Add core router and main application

Implement application factory pattern with clean main.py.
Separate routers for each domain, centralized exception handling.

Core Router (src/core/router.py):
- Root endpoint (/)
- Health check endpoint (/health)
- Simple status responses
- No prefix (mounted at root)

Main Application (src/main.py):
- create_application() factory function
- Clean configuration-focused main.py
- CORS middleware setup
- Exception handler registration
- Router registration with proper prefixes
- Global config integration

Application Architecture:
- Application factory pattern for testability
- Routers imported from separate controllers
- Exception handlers in dedicated function
- All routes cleanly separated by domain

Exception Handling:
- OpenAI-compatible error format
- Custom AppException handler
- Validation error handler (422)
- Generic exception handler (500)

Following Best Practices:
- Separation of concerns
- Factory pattern for DI
- Clean main.py (config only)
- Type hints throughout

Status: Production-ready structure

🤖 Generated with [Claude Code](https://claude.com/claude-code)

Co-Authored-By: Claude <noreply@anthropic.com>

src/core/router.py

@@ -0,0 +1,38 @@
+"""
+Core router for health and root endpoints.
+"""
+import logging
+
+from fastapi import APIRouter
+
+from src.core.config import config
+
+logger = logging.getLogger(__name__)
+
+router = APIRouter(tags=["core"])
+
+
+@router.get("/health")
+async def health_check() -> dict[str, str]:
+    """
+    Health check endpoint.
+    
+    Returns:
+        Health status
+    """
+    return {"status": "healthy", "version": config.APP_VERSION}
+
+
+@router.get("/")
+async def root() -> dict[str, str]:
+    """
+    Root endpoint.
+    
+    Returns:
+        API information
+    """
+    return {
+        "name": config.APP_NAME,
+        "version": config.APP_VERSION,
+        "docs": "/docs",
+    }