Split the web UI into a page per section

The dashboard was a single route that swapped section components via a
`section` state field. Convert it to the App Router's multi-page model so
each left-nav selection is its own route (/, /repositories, /agents,
/schedules, /approvals, /servers, /activity, /shared, /login), making the
pages modular and independently updatable.

- Move the token gate + store provider + sidebar into a persistent frame
  (AppFrame + Shell) rendered by the root layout, so auth, the polling
  loop, and shared state survive client-side navigation.
- Sidebar items are now <Link> routes; the active item and the store's
  polled section are derived from the URL (lib/nav).
- Each section gets an app/<section>/page.tsx; Runs stays at root and keeps
  its full-height split layout, the rest render in the shared scroll frame.
- Emit per-route index.html (trailingSlash) so the FastAPI StaticFiles
  mount serves clean slash-terminated URLs with no SPA rewrite.
- Regenerate the bundled static export.

Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01PsAeGVadULRzPhV2PRDttM

frontend/next.config.mjs

@@ -5,6 +5,10 @@ const nextConfig = {
   // authed API with relative paths, so there is no separate frontend server to run.
   output: "export",
   reactStrictMode: true,
+  // Emit each route as `<route>/index.html` (not `<route>.html`) so the FastAPI static
+  // mount — Starlette's StaticFiles(html=True) — serves clean, slash-terminated URLs like
+  // `/repositories/` straight from disk, with no SPA rewrite or per-route server config.
+  trailingSlash: true,
   // The export is served from disk with no image optimizer behind it.
   images: { unoptimized: true },
 };