(orchestration): Sync task context into operator IDE workspaces

## Component

API or orchestration

## Describe the feature

When a background task is running or recently updated, expose a **machine-readable context bundle** (summary, evidence links, relevant file paths, branch name, latest milestones) that IDE plugins or local agent configs can poll—so engineers debugging alongside the task do not re-read Slack/GitHub manually.

## Use case

Responders and tech leads often work in an IDE while a background agent runs elsewhere. Copy-pasting status from notifications is friction. A stable, low-latency context endpoint enables **side-by-side collaboration** without turning every task into a live pair-programming session.

## Proposed solution

1. `GET /tasks/{id}/workspace-context` returning JSON: `branch`, `repo`, `summary`, `milestones[]`, `open_questions[]`, `hydration_digest` (bounded).
2. Optional MCP tool or documented `bgagent context export --task-id` for local agents.
3. Auth same as `GET /tasks/{id}`; rate-limited.
4. Align fields with roadmap **Smart progress updates** event schema to avoid duplication.

## Other information

- Does not require hosting a shared cloud IDE session—local consumption only in v1.
- Browser extension channel (roadmap) may consume the same payload.

## Acknowledgements

- [ ] I may be able to implement this feature
- [ ] This might be a breaking change

## Acceptance criteria

- [ ] Endpoint documented in `API_CONTRACT.md`; types in `cli/src/types.ts`.
- [ ] CLI export command writes JSON to stdout or file.
- [ ] Payload size capped; large traces remain on S3 trace URI.
- [ ] Handler tests cover running and terminal tasks.

まず API_CONTRACT.md と cli/src/types.ts から始め、既存の GET /tasks/{id} ハンドラーと Smart progress updates のイベントスキーマを調べて、認証とフィールドの整合性を確認します。文書化されたコンテキストエンドポイント、CLI の JSON エクスポート、ペイロードを制限する動作を追加し、実行中およびターミナル状態のタスクを対象とするテストを作成します。すべての受け入れ条件を満たせば完了です。

索引モデルが issue の本文から書いたものです。