[FEATURE]: Web Push notifications for paired browsers #58428

Feature hasn't been suggested before.

• I have verified this feature I'm about to submit hasn't been suggested before.

Describe the enhancement you want to request

Problem

The web app can notify while a tab or installed PWA is open, but not after it is closed or suspended. A session that finishes, asks for permission, or asks a question while the user is away stays silent until they reopen the app. This is most noticeable on phones paired through opencode pair, where the browser tab is usually backgrounded.

Proposal

Add opt-in Web Push from the local OpenCode server to paired browsers:

• The server stores a VAPID key and one subscription per browser installation, then sends encrypted notifications for session completion, errors, permission requests, and questions.
• The app registers a small service-worker scope under /push/<server-key>/, so it does not touch the existing offline worker.
• Settings gets a per-server Enable/Disable control, and the existing notification switches choose which events are pushed.
• No OpenCode relay or paid infrastructure: the server sends directly to the browser vendor's push service, and payloads omit prompts, code, session titles, errors, and tool output.

Related

#47479 asks for in-app web notifications; #49961 reports the Android PWA bug. This proposal is about background delivery after the app is closed or suspended.

Scope

• Local server only; it must stay awake and online.
• iOS/iPadOS requires the app to be installed on the Home Screen.
• Failed deliveries are not retried.

I have a draft implementation in #58400. I am opening this first to ask whether the approach belongs in core and whether the scope above is acceptable. I will keep the PR as a draft until there is design approval.