feat: Implement cross-tab authentication synchronization in AuthProvider

- Add BroadcastChannel API for real-time logout synchronization across tabs
- Implement storage event listener as fallback for older browsers
- When a user logs out in one tab, all other tabs immediately reflect the logout state
- Update tests to verify storage event and BroadcastChannel behavior
- Update Architecture.md to document cross-tab synchronization
- Update Web-Development.md with authentication state management notes

The provider now broadcasts logout messages to other tabs so they immediately
reflect the logout state without requiring a page refresh or additional API calls.
The implementation uses BroadcastChannel as the primary sync mechanism with
storage events as a fallback for older browsers.

Co-authored-by: Copilot <223556219+Copilot@users.noreply.github.com>

Docs/Tasks.md

@@ -1,36 +1,3 @@
-## [Backend] `re` module imported inside function body
-
-**Where found**
-
-- `backend/app/main.py:198-199` — `import re` inside `_get_error_code()`
-
-**Why this is needed**
-
-Importing inside a function is a code smell. Standard practice is to import modules at the top of the file.
-
-**Goal**
-
-Move `import re` to the module-level imports at the top of `main.py`.
-
-**What to do**
-
-1. Add `import re` to existing imports section at top of `backend/app/main.py`
-2. Remove `import re` line from inside `_get_error_code()`
-
-**Possible traps and issues**
-
-- None — straightforward refactoring with no behavioral change
-
-**Docs changes needed**
-
-- No documentation changes needed
-
-**Doc references**
-
-- `backend/app/main.py`
-
----
-
 ## [Frontend] AuthProvider sessionStorage not synchronized across tabs
 
 **Where found**