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

ytLive.Tests/YouTubeStreamServiceTests.cs

@@ -18,6 +18,7 @@ public class YouTubeStreamServiceTests
         public string? LastBody;
         public string ResponseBody = """{"id":"BC123","snippet":{"title":"t"}}""";
         public Func<HttpRequestMessage, string?>? ResponseSelector;
+        public HttpStatusCode StatusCode = HttpStatusCode.OK;
         public readonly List<string> Requests = new();
 
         protected override Task<HttpResponseMessage> SendAsync(
@@ -26,7 +27,7 @@ public class YouTubeStreamServiceTests
             Requests.Add($"{request.Method.Method} {request.RequestUri}");
             LastBody = request.Content?.ReadAsStringAsync().GetAwaiter().GetResult();
             var body = ResponseSelector?.Invoke(request) ?? ResponseBody;
-            return Task.FromResult(new HttpResponseMessage(HttpStatusCode.OK)
+            return Task.FromResult(new HttpResponseMessage(StatusCode)
             {
                 Content = new StringContent(body, Encoding.UTF8, "application/json"),
             });
@@ -236,4 +237,32 @@ public class YouTubeStreamServiceTests
         Assert.Equal("bitrateHigh", report.Text);
         Assert.True(report.IsError);
     }
-}
+    // TASK 9 close-out (2026-09-01): the ONE integration test — the transition URL
+    // shape (broadcastStatus=complete + id + part, POST) and the never-throws
+    // contract when YouTube already auto-completed the broadcast (invalidTransition).
+    [Fact]
+    public async Task EndBroadcast_Transitions_Complete_And_Never_Throws_On_InvalidTransition()
+    {
+        var handler = new RecordingHandler();
+        var service = new YouTubeStreamService(CreateAuthed(), new HttpClient(handler));
+
+        var error = await service.EndBroadcastAsync("BC-42");
+
+        Assert.Null(error);
+        var request = Assert.Single(handler.Requests);
+        Assert.StartsWith("POST ", request);
+        Assert.Contains("/liveBroadcasts/transition?", request);
+        Assert.Contains("broadcastStatus=complete", request);
+        Assert.Contains("id=BC-42", request);
+        Assert.Contains("part=status", request);
+
+        // Already auto-ended: 403 invalidTransition must surface as an error string,
+        // never an exception — the stop path must complete regardless (autoStop backstop).
+        handler.StatusCode = HttpStatusCode.Forbidden;
+        handler.ResponseBody = """{"error":{"code":403,"message":"Invalid transition"}}""";
+        var closeOutError = await service.EndBroadcastAsync("BC-42");
+
+        Assert.NotNull(closeOutError);
+        Assert.Contains("403", closeOutError);
+    }
+}