Depends on #1487
## Summary
- Introduce `packages/core-agent` — a new pure-runtime package that
extracts zero-Vue/Pinia algorithmic logic from `@proj-airi/stage-ui`,
following a "compatible facade + core sinking" strategy
- Migrate shared chat types, LLM streaming types, chat hook registry,
session message merge logic, context registry algorithm, and LLM service
utilities into `core-agent`
- `stage-ui` files are preserved as re-exports / thin wrappers — all
external imports, store IDs, and public APIs remain unchanged
## Motivation
`stage-ui` currently mixes pure agent runtime logic (algorithms, type
definitions, stateless utilities) with Vue/Pinia state management and
browser-specific adapters. This coupling makes it hard to:
- Test agent logic in isolation
- Reuse agent algorithms outside of Vue contexts (e.g., server-side,
CLI, other frameworks)
- Reason about the boundary between "what the agent does" vs "how the UI
manages state"
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