feat(nav): the live YouTube search and Home's budget are named in the address

A YouTube search is `?yt=<term>` (+ `ytn` for a non-default count), so a reload, a link
or Back/Forward reopens it. It runs by itself only where it costs no quota: the free page
scrape, a search asked for in this document, or results still cached. An API search
reopened any other way shows its term and waits for "Search YouTube again".

Home's time budget is `hb`; the question's draw stays on the history entry.

An e2e spec opens every page's bookmark in a fresh browser context; CLAUDE.md and
docs/development.md name the module URL codec as the place for view state.

frontend/src/moduleUrl.contract.test.ts

@@ -16,6 +16,8 @@ import { plexParamUpdates, plexSearchParamUpdates } from "./lib/plexUrlState";
 import { activityParamUpdates, DEFAULT_ACTIVITY_STATE } from "./lib/activityUrlState";
 import { DEFAULT_LOGS_STATE, logsParamUpdates } from "./lib/logsUrlState";
 import { FILES_KEYS } from "./lib/filesUrlState";
+import { ytSearchParams } from "./lib/ytSearchUrlState";
+import { budgetParams } from "./lib/homeUrlState";
 import { PLAYER_KEYS, playerParamUpdates } from "./lib/playerUrlState";
 import { playlistViewParamUpdates } from "./lib/playlistsUrlState";
 import { SETTINGS_TAB_KEY } from "./lib/settingsTab";
@@ -43,7 +45,8 @@ const FEED: FeedFilters = {
 // What each module's own encoders write, from the encoders themselves. Every module that declares a
 // key has a row, so a new one cannot declare keys its codec does not write, or the other way round.
 const WRITES: Partial<Record<Page, () => string[]>> = {
-  feed: () => Object.keys(filterParamUpdates(FEED)),
+  feed: () => [...Object.keys(filterParamUpdates(FEED)), ...Object.keys(ytSearchParams("x", 40))],
+  home: () => Object.keys(budgetParams("short")),
   channels: () =>
     Object.keys(channelsParamUpdates({ view: "subscribed", sort: null, filters: {}, search: "" })),
   plex: () => [