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/modules/feed/module.ts

@@ -2,6 +2,7 @@ import { YouTubeIcon } from "../../lib/brandIcons";
 import { lazyWithPreload, lazyWithPreloadAlongside } from "../../lib/lazyWithPreload";
 import type { ModuleDef } from "../../lib/moduleDef";
 import { FEED_KEYS } from "../../lib/urlState";
+import { YT_SEARCH_KEYS } from "../../lib/ytSearchUrlState";
 import { FeedStore } from "./FeedStore";
 
 // The rail loads with the page (lazyWithPreloadAlongside), so it never arrives after it.
@@ -16,7 +17,8 @@ export const feedModule: ModuleDef = {
   Rail,
   Page: lazyWithPreloadAlongside(() => import("./FeedPage"), Rail),
   Store: FeedStore,
-  urlKeys: FEED_KEYS,
+  // Its filters, and the live YouTube search shown in its place (lib/ytSearchUrlState).
+  urlKeys: [...FEED_KEYS, ...YT_SEARCH_KEYS],
   playsVideos: true,
 };