feat(mall): serve catalog and currency from the live API

Wave 1 of replacing the fixed-data mock adapter. The mall now selects its API
adapter per domain, with catalog and currency served live while auth, cart,
orders, shipments and invoices stay on fixed data.

Backend:
- seed the 6 x 2 x 2 category tree as reference data (migration 0005). The API
  exposes no category write route, so this cannot come from the seed script
- filter public product listing by the category subtree with a recursive CTE,
  matching the mock's existing behaviour instead of exact-match
- add sort=price with order=asc|desc, validated by hand so an unsupported value
  returns the project's ApiError 400 shape rather than axum's own rejection

Mall:
- replace the all-or-nothing mockApi boolean with a liveDomains list composed
  through a typed per-domain pick map
- source home floors, the category menu, search and product detail from the
  catalog API; banners, promos, quick links, store card and comment/coupon
  content stay local display-only content
- drop the brand facet and the sales/comments sorts: no backend model backs them
- fix salesOf/commentCountOf, which parsed digits out of the product id and so
  rendered "NaN sold" for live UUID ids; they now hash the id

Seed: 24 products across 4 shops, idempotent on re-run.

Note: the mall defaults to a live catalog, so pnpm dev:mall now expects the API
to be running; set NUXT_PUBLIC_LIVE_DOMAINS to an empty array for all-mock work.

OpenSpec change: openspec/changes/replace-mock-api-wave-1

openspec/changes/replace-mock-api-wave-1/tasks.md

@@ -0,0 +1,34 @@
+# Tasks
+
+## 1. Backend: catalog browse
+
+- [x] 1.1 Add an append-only migration seeding child and grandchild categories under the existing `electronics`, `fashion` and `home-living` rows, matching the mock's 3-level shape; verify `GET /api/categories` returns more than 3 rows with populated `parent_id`
+- [x] 1.2 Replace the exact `p.category_id = $1` filter in `apps/api/src/routes/catalog.rs` with a `WITH RECURSIVE` subtree CTE shared by the count and page queries; verify a parent-category request returns its descendants' products
+- [x] 1.3 Add optional `sort=price` and `order=asc|desc` to `list_products`, ordering by each product's lowest active SKU price with `NULLS LAST`, and reject any other value with an `ApiError` 400 whose body is `{"error":{"code","message"}}`; verify the 400 shape with curl
+- [x] 1.4 Extend `apps/api/tests/catalog.rs` with cases for subtree listing, ascending/descending price order and the rejected-sort 400; verify `cargo test -p vmall-api` is green and repeatable
+
+## 2. Shared contract
+
+- [x] 2.1 Add optional `sort` and `order` to `ProductListQuery` in `packages/shared/src/api.ts` and pass them through in `createApi.listProducts`; verify `pnpm --filter @vmall/mall build` still type-checks
+
+## 3. Demo data
+
+- [x] 3.1 Grow `scripts/seed-demo.mjs` to roughly 24 bilingual products spread across the new category tree plus 4 shops, keeping the slug-lookup idempotency; verify a second run reports no duplicate creates and `GET /api/products?per_page=100` returns a full first page
+
+## 4. Mall: per-domain adapter switch
+
+- [x] 4.1 Replace the `mockApi` boolean in `apps/mall/plugins/api.ts` with a `liveDomains` list defaulting to `["catalog", "currency"]`, composing the live client over the fixed-data base through a typed per-domain pick map (no `any`); verify catalog requests hit `:8080` while `getCart` still resolves from fixed data with the backend stopped
+
+## 5. Mall: rewire browse surfaces
+
+- [x] 5.1 `apps/mall/pages/index.vue`: build floors from `listCategories` plus `listProducts` per category, leaving banners, promos, quick links and floor advert art local; verify every non-empty floor renders live products
+- [x] 5.2 `apps/mall/components/shell/CategoryMenu.vue`: source the tree from `listCategories`, keeping the header dropdown mode and the pinned home mode intact; verify the pinned sidebar shows children from the API
+- [x] 5.3 `apps/mall/pages/search.vue`: list through `listProducts` with the selected category's subtree and the new sort, and delete the brand facet plus the `sales`/`comments` sort options; verify filtering by a parent category lists descendant products
+- [x] 5.4 `apps/mall/pages/goods/[id].vue`: load the product and SKUs through `getProduct`, keeping the comment, coupon and sales-rail sections on local display-only content; verify an in-stock SKU selection updates price, stock and the cart target
+- [x] 5.5 Remove only the catalog helpers that lost their last consumer (`topCategories`, `childCategories`, `homeFloors`, `MOCK_BRANDS`, `brandOf`, `HomeFloor`, `MockBrand`) from `apps/mall/mock/data.ts`; amended from the original wording, which also removed `MOCK_PRODUCTS` and would have broken the rollback the `Mock API adapter` requirement promises, so `MOCK_PRODUCTS`/`searchMockProducts`/`productById` stay; verify `pnpm --filter @vmall/mall build` passes with no page importing a removed symbol
+
+## 6. Verification
+
+- [x] 6.1 Run `pnpm --filter @vmall/mall build`, `pnpm --filter @vmall/shop-admin build` and `pnpm --filter @vmall/admin build`, since the shared contract changed; verify all three pass
+- [x] 6.2 With the live backend and seeds running, verify in a browser that home, search, product detail and the pinned category menu render live data, and that no console errors appear
+- [x] 6.3 Stop the backend and confirm the mall still starts and the unmigrated domains (auth, cart, orders, invoices) keep working from fixed data; verify the browse pages degrade without crashing