The home timeline draws a day divider before the first post of each UTC day, never twice #6065

Last part of #5821. It uses the day_divider module.

Done when:

• [ ] feedRegion in src/web/studio/feed.ts puts a divider before each item dividers() names, built from §5's existing parts (a muted small line). Any new class goes in docs/design/areas/feed.md, and bun scripts/design-lint.ts passes.
• [ ] The live update (/live/feed) and the ?cursor= page add no divider that is already on the page. The fragment carries the last day it drew in a data attribute, and the next one passes that day as lastDayOnPage.
• [ ] A test renders the home with posts on three days and finds the three dividers in order, plus no repeated divider in a live fragment that continues the same day.
• [ ] The preview deploys and / answers 200 signed in. #5821 died on a failed preview deploy, so the builder loads the preview once before finishing.

Risk: the feed is ordered by score, not strictly by time. A divider marks a change of day between neighbours and must never reorder posts.

Parked by the steward, 2026-10-26 (owner: "get us unclogged now then resume everything else"). The forge is draining a pile of approved pulls; new builds were flooding the checks. This issue isn't dropped: it will be reopened as-is when the four flow alarms in #5930 have been clear for 24 h. The list is kept in the steward's parked-2026-10-26 file.