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/services/ocr_service.dart

@@ -1,3 +1,6 @@
+import 'dart:io';
+
+import '../editor/persistence/sidecar_repository.dart';
 import '../models/note.dart';
 import '../models/pen_tool.dart';
 import 'database_service.dart';
@@ -6,7 +9,12 @@ import 'stroke_rasterizer.dart';
 
 /// Runs OCR locally: typed text from strokes + handwriting via platform OCR.
 class OcrService {
-  /// Extract searchable text from [note] and merge into the local FTS index.
+  /// Extract searchable text from [note] and persist it for search. The text is
+  /// written to the note's `*.badnote.json` sidecar `ocrText` field — the
+  /// vault-scan source of truth the Phase 6 search index reads — and also
+  /// appended to the (rebuildable) SQLite FTS cache so legacy callers keep
+  /// working. [note.id] is the synthetic note path, which is exactly the
+  /// `sourceFilePath` the editor opened its [SidecarRepository] with.
   Future<void> processNote(Note note) async {
     final parts = <String>[];
 
@@ -40,6 +48,22 @@ class OcrService {
     final combined = parts.join(' ').trim();
     if (combined.isEmpty) return;
 
+    // Persist into the note's sidecar so the vault-scan search index finds it.
+    // Prefer the editor's already-open repo (same in-memory sidecar — no race);
+    // if the note is closed, open/flush/dispose a transient handle.
+    final open = SidecarRepositoryRegistry.forPath(note.id);
+    if (open != null) {
+      open.scheduleOcrTextSave(combined);
+      await open.flush();
+    } else if (await File('${note.id}$kSidecarSuffix').exists()) {
+      final repo = await SidecarRepository.open(note.id, docType: 'notebook');
+      repo.scheduleOcrTextSave(combined);
+      await repo.flush();
+      repo.dispose();
+    }
+
+    // Also keep the legacy SQLite FTS cache warm (rebuildable; not the source
+    // of truth). Harmless if the row is never read.
     final db = await DatabaseService.getInstance();
     await db.appendOcrToFts(note.id, combined);
   }