[Mobile] Build first end-to-end Activities vertical slice in Expo

Goal

Validate the complete multi-client architecture with one production-quality vertical slice before expanding mobile scope.

Scope

• Create Expo mobile app in the same repository/workspace.
• Implement mobile authentication using the approved auth design.
• Fetch and display Activities/Tasks through the mobile API.
• Support offline read persistence.
• Create/update/complete activities offline.
• Queue mutations locally and synchronize after reconnect.
• Handle duplicate delivery, retries, errors, and conflict reconciliation.
• Keep UI state separate from domain/sync logic.

Explicit non-goals

• Do not build all modules yet.
• Do not rewrite the web app.
• Do not introduce tRPC, microservices, or a second backend.

Acceptance Criteria

• Login, fetch, offline mutation, reconnect, sync, and refresh all work end-to-end.
• Automated tests cover the critical path.
• Architecture gaps discovered during implementation are documented as follow-up issues.

No files, tests, or entry points are named. First map the existing repository/workspace and approved auth design, then define the Expo app, Activities/Tasks API path, offline persistence and sync boundaries; done means the listed login, offline mutation, reconnect, synchronization, refresh, and critical-path test criteria work end-to-end.