feat(ui): display current git branch in browser tab, header, and nav tabs

- Add branch indicator in browser tab title ([test-gay] GaC-Srtool...)
- Add dynamic tab switching updates preserving branch prefix in browser title
- Add branch badge in brand header and right-aligned branch pill in navigation tabs
- Add scripts/sync-branch.js and version.json for automated branch detection

version.json

@@ -0,0 +1,5 @@
+{
+  "branch": "test-gay",
+  "version": "1.0.0",
+  "updatedAt": "2026-10-17T04:56:43.360Z"
+}