test: add xUnit test coverage for API controllers and client services

Create ChatAgent.Api.Tests and ChatAgent.Client.Tests projects with xUnit
and Moq. Test HealthController (200 + valid response), ChatController
(SSE streaming with mocked upstream, error handling), and ChatApiClient
(delta parsing, error events, health endpoint). 6 tests, all passing.

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>

openspec/changes/archive/2026-04-04-add-test-coverage/design.md

@@ -0,0 +1,45 @@
+## Context
+
+No test projects exist. The solution has 3 projects (Client, Api, Shared) under `src/`. Tests need to cover the API controllers and the client's ChatApiClient service, both of which involve HTTP and SSE streaming.
+
+## Goals / Non-Goals
+
+**Goals:**
+- Establish test infrastructure (xUnit + Moq)
+- Test API controllers using WebApplicationFactory (integration-style)
+- Test ChatApiClient using a mock HttpMessageHandler (unit-style)
+- All tests runnable via `dotnet test` from solution root
+
+**Non-Goals:**
+- Blazor component tests (bUnit) — Chat.razor is UI-heavy, defer to a future phase
+- End-to-end browser tests (Playwright/Selenium)
+- Testing the upstream Responses API itself
+
+## Decisions
+
+### Decision 1: Test framework — xUnit + Moq
+
+xUnit is the .NET standard. Moq for mocking HttpMessageHandler so we can control HTTP responses in tests without hitting real servers.
+
+### Decision 2: Test project layout
+
+```
+tests/
+├── ChatAgent.Api.Tests/       → references Api project
+└── ChatAgent.Client.Tests/    → references Client project + Shared
+```
+
+Both added to the `ChatAgent.sln` solution file under a `tests` solution folder.
+
+### Decision 3: API tests use WebApplicationFactory
+
+`Microsoft.AspNetCore.Mvc.Testing` provides `WebApplicationFactory<Program>` for integration-style tests. For ChatController, we inject a mock `IHttpClientFactory` that returns a handler with canned SSE responses — no real Responses API needed.
+
+### Decision 4: Client tests mock HttpMessageHandler
+
+ChatApiClient takes an HttpClient. Tests create an HttpClient with a custom `DelegatingHandler` that returns canned SSE response streams. This tests the SSE parsing logic in isolation.
+
+## Risks / Trade-offs
+
+- [WebApplicationFactory requires Api's Program to be accessible] → Add `InternalsVisibleTo` or use the `public partial class Program {}` trick in Api's Program.cs
+- [SSE stream mocking is verbose] → Create a small helper method that builds SSE response content from a list of events