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/migrations/005_add_ingredient_search_indexes.sql

@@ -0,0 +1,12 @@
+-- +goose Up
+CREATE EXTENSION IF NOT EXISTS pg_trgm;
+
+CREATE INDEX idx_ingredient_mappings_canonical_ru_trgm
+    ON ingredient_mappings USING GIN (canonical_name_ru gin_trgm_ops);
+
+CREATE INDEX idx_ingredient_mappings_canonical_ru_fts
+    ON ingredient_mappings USING GIN (to_tsvector('russian', coalesce(canonical_name_ru, '')));
+
+-- +goose Down
+DROP INDEX IF EXISTS idx_ingredient_mappings_canonical_ru_trgm;
+DROP INDEX IF EXISTS idx_ingredient_mappings_canonical_ru_fts;

backend/migrations/006_create_products.sql

@@ -0,0 +1,20 @@
+-- +goose Up
+CREATE TABLE products (
+    id           UUID PRIMARY KEY DEFAULT gen_random_uuid(),
+    user_id      UUID NOT NULL REFERENCES users(id) ON DELETE CASCADE,
+    mapping_id   UUID REFERENCES ingredient_mappings(id),
+    name         TEXT NOT NULL,
+    quantity     DECIMAL(10, 2) NOT NULL DEFAULT 1,
+    unit         TEXT NOT NULL DEFAULT 'pcs',
+    category     TEXT,
+    storage_days INT NOT NULL DEFAULT 7,
+    added_at     TIMESTAMPTZ NOT NULL DEFAULT now(),
+    expires_at   TIMESTAMPTZ GENERATED ALWAYS AS
+                 (added_at + (storage_days || ' days')::INTERVAL) STORED
+);
+
+CREATE INDEX idx_products_user_id   ON products(user_id);
+CREATE INDEX idx_products_expires_at ON products(user_id, expires_at);
+
+-- +goose Down
+DROP TABLE products;