fix(backend): enforce gapless 1..N reindexing normalization across research areas CRUD

- Implement normalizeResearchAreasOrder inside transactions for Create, Update, and Delete operations
- Prevent duplicate or corrupted display_order overlaps when inserting/moving items
- Clean up existing database display_order sequence
- Pass all unit tests and E2E test gates cleanly

backend/internal/repository/home.go

@@ -126,6 +126,35 @@ func (r *HomeRepository) GetResearchAreas(ctx context.Context) ([]models.Researc
 	return areas, rows.Err()
 }
 
+func (r *HomeRepository) normalizeResearchAreasOrder(ctx context.Context, tx *sql.Tx) error {
+	rows, err := tx.QueryContext(ctx, `SELECT id FROM research_areas ORDER BY display_order ASC, id ASC`)
+	if err != nil {
+		return fmt.Errorf("query areas for normalization: %w", err)
+	}
+	defer rows.Close()
+
+	var ids []int
+	for rows.Next() {
+		var id int
+		if err := rows.Scan(&id); err != nil {
+			return fmt.Errorf("scan area id: %w", err)
+		}
+		ids = append(ids, id)
+	}
+	if err := rows.Err(); err != nil {
+		return err
+	}
+
+	for idx, id := range ids {
+		newOrder := idx + 1
+		_, err := tx.ExecContext(ctx, `UPDATE research_areas SET display_order = ? WHERE id = ?`, newOrder, id)
+		if err != nil {
+			return fmt.Errorf("update area %d to order %d: %w", id, newOrder, err)
+		}
+	}
+	return nil
+}
+
 func (r *HomeRepository) CreateResearchArea(ctx context.Context, a models.ResearchArea) (int, error) {
 	tx, err := r.db.BeginTx(ctx, nil)
 	if err != nil {
@@ -161,12 +190,20 @@ func (r *HomeRepository) CreateResearchArea(ctx context.Context, a models.Resear
 		return 0, fmt.Errorf("insert research area: %w", err)
 	}
 
+	id, err := res.LastInsertId()
+	if err != nil {
+		return 0, fmt.Errorf("get inserted area id: %w", err)
+	}
+
+	if err := r.normalizeResearchAreasOrder(ctx, tx); err != nil {
+		return 0, fmt.Errorf("normalize display_order: %w", err)
+	}
+
 	if err := tx.Commit(); err != nil {
 		return 0, fmt.Errorf("commit tx: %w", err)
 	}
 
-	id, err := res.LastInsertId()
-	return int(id), err
+	return int(id), nil
 }
 
 func (r *HomeRepository) UpdateResearchArea(ctx context.Context, id int, a models.ResearchArea) error {
@@ -202,6 +239,8 @@ func (r *HomeRepository) UpdateResearchArea(ctx context.Context, id int, a model
 		if err != nil {
 			return fmt.Errorf("shift research areas display_order on update: %w", err)
 		}
+	} else if a.DisplayOrder <= 0 {
+		a.DisplayOrder = oldOrder
 	}
 
 	res, err := tx.ExecContext(ctx, `
@@ -216,11 +255,22 @@ func (r *HomeRepository) UpdateResearchArea(ctx context.Context, id int, a model
 	if err == nil && rows == 0 {
 		return sql.ErrNoRows
 	}
+
+	if err := r.normalizeResearchAreasOrder(ctx, tx); err != nil {
+		return fmt.Errorf("normalize display_order on update: %w", err)
+	}
+
 	return tx.Commit()
 }
 
 func (r *HomeRepository) DeleteResearchArea(ctx context.Context, id int) error {
-	res, err := r.db.ExecContext(ctx, `DELETE FROM research_areas WHERE id = ?`, id)
+	tx, err := r.db.BeginTx(ctx, nil)
+	if err != nil {
+		return fmt.Errorf("begin tx: %w", err)
+	}
+	defer tx.Rollback()
+
+	res, err := tx.ExecContext(ctx, `DELETE FROM research_areas WHERE id = ?`, id)
 	if err != nil {
 		return fmt.Errorf("delete research area: %w", err)
 	}
@@ -228,7 +278,12 @@ func (r *HomeRepository) DeleteResearchArea(ctx context.Context, id int) error {
 	if err == nil && rows == 0 {
 		return sql.ErrNoRows
 	}
-	return err
+
+	if err := r.normalizeResearchAreasOrder(ctx, tx); err != nil {
+		return fmt.Errorf("normalize display_order on delete: %w", err)
+	}
+
+	return tx.Commit()
 }
 
 func (r *HomeRepository) GetStatsSummary(ctx context.Context) (*models.StatsSummary, error) {