API docs: `/chat` endpoint

[codebestia]

Write apps/ai_agent/docs/api-chat.md documenting POST /chat — request/response shape (ChatRequest/ChatResponse), the system prompt behavior, model used, and failure modes (missing API key, upstream OpenAI errors).

Acceptance criteria:

• Request and response fields are documented with types, sourced from the actual Pydantic models in main.py
• Every documented failure mode has its actual HTTP status code and response body shape verified against the code/tests
• A worked example request/response pair is included

[Kingvic300]

@Kingvic300 has applied to work on this issue as part of the Stellar Wave Program's 7th wave.

I would love to work on this kindly assign it to me

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[cythecode]

@cythecode has applied to work on this issue as part of the Stellar Wave Program's 7th wave.

Hello maintainer, i’ve got this covered! Ready to drop a high-impact, clean PR that gets this merged seamlessly

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[bamiebot-maker]

@bamiebot-maker has applied to work on this issue as part of the Stellar Wave Program's 7th wave.

Hi team! 🦊 I'm ready to tackle this task. My technical background aligns well with this issue, and I can have a clean, tested Pull Request ready for review within your expected timeframe. Please let me assign me so I can begin

ℹ️ Repo Maintainers: To accept this application, review their application or assign @bamiebot-maker to this issue.

[drips-wave]

Congratulations, @bamiebot-maker! 🎉 Your application was accepted by the repo's maintainers, and the issue is due on July 30, 2026.

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