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/editor/persistence/sidecar_flush_observer.dart

@@ -0,0 +1,58 @@
+// lib/editor/persistence/sidecar_flush_observer.dart
+//
+// Phase 6 / §F.3 of the file-based storage plan (docs/plans/2026-06-24-file-
+// based-storage.md): app-lifecycle flush hardening.
+//
+// The per-file SidecarRepository debounces writes by 800 ms. That window is the
+// data-loss gap on a Windows tablet: if the OS suspends or closes the app
+// before the timer fires, the last strokes never reach disk. This observer
+// listens for the app leaving the foreground and DRAINS every open repo's
+// pending write before the process can be frozen, so "never lose the last
+// strokes on app close" holds even when the editor's own dispose() doesn't run.
+//
+// Registered once in BadNoteApp; it delegates to
+// [SidecarRepositoryRegistry.flushAll], which awaits every repo's flush().
+
+import 'package:flutter/widgets.dart';
+
+import 'sidecar_repository.dart';
+
+/// A [WidgetsBindingObserver] that flushes all open sidecar repositories when
+/// the app leaves the foreground (`inactive`/`paused`/`detached`/`hidden`).
+class SidecarFlushObserver with WidgetsBindingObserver {
+  /// Whether the observer is currently registered with the binding.
+  bool get isAttached => _attached;
+  bool _attached = false;
+
+  /// Register with [WidgetsBinding.instance] so lifecycle changes are observed.
+  void attach() {
+    if (_attached) return;
+    WidgetsBinding.instance.addObserver(this);
+    _attached = true;
+  }
+
+  /// Stop observing lifecycle changes.
+  void detach() {
+    if (!_attached) return;
+    WidgetsBinding.instance.removeObserver(this);
+    _attached = false;
+  }
+
+  @override
+  void didChangeAppLifecycleState(AppLifecycleState state) {
+    switch (state) {
+      // Any transition out of the foreground is a potential suspend/kill point:
+      // drain pending sidecar writes now (the editors' own dispose() may never
+      // run when the OS freezes the process).
+      case AppLifecycleState.inactive:
+      case AppLifecycleState.hidden:
+      case AppLifecycleState.paused:
+      case AppLifecycleState.detached:
+        // Fire-and-forget at the framework boundary, but each write is atomic
+        // and awaited inside flushAll, so a half-written sidecar is impossible.
+        SidecarRepositoryRegistry.flushAll();
+      case AppLifecycleState.resumed:
+        break;
+    }
+  }
+}