fix(search): route the library header search to /search

Typing in the desktop header search bar ran library.search() in place and
relied on /library rendering the results inline. On every other /library/**
route nothing rendered them, so the search bar looked broken: results were
fetched and never shown.

Make /search the single surface that renders results. The header bar becomes a
navigator — it hands the query and route-derived scope to /search via ?q= and
?scope=, which seed the page and run the search on arrival. The inline result
block and the header's scope chips are removed; the chips live on /search,
which owns the results. The empty `all` scope is omitted from the URL, and
typing while already on /search does not push a history entry per keystroke.

src/lib/utils/searchScope.test.ts

@@ -7,6 +7,8 @@ import {
   normalizeGroupOrder,
   reorderGroups,
   resolveSearchScope,
+  searchRouteUrl,
+  shouldNavigateToSearch,
   scopeItemTypes,
   type SearchGroupId,
 } from "./searchScope";
@@ -370,4 +372,42 @@ describe("reorderGroups", () => {
     expect(reorderGroups(order, -1, 2)).toEqual(order);
     expect(reorderGroups(order, 0, 9)).toEqual(order);
   });
-});
+});
+
+describe("searchRouteUrl", () => {
+  it("encodes the query and the scope", () => {
+    expect(searchRouteUrl("miles davis", "music")).toBe("/search?q=miles%20davis&scope=music");
+  });
+
+  it("omits the scope key for the default `all` scope", () => {
+    expect(searchRouteUrl("dune", "all")).toBe("/search?q=dune");
+  });
+
+  it("targets bare /search for an empty query so the page shows its empty state", () => {
+    expect(searchRouteUrl("", "all")).toBe("/search");
+    expect(searchRouteUrl("   ", "music")).toBe("/search");
+  });
+});
+
+describe("shouldNavigateToSearch", () => {
+  it("navigates from any library page, which cannot render results itself", () => {
+    // The bug: the header search bar shows on every /library/** route but only
+    // /library rendered $library.searchResults, so typing did nothing on
+    // /library/music, /library/tv, /library/movies and detail pages.
+    expect(shouldNavigateToSearch("/library", "jazz")).toBe(true);
+    expect(shouldNavigateToSearch("/library/music", "jazz")).toBe(true);
+    expect(shouldNavigateToSearch("/library/tv", "jazz")).toBe(true);
+    expect(shouldNavigateToSearch("/library/movies", "jazz")).toBe(true);
+    expect(shouldNavigateToSearch("/library/abc123", "jazz")).toBe(true);
+  });
+
+  it("stays put when already on /search, so typing does not re-push history", () => {
+    expect(shouldNavigateToSearch("/search", "jazz")).toBe(false);
+    expect(shouldNavigateToSearch("/search?q=old", "jazz")).toBe(false);
+  });
+
+  it("does not navigate on an empty query", () => {
+    expect(shouldNavigateToSearch("/library/music", "")).toBe(false);
+    expect(shouldNavigateToSearch("/library/music", "  ")).toBe(false);
+  });
+});