feat: implement Iteration 2 — product management

Backend:
- migrations/005: add pg_trgm extension + search indexes on ingredient_mappings
- migrations/006: products table with computed expires_at column
- ingredient: add Search method (aliases + ILIKE + trgm) + HTTP handler
- product: full package — model, repository (CRUD + BatchCreate + ListForPrompt), handler
- gemini: add AvailableProducts field to RecipeRequest, include in prompt
- recommendation: add ProductLister interface, load user products for personalised prompts
- server/main: wire ingredient and product handlers with new routes

Flutter:
- models: Product, IngredientMapping with json_serializable
- ProductService: getProducts, createProduct, updateProduct, deleteProduct, searchIngredients
- ProductsNotifier: create/update/delete with optimistic delete
- ProductsScreen: expiring-soon section, normal section, swipe-to-delete, edit bottom sheet
- AddProductScreen: name field with 300ms debounce autocomplete, qty/unit/days fields
- app_router: /products/add route + Badge on Products nav tab showing expiring count
- MainShell converted to ConsumerWidget for badge reactivity

Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>

backend/internal/ingredient/repository.go

@@ -94,6 +94,32 @@ func (r *Repository) GetByID(ctx context.Context, id string) (*IngredientMapping
 	return m, err
 }
 
+// Search finds ingredient mappings matching the query string.
+// Uses a three-level strategy: exact aliases match, ILIKE, and pg_trgm similarity.
+func (r *Repository) Search(ctx context.Context, query string, limit int) ([]*IngredientMapping, error) {
+	if limit <= 0 {
+		limit = 10
+	}
+	q := `
+		SELECT id, canonical_name, canonical_name_ru, spoonacular_id, aliases,
+			category, default_unit,
+			calories_per_100g, protein_per_100g, fat_per_100g, carbs_per_100g, fiber_per_100g,
+			storage_days, created_at, updated_at
+		FROM ingredient_mappings
+		WHERE aliases @> to_jsonb(lower($1)::text)
+		   OR canonical_name_ru ILIKE '%' || $1 || '%'
+		   OR similarity(canonical_name_ru, $1) > 0.3
+		ORDER BY similarity(canonical_name_ru, $1) DESC
+		LIMIT $2`
+
+	rows, err := r.pool.Query(ctx, q, query, limit)
+	if err != nil {
+		return nil, fmt.Errorf("search ingredient_mappings: %w", err)
+	}
+	defer rows.Close()
+	return collectMappings(rows)
+}
+
 // Count returns the total number of ingredient mappings.
 func (r *Repository) Count(ctx context.Context) (int, error) {
 	var n int