chore: bump deps + adopt pydantic-ai conversation_id

[LukeMainwaring]

Summary

• Routine weekly dependency bump across backend (psycopg, pydantic-ai, pydantic-evals, transformers) and frontend (axios, lucide-react, @tanstack/react-query, ai/@ai-sdk/react, biome, ultracite, postcss, hey-api/openapi-ts, use-stick-to-bottom).
• Refresh docs/pydantic-ai-llms-full.txt.
• Adopt pydantic-ai 1.89's conversation_id parameter on VercelAIAdapter.dispatch_request — pass thread_id so runs are correlated with a stable id instead of a fresh UUID7 per call.
• Remove postProcess: ["biome:format"] from openapi-ts.config.ts — generate-client already runs pnpm run format (ultracite fix) right after openapi-ts, so this was a redundant format pass.

Notable changelog items checked but no action needed

• @hey-api/openapi-ts 0.97 had 15 breaking changes, but our config doesn't touch any of the affected surfaces (no runtimeConfigPath, no error interceptors, throwOnError: true already emitted).
• axios 1.16 now enforces maxBodyLength/maxContentLength on the fetch adapter — we don't override the defaults (-1 = unlimited), so behavior is unchanged.
• transformers 5.7 didn't touch CLAP.

Test plan

• uv run --directory backend pre-commit run --all-files clean
• pnpm -C frontend lint clean
• mypy --strict clean on edited router
• Manually verify a chat round-trip after deploy (no protocol change expected)

🤖 Generated with Claude Code