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/cmd/server/main.go

@@ -14,8 +14,10 @@ import (
 	"github.com/food-ai/backend/internal/config"
 	"github.com/food-ai/backend/internal/database"
 	"github.com/food-ai/backend/internal/gemini"
+	"github.com/food-ai/backend/internal/ingredient"
 	"github.com/food-ai/backend/internal/middleware"
 	"github.com/food-ai/backend/internal/pexels"
+	"github.com/food-ai/backend/internal/product"
 	"github.com/food-ai/backend/internal/recommendation"
 	"github.com/food-ai/backend/internal/savedrecipe"
 	"github.com/food-ai/backend/internal/server"
@@ -91,8 +93,16 @@ func run() error {
 	geminiClient := gemini.NewClient(cfg.GeminiAPIKey)
 	pexelsClient := pexels.NewClient(cfg.PexelsAPIKey)
 
+	// Ingredient domain
+	ingredientRepo := ingredient.NewRepository(pool)
+	ingredientHandler := ingredient.NewHandler(ingredientRepo)
+
+	// Product domain
+	productRepo := product.NewRepository(pool)
+	productHandler := product.NewHandler(productRepo)
+
 	// Recommendation domain
-	recommendationHandler := recommendation.NewHandler(geminiClient, pexelsClient, userRepo)
+	recommendationHandler := recommendation.NewHandler(geminiClient, pexelsClient, userRepo, productRepo)
 
 	// Saved recipes domain
 	savedRecipeRepo := savedrecipe.NewRepository(pool)
@@ -105,6 +115,8 @@ func run() error {
 		userHandler,
 		recommendationHandler,
 		savedRecipeHandler,
+		ingredientHandler,
+		productHandler,
 		authMW,
 		cfg.AllowedOrigins,
 	)