feat: wire chat UI to Responses API with streaming

Add ChatController that proxies POST /api/chat to the local Responses API
(localhost:8317/v1/responses) with SSE streaming. Client reads tokens via
SetBrowserResponseStreamingEnabled and renders them incrementally. Includes
thinking indicator, input disabled during streaming, and error handling.

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

openspec/changes/archive/2026-04-04-wire-responses-api/tasks.md

@@ -0,0 +1,29 @@
+## 1. Shared Models
+
+- [x] 1.1 Create ChatRequest.cs in ChatAgent.Shared/Models with a Messages list property
+
+## 2. API Backend
+
+- [x] 2.1 Add appsettings.json to ChatAgent.Api with ResponsesApi:BaseUrl and ResponsesApi:Model
+- [x] 2.2 Register an HttpClient for the Responses API proxy in Api Program.cs
+- [x] 2.3 Create ChatController with POST /api/chat that proxies to the Responses API with streaming
+- [x] 2.4 Parse Responses API SSE stream, extract response.output_text.delta events, re-emit as simplified SSE to client
+
+## 3. Client Streaming
+
+- [x] 3.1 Add a streaming SendChatAsync method to ChatApiClient that uses SetBrowserResponseStreamingEnabled and HttpCompletionOption.ResponseHeadersRead
+- [x] 3.2 Parse the simplified SSE stream line-by-line, yielding text deltas
+
+## 4. Chat Page Updates
+
+- [x] 4.1 Replace hardcoded response in Chat.razor with a call to ChatApiClient.SendChatAsync
+- [x] 4.2 Append tokens to the assistant message incrementally with StateHasChanged after each delta
+- [x] 4.3 Add a thinking indicator shown until the first token arrives
+- [x] 4.4 Disable input field and send button while streaming is in progress
+- [x] 4.5 Handle errors — display error message if API call fails
+- [x] 4.6 Auto-scroll during streaming (not just at the end)
+
+## 5. Verify
+
+- [x] 5.1 Run dotnet build to confirm no errors
+- [ ] 5.2 Manually verify: send a message, see streaming response from Claude