Per-project task count badge in the sidebar (queue↔project keystone)

[Eyalm321]

Type: AFK

What to build

Bind the durable work-queue to projects and surface each project's task volume in the sidebar. Establish the convention that a project's work queue is addressed by its project id — the worker controller defaults its queue to the registered project's id when launched inside that project's repo. The GUI host periodically reads each project's queue summary from the control API and shows a small task-count badge on the project row in the sidebar (alongside the existing worktree count).

This slice is the keystone for everything else (both the richer GUI lanes and the GitHub source target the project's queue by id).

Acceptance criteria

• A project's work queue is addressable by its project id (controller defaults the queue to the project id when run inside a registered project's repo)
• The sidebar shows a task-count badge on each project row reflecting its non-terminal queue depth, refreshed within ~2s
• Projects with no queue/tasks show no badge
• Enqueuing/completing a task updates the badge live, without restarting the app

Blocked by

None - can start immediately.