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.dom.test.tsx

@@ -174,8 +174,9 @@ function mountWithClient() {
 }
 
 beforeEach(() => {
-  // Home keeps its question on the history entry (Back restores it): each test starts a fresh one.
-  window.history.replaceState(null, "");
+  // Home keeps its question on the history entry (Back restores it) and its budget in the address:
+  // each test starts a fresh one of both.
+  window.history.replaceState(null, "", "/");
   discover.mockReset();
   discoverPick.mockReset();
   setPage.mockReset();
@@ -636,6 +637,37 @@ describe("Home", () => {
     await waitFor(() => expect(discoverPick.mock.lastCall).toEqual(asked));
   });
 
+  it("names the chosen budget in the address, and the default by leaving it out", async () => {
+    mount();
+    await screen.findByText("Hero v1");
+    fireEvent.click(screen.getByText("home.budget.long"));
+    await waitFor(() => expect(window.location.search).toBe("?hb=long"));
+    fireEvent.click(screen.getByText("home.budget.any"));
+    await waitFor(() => expect(window.location.search).toBe(""));
+  });
+
+  it("opens on the budget a link names — the question's draw is not the link's", async () => {
+    window.history.replaceState(
+      { sfHome: { budget: "short", seed: 7, press: 1, seen: ["youtube:v1"] } },
+      "",
+      "/?page=home&hb=long",
+    );
+    mount();
+    await waitFor(() => expect(discover).toHaveBeenLastCalledWith("long", 0, "en"));
+    expect(discoverPick).not.toHaveBeenCalled(); // a question left under another budget is not reused
+  });
+
+  it("an address naming no budget opens on the default, over a question left under another", async () => {
+    window.history.replaceState(
+      { sfHome: { budget: "short", seed: 7, press: 1, seen: [] } },
+      "",
+      "/?page=home",
+    );
+    mount();
+    await waitFor(() => expect(discover).toHaveBeenLastCalledWith("any", 0, "en"));
+    expect(discoverPick).not.toHaveBeenCalled();
+  });
+
   // Measured in the browser (2026-09-07), before the fix: the palette's "Meglepetés" arrived at Home
   // as `start=2` — two picks deep before the page had painted once — because StrictMode invokes a
   // mount effect twice with the same render snapshot, and a reroll is not idempotent. The token is

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);

frontend/src/modules/home/module.ts

@@ -1,6 +1,7 @@
 import { Home } from "lucide-react";
 import { lazyWithPreload } from "../../lib/lazyWithPreload";
 import type { ModuleDef } from "../../lib/moduleDef";
+import { HOME_KEYS } from "../../lib/homeUrlState";
 
 // This page as the shell and the navigation know it (lib/moduleDef). Its page loads lazily.
 export const homeModule: ModuleDef = {
@@ -11,8 +12,8 @@ export const homeModule: ModuleDef = {
   icon: Home,
   Page: lazyWithPreload(() => import("./Home")),
   maxWidth: "80rem",
-  // Nothing of its own: the question is an answer, not a setting — the same question draws other
-  // cards once what you have watched moves on.
-  urlKeys: [],
+  // Its time budget, a setting. Not the question's draw: the same question draws other cards once
+  // what you have watched moves on (lib/homeUrlState).
+  urlKeys: HOME_KEYS,
   playsVideos: true,
 };