Implement "Browse Product Catalog" (UC-010)

Problem Description

An online shop's engineering team has already run /flyway-migration and /entity-model for their Spring Boot + JPA + Angular project. The product table already exists (see the Flyway migration under inputs/backend/src/main/resources/db/migration/), the entity model is at docs/entity_model.md, and the use case is specified at docs/use-cases/UC-010-browse-product-catalog.md. Nobody has written any backend or frontend code for this use case yet.

The backend is a single flat Maven project at backend/ with package com.example.shop, using Spring Boot 3, Spring Data JPA, and Flyway with spring.jpa.hibernate.ddl-auto=validate — there is no hexagonal multi-module split in this project. The frontend is an Angular (standalone components) project at frontend/ with an existing empty ProductCatalog component already wired into app.routes.ts at /products.

Output Specification

Implement UC-010 end to end:

1. Backend (backend/src/main/java/com/example/shop/...): a Product JPA entity mapped onto the existing table, a Spring Data JPA repository, a service class implementing the "only in-stock, optional category filter" logic from BR-010, and a @RestController exposing the data as JSON through a DTO — never the raw @Entity.
2. Frontend (frontend/src/app/...): fill in (or replace) the ProductCatalog component so it fetches from the new endpoint and renders the product grid with a category filter dropdown, using Angular signal() state and a hand-written HttpClient service.

Do not create any test files or test classes — that is out of scope for this task.