Built-in browser Home button

Add a Home entry to the OpenProgram built-in browser toolbar, reusing the existing new-tab page without introducing a configurable home URL.

Behavior

Both the desktop native WebContentsView toolbar and the ordinary browser iframe toolbar display a House icon button.
The button title and accessible name follow the current language: Home or Home.
Clicking replaces the current web tab in place with the existing ntp page; it does not add a tab or navigate to an external URL.
Preserve tab order, active state and compound-tab member position; remove the old web tab id and clear legacy split state that references it.

Boundaries

Do not add custom home settings, search-engine settings, dependencies, background processes or a new page implementation. Back, forward, address input, reload, bookmarks, split view and external opening remain unchanged.

Task files

Production code:apps/web/components/center-tabs/web-tab-pane.tsx, apps/web/lib/tabs/center-tabs-store.ts. Tests:apps/web/scripts/runtime/check-web-split.mjs. Design and acceptance records: this file.

Acceptance

RED: the existing code lacks the store action, House button and wiring for both toolbars.
GREEN: state tests demonstrate in-place replacement, group updates and native web-view id cleanup; source checks verify Home buttons in both toolbars.
After building, use only the installed /Applications/OpenProgram.app for visible page acceptance; do not start Electron from source.

Implementation record

Implemented and passed automated checks and installed-App acceptance.

All Web checks, TypeScript checks and the production build passed.
The installed /Applications/OpenProgram.app starts the default worker with its bundled Python runtime; Electron was not started from source.
DOM/ARIA acceptance confirms exactly one Home button in the toolbar;title and the accessible name are correct, and the address bar displays https://example.com/.
Clicking Home displays the existing New tab page, preserves the tab count and removes the original web tab's CDP target.
One screenshot of the current viewport confirms button placement and visual consistency; OCR, candidate-box detection and repeated screenshots were not used.