Add section grouping to shared note tabs

Notes can now be clustered into named sections in the tab bar: each tab
gets a 🏷 button to assign it to a section, and clicking a section's
label renames (or, given an empty name, ungroups) every page in it.

note_pages gains a `section` column. Since the app has no migration
framework, database.py adds the column to existing SQLite files at
startup via PRAGMA table_info + ALTER TABLE. core/notes.set_section()
keeps same-section pages contiguous by renumbering position whenever a
page's section changes, so the tab bar and the AI-facing title list
(now grouped under "## section" headings) can both treat "adjacent" as
"same group" without a separate ordering field.

Also fixes a bug in the tab rename input: ref="renameInput" sits inside
a nested v-for, which Vue 3 binds as an array rather than a single
element, so renameInput.value?.select() was throwing on every rename.

Co-Authored-By: Claude Opus 5 <noreply@anthropic.com>

backend/app/db/database.py

@@ -1,4 +1,4 @@
-from sqlalchemy import create_engine
+from sqlalchemy import create_engine, text
 from sqlalchemy.orm import sessionmaker, DeclarativeBase
 from app.core.config import DATABASE_URL
 
@@ -18,6 +18,17 @@ def get_db():
         db.close()
 
 
+def _add_missing_columns() -> None:
+    """マイグレーションフレームワークを使っていないため、既存DBに後から追加されたカラムを
+    ここで簡易的に補う。create_all() は既存テーブルへのカラム追加は行わないので必要。"""
+    with engine.connect() as conn:
+        existing = {row[1] for row in conn.execute(text("PRAGMA table_info(note_pages)"))}
+        if existing and "section" not in existing:
+            conn.execute(text("ALTER TABLE note_pages ADD COLUMN section VARCHAR"))
+            conn.commit()
+
+
 def init_db():
     from app.db import models  # noqa: F401
     Base.metadata.create_all(bind=engine)
+    _add_missing_columns()