fix: close Wave 1.1 completion-gate audit findings

- Headless consumers (task scheduler, background follow-up) now treat a
  completion-gate final_response as the authoritative answer instead of
  collecting deltas only. A gated replacement no longer leaves scheduled
  output empty, which used to trigger an extra, ungated grace-summary
  model call.
- The scheduler closes the agent stream with contextlib.aclosing, so the
  approval-pause break unwinds the gate's journal and teacher-takeover
  context in its own task. Chained runs no longer inherit a stale
  parent_run_id, and later finalization no longer raises ContextVar
  reset errors.
- On provider error, the completion gate applies the live answer's
  statement filter to persisted round_texts. Diagnostics and the failure
  note survive; claims rejected by the gate cannot reappear on reload.

src/bg_monitor.py

@@ -42,6 +42,7 @@ async def _drain_agent(sess, messages):
     saves, so the frontend rebuilds them as standard agent-thread tool cards."""
     from src.agent_loop import stream_agent_loop
     full = ""
+    final_replaced = False
     tool_events = []
     round_num = 1
     async for chunk in stream_agent_loop(
@@ -68,7 +69,17 @@ async def _drain_agent(sess, messages):
             if isinstance(delta, str):
                 if d.get("thinking"):
                     continue
+                if final_replaced:
+                    # A later answer supersedes the replacement, as the
+                    # completion gate treats it.
+                    full = ""
+                    final_replaced = False
                 full += delta
+        elif d.get("type") == "final_response":
+            # The completion gate may present its sanitized answer as one
+            # replacement instead of deltas.
+            full = str(d.get("content") or "")
+            final_replaced = True
         elif d.get("type") == "agent_step":
             round_num = d.get("round", round_num)
         elif d.get("type") == "tool_output":