Cache and centralize UT Direct login-state detection

[spotta85]

Problem

We dont know if the user is logged in. We can check but thats slow
#178

Solution

Explicity manage this stuff and cache for performance :)

flowchart LR
    subgraph login-state.ts [login-state.ts — single owner]
        cache[(local:utdLoggedIn<br/>true / false / null)]
        probe[isLoggedIn probe fetch]
    end

    CS[Content script<br/>recordLoginStateFromPage] -->|DOM: login form?| cache
    CK[Cookie watcher<br/>sessionid-degree-audits-production] -->|removed → false<br/>set → verify| cache
    BG[Background runNewAudit] -->|refreshLoginState guard| probe
    POP[Popup] -->|getCachedLoginState → instant paint| cache
    POP -->|refreshLoginState → verify| probe
    cache -->|watchLoginState → live updates| POP
    probe --> cache

Loading

The interface (everything else — probe URL, cookie name, storage key, redirect semantics — is private):

Export	Used by	Purpose
getCachedLoginState()	popup	instant paint (ms, may be stale)
refreshLoginState()	popup, background	live truth + cache write
watchLoginState(cb)	popup	react to writes from other contexts
recordLoginStateFromPage(doc)	content script	definitive DOM signal on real UT pages
registerSessionCookieWatcher()	background	instant invalidation when the session cookie dies
openLoginTab()	popup, background	the one way to send a user to log in

Behavior

You can basically tell with certainty if user is not logged in or what.
Use the formatting described in teh code -> first check, then do the refresh with the auxillary functions

If the session expires adn coookie not deleted, we kinda fried but tahts why you ahve to check first before doing any action that requires auth.

[linear]

DAP-99