The streaming API currently logs backend exceptions but still emits the
same `finish_reason: "stop"` + `[DONE]` terminator used for successful
responses. That makes a failed streamed request look successful to
OpenAI-compatible clients.
This keeps the fix narrow: track whether the stream backend failed and
suppress the success terminator in that case. A regression test locks in
the expected behavior.
Constraint: Keep the non-streaming response path untouched
Constraint: Follow up on the known limitation called out during PR #3222 review without redesigning the SSE protocol
Rejected: Introduce a custom SSE error event shape in the same patch | expands API surface and review scope
Confidence: high
Scope-risk: narrow
Reversibility: clean
Directive: If explicit streamed error events are added later, keep them distinct from the success stop+[DONE] terminator to preserve client retry semantics
Tested: PYTHONPATH=$PWD pytest -q tests/test_api_stream.py /Users/jh0927/Workspace/nanobot-validation-artifacts-2026-04-18/test_api_stream_error_regression.py
Not-tested: Full repository test suite
Related: #3260
Related: #3222
