fix: vault home recursive + backup viewer fixes + AI status at startup

- Vault home (#12): recursive file listing (rglob), rel_dir in metadata,
  'recursif' badge, updated roadmap description
- Backup viewer: fix close button (inline onclick + event delegation
  fallback for text nodes), improve error messages
- Backup API: remove response_model to prevent Pydantic 500 errors,
  add try/except + logging on backups/diff/restore endpoints,
  per-entry error handling in _list_backup_files,
  encoding safety (errors='replace') on read_text
- AI status: call /api/ai/status at startup in _initAIStatus
  instead of waiting for user click on sync badge

frontend/js/sync.js

@@ -1,6 +1,7 @@
 /* ObsiGate — Sync: SSE client + PWA registration */
 import { state } from './state.js';
 import { showToast } from './ui.js';
+import { api } from './auth.js';
 import { loadVaults, loadTags, refreshTagsForContext, refreshSidebarTreePreservingState } from './sidebar.js';
 import { loadRecentFiles } from './config.js';
 import { openFile } from './viewer.js';
@@ -339,16 +340,16 @@ function _renderSyncPanel(panel) {
 let _aiProcessing = false;
 let _aiConfigured = null; // null = unknown, true/false
 
-/** Check if AI is configured by looking for the AI toolbar in the DOM.
- *  The toolbar is only rendered when the AI API responds successfully. */
-function _checkAIStatus() {
-  const container = document.getElementById('ai-toolbar-container');
-  if (container && container.children.length > 0) {
-    _aiConfigured = true;
-  } else if (_aiConfigured === null) {
-    // Keep unknown state until we have evidence
-    _aiConfigured = null;
+/** Check if AI is configured by calling the /api/ai/status endpoint.
+ *  Updates _aiConfigured and refreshes the panel if open. */
+async function _checkAIStatus() {
+  try {
+    const status = await api('/api/ai/status');
+    _aiConfigured = status.configured === true;
+  } catch {
+    _aiConfigured = false;
   }
+  _updateAIStatusInPanel();
 }
 
 function _renderAIStatusSection(panel) {
@@ -420,6 +421,9 @@ function _setBadgeAIProcessing(active) {
 
 /** Initialize AI status tracking. */
 function _initAIStatus() {
+  // Check AI status immediately at startup
+  _checkAIStatus();
+
   // Listen for AI processing events from ai.js
   document.addEventListener('ai-processing-start', () => {
     _aiProcessing = true;
@@ -433,13 +437,11 @@ function _initAIStatus() {
     _updateAIStatusInPanel();
   });
 
-  // Check AI status on first sync panel open
+  // Re-check AI status when sync panel opens (in case config changed)
   const badge = document.getElementById('sync-badge');
   if (badge) {
-    const origClick = badge.onclick;
     badge.addEventListener('click', () => {
       _checkAIStatus();
-      _updateAIStatusInPanel();
     }, { once: false });
   }
 }