feat(server,watch): threads remember the dispatch origin header and report it back

tests/test_server.py

@@ -625,6 +625,29 @@ async def test_dispatch_tracks_thread(client: Client, registry: Registry) -> Non
     )
 
 
+async def test_dispatch_remembers_its_origin(
+    client: Client, registry: Registry, monkeypatch: pytest.MonkeyPatch
+) -> None:
+    from t3code_mcp import server
+    from t3code_mcp.watch import Event, Watcher
+
+    monkeypatch.setattr(
+        server, "get_http_headers", lambda: {"x-conversation-origin": "conv-7"}
+    )
+    async with client:
+        started = data(
+            await client.call_tool(
+                "t3_dispatch",
+                {"machine": "mac", "project": "t3-smoke", "prompt": "from a branch"},
+            )
+        )
+    tid = started["thread_id"]
+    assert registry.tracker.threads[tid].origin == "conv-7"
+    watcher = Watcher(registry, registry.tracker, None)
+    payload = await watcher.payload(Event("completed", tid, "mac"))
+    assert payload["origin"] == "conv-7"
+
+
 def test_tracker_watch_flag(tmp_path) -> None:
     from t3code_mcp.watch import Tracker