## Summary
- add an opt-in, device-local PWA app-icon badge driven only by the authoritative unread Updates total
- expose the preference in My Work settings and the mobile Device Setup flow
- preserve offline truthfulness, suppress duplicate Badging API calls, and report platform failures without disrupting live refresh
- package and precache the controller within the mandatory mobile workspace bundle
## Verification
- `pytest -q` — 2407 passed, 12 skipped
- focused controller/setup/bundle/service-worker tests — 118 passed
- Chromium at 390×844 — both badge controls are 44px high with no horizontal overflow
Closes #1278
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Summary
Verification
pytest -q— 2407 passed, 12 skippedCloses #1278