feat(navigation): persist last-port for next-login + root → /dashboard

Login routing previously always landed at the user's first port-role.
With a multi-port operator (super-admins, multi-tenant ops) the active
port reverted on every login, breaking the "I was working in X
yesterday" continuity.

- PortProvider PATCHes `/api/v1/me` with `preferences.defaultPortId =
  currentPort.id` whenever the active port changes (URL or explicit
  switch). Ref-keyed dedupe; fire-and-forget so navigation isn't
  blocked by a transient PATCH failure.
- UserMenu's port-switcher also writes the preference on click so the
  preference is captured even for users who never re-render through
  PortProvider.
- /dashboard resolver checks `preferences.defaultPortId` first, falling
  back to first-port-by-name (super-admin) or first-role (everyone
  else). The preference is verified against current access before being
  honoured — a stale id from a revoked role or archived port can't
  strand the user on a 403.
- Add /src/app/page.tsx that redirects `/` → `/dashboard` so the
  middleware's `redirect=/` post-login parameter doesn't dump users on
  an empty 404. The existing /dashboard handler then routes them on to
  their resolved port.
- UserMenu sign-out: replace `router.push('/api/auth/sign-out')` (which
  issued a GET against better-auth's POST-only endpoint, causing Safari
  and Comet/Arc to land the JSON response as a `sign_out` download)
  with `signOut()` from the auth client + an explicit redirect.

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>

src/providers/port-provider.tsx

@@ -1,9 +1,10 @@
 'use client';
 
-import { createContext, useContext, useEffect, type ReactNode } from 'react';
+import { createContext, useContext, useEffect, useRef, type ReactNode } from 'react';
 import { useParams } from 'next/navigation';
 
 import { useUIStore } from '@/stores/ui-store';
+import { apiFetch } from '@/lib/api/client';
 import type { Port } from '@/lib/db/schema/ports';
 
 interface PortContextValue {
@@ -47,6 +48,26 @@ export function PortProvider({ children, ports, defaultPortId }: PortProviderPro
     }
   }, [currentPort, currentPortId, currentPortSlug, setPort]);
 
+  // Remember the last port the user landed on (URL-derived or
+  // explicit-switch) so the next login routes here automatically. Tracked
+  // in a ref-keyed dedupe so we only PATCH when the active port actually
+  // changes — re-renders inside the same port don't write. Fire-and-forget;
+  // a transient network failure shouldn't block navigation, and the
+  // post-login resolver verifies access so a stale value can't strand the
+  // user on a 403.
+  const lastPersistedPortIdRef = useRef<string | null>(null);
+  useEffect(() => {
+    if (!currentPort) return;
+    if (lastPersistedPortIdRef.current === currentPort.id) return;
+    lastPersistedPortIdRef.current = currentPort.id;
+    void apiFetch('/api/v1/me', {
+      method: 'PATCH',
+      body: { preferences: { defaultPortId: currentPort.id } },
+    }).catch(() => {
+      /* silent — best-effort */
+    });
+  }, [currentPort]);
+
   return (
     <PortContext.Provider
       value={{