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.ts

@@ -62,6 +62,42 @@ export function resolveSearchScope(pathname: string): SearchScope {
   return "all";
 }
 
+/**
+ * The URL of the single search surface for a query + scope.
+ *
+ * `/search` is the *only* route that renders results, so every other search
+ * affordance (the desktop header bar) is a navigator to this URL rather than a
+ * second result renderer. The `all` scope is the page's own default, so it is
+ * omitted to keep shared/back-navigated URLs clean.
+ *
+ * TRACES: UR-049 | DR-063
+ */
+export function searchRouteUrl(query: string, scope: SearchScope): string {
+  const trimmed = query.trim();
+  if (!trimmed) return "/search";
+
+  const params = new URLSearchParams({ q: trimmed });
+  if (scope !== "all") params.set("scope", scope);
+  // URLSearchParams renders spaces as "+", valid in a query but noisier to
+  // read; %20 is equally valid and matches how the app builds other links.
+  return `/search?${params.toString().replace(/\+/g, "%20")}`;
+}
+
+/**
+ * Whether a search typed on `pathname` must navigate to `/search` to be seen.
+ *
+ * True for every route except `/search` itself: no other page renders
+ * `searchResults`, so a search performed there is invisible. Guarding on
+ * `/search` keeps typing from pushing a history entry per keystroke.
+ *
+ * TRACES: UR-049 | DR-063
+ */
+export function shouldNavigateToSearch(pathname: string, query: string): boolean {
+  if (!query.trim()) return false;
+  const path = pathname.split(/[?#]/)[0].replace(/\/+$/, "") || "/";
+  return path !== "/search";
+}
+
 // ---------------------------------------------------------------------------
 // Result groups
 // ---------------------------------------------------------------------------