[Feature Request] AI-powered automatic categorization for inbox management

Summary
Add AI-powered automatic categorization to help organize a cluttered/mixed inbox, similar to how Arc groups items into labeled folders in its sidebar.

Current Behavior
The inbox is currently a flat, unorganized list. As the number of items grows (PRs, triggers, tool notifications, OAuth requests, etc.), it becomes hard to scan and find what matters, and users must manually sort or search through everything.

Expected Behavior
The system should use AI to automatically classify and group inbox items into meaningful categories/folders (e.g. by source, topic, or type — such as "Pull Requests", "Trigger", "Dify", "OAuth" as shown in the reference screenshot), so users can navigate a large mixed inbox at a glance instead of scrolling through a single flat list.

Why This Matters
- Reduces manual triage time as inbox volume grows.
- Improves findability — users can jump straight to the category they care about.
- Makes it easier to spot important/urgent items (e.g. warnings) that would otherwise get buried in a noisy inbox.
- This is a proven UX pattern (see Arc browser sidebar) that maps well onto our own multi-source inbox.

Suggested Implementation
- Auto-generate folder/category labels from message content and source metadata.
- Allow items to surface status indicators per category (e.g. warning icon on "Pull Requests" folder in the reference screenshot).
- Categories can be dynamic (created/merged by AI) rather than a fixed static taxonomy, with manual override/rename support.

Screenshot

Reference: Arc browser sidebar showing AI-categorized folders (Pull Requests, Trigger, Dify, OAuth) for a mixed inbox.

The issue does not identify files, tests, or an entry point. Start by locating the existing inbox UI and item/source model, then clarify the categorization scope, AI approach, category editing behavior, and acceptance criteria before implementation; done should include usable grouping of mixed inbox items and a defined way to verify the result.

Written by the indexing model from the issue text.