feat(note): page background templates (rnote-style)

A blank note can show a page-background template painted behind the
ink, picked from the toolbar and persisted per notebook.

- NoteBackground: blank / dots / ruled / grid / cornell, drawn in
  page space (scales with zoom), subtle grey. Cornell = left margin +
  bottom summary rule over a ruled body.
- Stored as the enum name in the notebook sidecar (back-compat:
  missing/unknown -> blank), saved/loaded via SidecarRepository so it
  restores on reopen.
- Picker added to the note tool palette.

PDF backgrounds skipped (PDFs have their own page content). analyze
clean, 386 tests green.

test/badnote_sidecar_test.dart

@@ -231,6 +231,22 @@ void main() {
     expect(reparsed.strokes[0]!.single.id, 's0');
   });
 
+  test('background name round-trips through the sidecar', () {
+    final original = BadnoteSidecar(docType: 'notebook', background: 'cornell');
+    final reparsed = _roundTrip(original);
+    expect(reparsed.background, 'cornell');
+    // Omitted when null (no key bloat for blank/legacy sidecars).
+    expect(BadnoteSidecar().toJson().containsKey('background'), isFalse);
+  });
+
+  test('missing background decodes to null (back-compat → blank)', () {
+    final json =
+        jsonDecode(jsonEncode(BadnoteSidecar().toJson())) as Map<String, dynamic>;
+    expect(json.containsKey('background'), isFalse);
+    final reparsed = BadnoteSidecar.fromJson(json);
+    expect(reparsed.background, isNull);
+  });
+
   test('missing scratchpad defaults to 4000x4000 empty pad', () {
     final json = SidecarScratchLink(
       link: const ScratchLink(

test/note_background_test.dart

@@ -0,0 +1,41 @@
+// test/note_background_test.dart
+//
+// Covers the NoteBackground template enum decode (back-compat: missing/unknown →
+// blank) and the painter's shouldRepaint reacting to a background change.
+
+import 'package:flutter_test/flutter_test.dart';
+
+import 'package:badnote/editor/canvas/note_background.dart';
+
+void main() {
+  group('noteBackgroundFromName', () {
+    test('decodes each enum name round-trip', () {
+      for (final b in NoteBackground.values) {
+        expect(noteBackgroundFromName(b.name), b);
+      }
+    });
+
+    test('null → blank (back-compat for legacy sidecars)', () {
+      expect(noteBackgroundFromName(null), NoteBackground.blank);
+    });
+
+    test('unknown name → blank (forward-compat)', () {
+      expect(noteBackgroundFromName('isometric'), NoteBackground.blank);
+      expect(noteBackgroundFromName(''), NoteBackground.blank);
+    });
+  });
+
+  group('NoteBackgroundPainter.shouldRepaint', () {
+    test('repaints when the background changes', () {
+      const a = NoteBackgroundPainter(NoteBackground.blank);
+      const b = NoteBackgroundPainter(NoteBackground.dots);
+      expect(a.shouldRepaint(b), isTrue);
+    });
+
+    test('does not repaint when the background is identical', () {
+      const a = NoteBackgroundPainter(NoteBackground.grid);
+      const b = NoteBackgroundPainter(NoteBackground.grid);
+      expect(a.shouldRepaint(b), isFalse);
+    });
+  });
+}