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/routes/search/+page.svelte

@@ -5,15 +5,41 @@
   import Search from "$lib/components/Search.svelte";
   import SearchResults from "$lib/components/search/SearchResults.svelte";
   import SearchScopeChips from "$lib/components/search/SearchScopeChips.svelte";
-  import { resolveSearchScope, type SearchScope } from "$lib/utils/searchScope";
+  import { resolveSearchScope, SEARCH_SCOPES, type SearchScope } from "$lib/utils/searchScope";
   import type { MediaItem } from "$lib/api/types";
 
-  let searchQuery = $state("");
+  // `?q=` / `?scope=` seed the page so the desktop header search bar can hand
+  // a query over by navigating here — /search is the only surface that renders
+  // results, so every other search affordance routes into it.
+  // TRACES: UR-049 | DR-063
+  const initialQuery = $page.url.searchParams.get("q") ?? "";
+  const initialScope = $page.url.searchParams.get("scope");
+
+  let searchQuery = $state(initialQuery);
 
   // Route resolves the *initial* scope only. Deriving it reactively would snap
   // a user who widened to All back to the route's scope on any navigation.
   // TRACES: UR-049 | DR-064
-  let scope = $state<SearchScope>(resolveSearchScope($page.url.pathname));
+  let scope = $state<SearchScope>(
+    SEARCH_SCOPES.includes(initialScope as SearchScope)
+      ? (initialScope as SearchScope)
+      : resolveSearchScope($page.url.pathname)
+  );
+
+  // A query arriving in the URL must actually run — mounting with a seeded
+  // input alone would render the empty state with a filled box.
+  $effect(() => {
+    const q = $page.url.searchParams.get("q") ?? "";
+    if (!q.trim()) return;
+    const urlScope = $page.url.searchParams.get("scope");
+    const nextScope = SEARCH_SCOPES.includes(urlScope as SearchScope)
+      ? (urlScope as SearchScope)
+      : "all";
+    if (q === $library.searchQuery && nextScope === scope) return;
+    searchQuery = q;
+    scope = nextScope;
+    library.search(q, nextScope);
+  });
 
   async function handleSearch(query: string) {
     if (query.trim()) {