feat(voice): add first_transcript_after_eos_delay metric

[anshulkulhari7]

Summary

Closes #4795.

Adds a new end-of-turn latency metric, first_transcript_after_eos_delay, computed as:

first_transcript_after_eos_delay = first_transcript_time - end_of_speech_time

where end_of_speech_time is captured on VAD or STT end-of-speech, and first_transcript_time is the first interim/final transcript event received after that EOS.

Why

The existing transcription_delay is computed from the last speaking anchor to the final transcript, so it reflects end-to-end turn latency rather than "speech end → first result". As noted in the issue, a "speech end → first transcript" metric is needed for comparing provider latency and aligning with backend SLA definitions. This new metric is distinct from and complementary to transcription_delay.

What changed

AudioRecognition now tracks two new per-turn anchors:

• _end_of_speech_time — set on VAD END_OF_SPEECH and STT END_OF_SPEECH (stt turn-detection mode).
• _first_transcript_after_eos_time — set to the arrival time of the first transcript event (interim / preflight / final) that occurs after an end-of-speech, and only set once per turn so later transcripts don't move it.

The computed delay is plumbed through additively:

• _EndOfTurnInfo.first_transcript_after_eos_delay
• EOUMetrics.first_transcript_after_eos_delay (defaults to 0.0)
• the user MetricsReport
• log_metrics output for EOUMetrics
• the user_turn telemetry span (lk.first_transcript_after_eos_delay)

The metric is computed independently of the VAD speaking anchors used by transcription_delay / end_of_turn_delay, and falls back to 0.0 (VAD EOS) when STT EOS is unavailable, per the issue's note. When no transcript arrives strictly after EOS the value is left unset (None internally, 0.0 on the public model), which is honest rather than reporting a misleading value. The change is additive and backward-compatible — no existing field or default changes.

Testing

New unit test tests/test_first_transcript_after_eos.py drives AudioRecognition directly with crafted VAD/STT events (no audio, no network) and asserts:

• a VAD END_OF_SPEECH event anchors _end_of_speech_time;
• the first transcript after EOS anchors the time and later transcripts don't overwrite it;
• a transcript arriving before any EOS does not set the anchor;
• the value reported on _EndOfTurnInfo equals first_transcript_time - end_of_speech_time.

$ uv run pytest tests/test_first_transcript_after_eos.py --unit
4 passed

Also verified no regressions in the existing audio-recognition / session suites:

$ uv run pytest tests/test_speech_start_time_persistence.py \
    tests/test_audio_recognition_push_audio.py \
    tests/test_audio_recognition_aclose.py \
    tests/test_agent_session.py tests/test_session_host.py --unit

Lint / format / types on the changed files:

$ uv run ruff check <changed files>          # All checks passed!
$ uv run ruff format --check <changed files>  # already formatted
$ mypy --strict (pydantic plugin) -p livekit.agents  # no errors in changed modules

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AI-assisted: this change was prepared with AI assistance and reviewed/verified by me.

[devin-ai-integration]

✅ Devin Review: No Issues Found

Devin Review analyzed this PR and found no bugs or issues to report.