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/product/model.go

@@ -0,0 +1,39 @@
+package product
+
+import "time"
+
+// Product is a user's food item in their pantry.
+type Product struct {
+	ID           string    `json:"id"`
+	UserID       string    `json:"user_id"`
+	MappingID    *string   `json:"mapping_id"`
+	Name         string    `json:"name"`
+	Quantity     float64   `json:"quantity"`
+	Unit         string    `json:"unit"`
+	Category     *string   `json:"category"`
+	StorageDays  int       `json:"storage_days"`
+	AddedAt      time.Time `json:"added_at"`
+	ExpiresAt    time.Time `json:"expires_at"`
+	DaysLeft     int       `json:"days_left"`
+	ExpiringSoon bool      `json:"expiring_soon"`
+}
+
+// CreateRequest is the body for POST /products.
+type CreateRequest struct {
+	MappingID   *string  `json:"mapping_id"`
+	Name        string   `json:"name"`
+	Quantity    float64  `json:"quantity"`
+	Unit        string   `json:"unit"`
+	Category    *string  `json:"category"`
+	StorageDays int      `json:"storage_days"`
+}
+
+// UpdateRequest is the body for PUT /products/{id}.
+// All fields are optional (nil = keep existing value).
+type UpdateRequest struct {
+	Name        *string  `json:"name"`
+	Quantity    *float64 `json:"quantity"`
+	Unit        *string  `json:"unit"`
+	Category    *string  `json:"category"`
+	StorageDays *int     `json:"storage_days"`
+}