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/home/Home.tsx

@@ -79,6 +79,8 @@ import {
 } from "../../components/ui/mediaCardStyles";
 import RemoteImage from "../../components/ui/RemoteImage";
 import { useOpenLinkedPlayer } from "../../features/player/useOpenLinkedPlayer";
+import { budgetFrom, budgetParams, DEFAULT_BUDGET } from "../../lib/homeUrlState";
+import { useUrlMirror } from "../../lib/useUrlMirror";
 
 // The Home surface — "Mit nézzek most?".
 //
@@ -701,6 +703,15 @@ function readSavedQuestion(): SavedQuestion | null {
   };
 }
 
+/** The question the page opens with: the budget its address names (a link wins, and an address
+ *  naming none means the default), with the rest of what the entry was left with only when that was
+ *  asked under the same budget. */
+function openingQuestion(): { budget: DiscoverBudget; saved: SavedQuestion | null } {
+  const budget = budgetFrom(new URLSearchParams(window.location.search)) ?? DEFAULT_BUDGET;
+  const saved = readSavedQuestion();
+  return { budget, saved: saved?.budget === budget ? saved : null };
+}
+
 export default function Home() {
   const { t, i18n } = useTranslation();
   const qc = useQueryClient();
@@ -710,8 +721,10 @@ export default function Home() {
   // The question on screen lives on this page's history ENTRY as well: opening a card leaves Home
   // (it unmounts), and Back used to come back to "Doesn't matter" and a different hero — whose card
   // then left the Continue rail, so the keyboard landed on its neighbour (prod UAT 2026-09-19).
-  const [saved] = useState(readSavedQuestion);
-  const [budget, setBudget] = useState<DiscoverBudget>(saved?.budget ?? "any");
+  // The budget alone is also in the address (lib/homeUrlState): a setting, unlike the draw under it.
+  const [{ budget: openedBudget, saved }] = useState(openingQuestion);
+  const [budget, setBudget] = useState<DiscoverBudget>(openedBudget);
+  useUrlMirror(budgetParams(budget));
   // 0 = "the hero the page loaded with". Any other value is a reroll, answered by its own query so
   // pressing the button does not refetch the rails under the user's cursor.
   const [seed, setSeed] = useState(saved?.seed ?? 0);