoStop fires (~1 min). Log-only failure:
+        // the backstop still ends the broadcast; a stop never fails over cosmetics.
+        if (wasLive && _currentBroadcastId != null)
+        {
+            var closeOutError = await _youtubeStream.EndBroadcastAsync(_currentBroadcastId);
+            if (closeOutError != null)
+                AppLog.Write($"End close-out: {closeOutError} — enableAutoStop will finish the broadcast");
+        }
+
         _currentBroadcastId = null;
         OnPropertyChanged(nameof(CurrentBroadcastId));