feat: replace Claude with GPT for AI chat integration

- Replace ClaudeAdapter with GPTAdapter using OpenAI GPT (gpt-5-mini)
- Implement function calling for calendar operations (getDay, proposeCreate/Update/Delete, searchEvents)
- Add provider-agnostic AI utilities in ai/utils/ (systemPrompt, toolDefinitions, toolExecutor, eventFormatter)
- Add USE_TEST_RESPONSES env var to toggle between real AI and test responses
- Switch ChatService.processMessage to use real AI provider
- Add npm run format command for Prettier
- Update CLAUDE.md with new architecture

CLAUDE.md

@@ -13,6 +13,7 @@ This is a fullstack TypeScript monorepo with npm workspaces.
 ### Root (monorepo)
 ```bash
 npm install          # Install all dependencies for all workspaces
+npm run format       # Format all TypeScript files with Prettier
 ```
 
 ### Client (apps/client) - Expo React Native app
@@ -44,7 +45,7 @@ npm run start -w @caldav/server        # Run compiled server (port 3000)
 | Backend | Express.js | Web framework |
 | | MongoDB | Database |
 | | Mongoose | ODM |
-| | Claude (Anthropic) | AI/LLM for chat |
+| | GPT (OpenAI) | AI/LLM for chat |
 | | JWT | Authentication |
 | Planned | iCalendar | Event export/import |
 
@@ -134,7 +135,14 @@ src/
 │       ├── MongoEventRepository.ts
 │       └── MongoChatRepository.ts
 ├── ai/
-│   └── ClaudeAdapter.ts      # Implements AIProvider
+│   ├── GPTAdapter.ts         # Implements AIProvider using OpenAI GPT
+│   ├── index.ts              # Re-exports GPTAdapter
+│   └── utils/                # Shared AI utilities (provider-agnostic)
+│       ├── index.ts          # Re-exports
+│       ├── eventFormatter.ts # formatExistingEvents() for system prompt
+│       ├── systemPrompt.ts   # buildSystemPrompt() - German calendar assistant prompt
+│       ├── toolDefinitions.ts # TOOL_DEFINITIONS - provider-agnostic tool specs
+│       └── toolExecutor.ts   # executeToolCall() - handles getDay, proposeCreate/Update/Delete, searchEvents
 └── utils/
     ├── jwt.ts                # signToken(), verifyToken()
     ├── password.ts           # hash(), compare()
@@ -264,6 +272,8 @@ Server requires `.env` file in `apps/server/`:
 JWT_SECRET=your-secret-key
 JWT_EXPIRES_IN=1h
 MONGODB_URI=mongodb://root:mongoose@localhost:27017/calchat?authSource=admin
+OPENAI_API_KEY=sk-proj-...
+USE_TEST_RESPONSES=false    # true = static test responses, false = real GPT AI
 ```
 
 ## Current Implementation Status
@@ -284,15 +294,15 @@ MONGODB_URI=mongodb://root:mongoose@localhost:27017/calchat?authSource=admin
   - `utils/eventFormatters`: getWeeksOverview(), getMonthOverview() with German localization
   - `utils/recurrenceExpander`: expandRecurringEvents() using rrule library for RRULE parsing
   - `ChatController`: getConversations(), getConversation() with cursor-based pagination support
-  - `ChatService`: getConversations(), getConversation(), processMessage() now persists user/assistant messages to DB, confirmEvent()/rejectEvent() update respondedAction and persist response messages
+  - `ChatService`: getConversations(), getConversation(), processMessage() uses real AI or test responses (via USE_TEST_RESPONSES), confirmEvent()/rejectEvent() update respondedAction and persist response messages
   - `MongoChatRepository`: Full CRUD implemented (getConversationsByUser, createConversation, getMessages with cursor pagination, createMessage, updateMessage)
   - `ChatRepository` interface: updateMessage() added for respondedAction tracking
+  - `GPTAdapter`: Full implementation with OpenAI GPT (gpt-5-mini model), function calling for calendar operations
+  - `ai/utils/`: Provider-agnostic shared utilities (systemPrompt, toolDefinitions, toolExecutor, eventFormatter)
 - **Stubbed (TODO):**
   - `AuthMiddleware.authenticate()`: Currently uses fake user for testing
   - `AuthController`: refresh(), logout()
   - `AuthService`: refreshToken()
-- **Not started:**
-  - `ClaudeAdapter` (AI integration - currently using test responses)
 
 **Shared:** Types, DTOs, constants (Day, Month with German translations), ExpandedEvent type, and date utilities defined and exported.