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

Services/YouTubeStreamService.cs

@@ -124,6 +124,28 @@ public class YouTubeStreamService
         return $"YouTube rejected the update ({(int)response.StatusCode})";
     }
 
+    /// <summary>Proper close-out (TASK 9 design decision 6 — implemented 2026-09-01;
+    /// until then we relied entirely on enableAutoStop, leaving viewers on a frozen
+    /// "stream offline" screen for ~a minute): POST liveBroadcasts.transition
+    /// broadcastStatus=complete. MUST be called AFTER the encoder closed the RTMP
+    /// push so no frames post-date the end. A broadcast YouTube already auto-completed
+    /// answers 403 invalidTransition / 410 — logged and surfaced as an error string,
+    /// never thrown: the stop path must not fail over a cosmetic close-out.</summary>
+    public async Task<string?> EndBroadcastAsync(string broadcastId)
+    {
+        if (!await EnsureToken()) return "not signed in";
+
+        _http.DefaultRequestHeaders.Authorization = new("Bearer", _auth.CurrentChannel!.AccessToken);
+        var response = await _http.PostAsync(
+            $"{ApiBase}/liveBroadcasts/transition?broadcastStatus=complete&id={Uri.EscapeDataString(broadcastId)}&part=status",
+            content: null);
+
+        if (response.IsSuccessStatusCode) return null;
+        var body = await response.Content.ReadAsStringAsync();
+        AppLog.Write($"Broadcast transition(complete) failed ({(int)response.StatusCode}): {body}");
+        return $"YouTube rejected the end transition ({(int)response.StatusCode})";
+    }
+
     /// <summary>Returns the channel's reusable stream (TASK 5 design decision 2):
     /// lists existing streams first and reuses the one with cdn.isReusable=true,
     /// creating it with variable resolution/frame rate on first use. Binding to a