Improve: restyle the home screen and app icons on the iOS 27 kit

[SandObserver]

Summary

• Draw app icon plates with the kit's material: a centred top and bottom highlight, darker side bands and a lighter-to-darker body gradient, as percentage gradients plus a half-point hairline so they scale with the icon.
• Share ICON_R (0.26, the kit's icon grid) from utils.js for grid, dock, folder and phone icons.
• Switch the dock, folders and search pill to the kit's dock material, and draw the dock at radius 38 with 0.6 smoothing.
• Match badges to the kit's size and offset, widen the open folder's badge inset to match, and restyle the badge popover as glass.
• Put the page dots on a capsule platter, and start the pager hidden so the API error screen shows no empty bar.
• Size the phone search pill to 30 with 12 semibold text under the label shadow, and phone icon labels to 12.
• Restyle Spotlight's field and results as glass capsules with sentence-case section headings.
• Restyle the first-run password prompt as a kit alert, and the empty-state and error buttons as glass capsules.

Testing
All green, including Docker build and e2e. Updated dock, pager, Spotlight and icon-radius tests; added checks that the pager starts hidden and that phone icon corners stay derived.

Changelog

Changed

• Restyle the home screen and app icons to match iOS and iPadOS 27.

Fixed

• Hide the empty page indicator on the connection error screen.

Checklist

• Stays within the project constraints: one container, no runtime dependencies, no build step (see CONTRIBUTING.md).
• Tests added or updated for new logic, and the full suite, lint, and typechecks pass.
• Docs updated if behavior or configuration changed.