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(