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/library/+layout.svelte

@@ -6,8 +6,12 @@
   import { library } from "$lib/stores/library";
   import { useScrollGuard } from "$lib/composables/useScrollGuard";
   import Search from "$lib/components/Search.svelte";
-  import SearchScopeChips from "$lib/components/search/SearchScopeChips.svelte";
-  import { resolveSearchScope, type SearchScope } from "$lib/utils/searchScope";
+  import {
+    resolveSearchScope,
+    searchRouteUrl,
+    shouldNavigateToSearch,
+    type SearchScope,
+  } from "$lib/utils/searchScope";
   import AppHeader from "$lib/components/AppHeader.svelte";
   import BottomUi from "$lib/components/BottomUi.svelte";
   import SleepTimerModal from "$lib/components/player/SleepTimerModal.svelte";
@@ -48,18 +52,22 @@
     }
   });
 
+  // The header bar is a *navigator*, not a second results surface: /search is
+  // the only route that renders searchResults, so searching here routes there
+  // with the query + route-derived scope in the URL. Previously this ran
+  // library.search() in place, which was invisible on every /library/** page
+  // except /library itself.
+  // TRACES: UR-049 | DR-063
   async function handleSearch(query: string) {
-    if (query.trim()) {
-      await library.search(query, searchScope);
-    } else {
+    if (!query.trim()) {
       library.clearSearch();
+      return;
     }
-  }
-
-  async function handleScopeChange(next: SearchScope) {
-    searchScope = next;
-    if (searchQuery.trim()) {
-      await library.search(searchQuery, next);
+    if (shouldNavigateToSearch($page.url.pathname, query)) {
+      await goto(searchRouteUrl(query, searchScope));
+      // The query now lives in the URL; clear the header input so returning to
+      // a library page does not leave a stale term sitting in the box.
+      searchQuery = "";
     }
   }
 </script>
@@ -74,14 +82,12 @@
     <AppHeader search={librarySearch} />
 
     {#snippet librarySearch()}
+      <!-- Scope chips live on /search, which owns the results. -->
       <Search
         bind:value={searchQuery}
         placeholder="Search your library..."
         onSearch={handleSearch}
       />
-      {#if searchQuery.trim()}
-        <SearchScopeChips scope={searchScope} onChange={handleScopeChange} />
-      {/if}
     {/snippet}
 
     <!-- Main content. The BottomUi below is an in-flow flex sibling, so this

src/routes/library/+page.svelte

@@ -12,8 +12,9 @@
   // Scroll guard from layout - prevents accidental taps during scrolling (Android)
   const scrollGuard = getContext<ReturnType<typeof useScrollGuard>>("scrollGuard");
 
-  let searchResults = $derived($library.searchResults);
-  let searchQuery = $derived($library.searchQuery);
+  // Search results are rendered exclusively by /search — this page used to
+  // render them inline, which made the header search bar appear broken on every
+  // other /library/** route. TRACES: UR-049 | DR-063
 
   const isMusicLibrary = $derived($currentLibrary?.collectionType === "music");
 
@@ -169,28 +170,7 @@
 </script>
 
 <div class="space-y-8">
-  {#if searchQuery}
-    <!-- Search results -->
-    <div>
-      <div class="flex items-center justify-between mb-4">
-        <h1 class="text-2xl font-bold text-white">
-          Search results for "{searchQuery}"
-        </h1>
-        <button
-          onclick={() => library.clearSearch()}
-          class="text-sm text-gray-400 hover:text-white"
-        >
-          Clear search
-        </button>
-      </div>
-
-      <LibraryGrid
-        items={searchResults}
-        loading={$isLibraryLoading}
-        onItemClick={handleItemClick}
-      />
-    </div>
-  {:else if showInlineLibraryContent}
+  {#if showInlineLibraryContent}
     <!-- Library content (live TV / channels / other inline-rendered types) -->
     <div class="space-y-6">
       <div class="flex items-center gap-4">

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()) {