feat(storage): app-pause flush + vault search index

Phase 6 (final storage phase).

- SidecarRepositoryRegistry tracks every open repo; SidecarFlushObserver
  (a WidgetsBindingObserver in main) flushes them all on
  inactive/hidden/paused/detached, awaiting each flush — the last
  strokes can't be lost on app close, not just on the 800ms timer.
- VaultSearchIndex rebuilds by scanning vault sidecars (the source of
  truth) — note titles, OCR text and document names — and search_provider
  queries it, so search spans notes + PDFs. Rebuilt on launch / after
  import.

The vault file-based storage migration (Phases 0-6) is complete:
annotations travel with the file, picked vault folder, atomic autosave,
one Import-file entry, SQLite migrated to sidecars. analyze clean,
tests green.

lib/storage/badnote_sidecar.dart

@@ -222,6 +222,7 @@ class BadnoteSidecar {
     List<Bookmark>? bookmarks,
     List<SidecarScratchLink>? scratchLinks,
     Map<int, String>? legacyAnnotations,
+    this.ocrText,
     this.legacyId,
   })  : strokes = strokes ?? <int, List<EditorStroke>>{},
         highlights = highlights ?? <int, List<SidecarHighlight>>{},
@@ -263,6 +264,14 @@ class BadnoteSidecar {
   /// sidecars.
   final Map<int, String> legacyAnnotations;
 
+  /// Searchable text recovered from this notebook's handwriting via local OCR
+  /// (Phase 6 search index). Persisted in the sidecar — the source of truth —
+  /// so the vault-scan search index can find handwritten notes WITHOUT the
+  /// (rebuildable, per-device) SQLite cache. Typed text already lives in the
+  /// strokes' `textContent`, so this holds ONLY the OCR'd handwriting. Null when
+  /// the notebook has no handwriting or OCR hasn't run.
+  final String? ocrText;
+
   /// The legacy SQLite row id this sidecar was migrated from (a `documents.id`
   /// or `notes.id`). Set ONLY by the one-time migration; it makes the migration
   /// idempotent (a re-run recognizes an already-migrated item by this id even if
@@ -295,6 +304,7 @@ class BadnoteSidecar {
             for (final entry in legacyAnnotations.entries)
               entry.key.toString(): entry.value,
           },
+        if (ocrText != null && ocrText!.isNotEmpty) 'ocrText': ocrText,
         if (legacyId != null) 'legacyId': legacyId,
       };
 
@@ -349,6 +359,7 @@ class BadnoteSidecar {
         }
         return out;
       }(),
+      ocrText: json['ocrText'] as String?,
       legacyId: json['legacyId'] as String?,
     );
   }