T-06: Remove AppState Protocol, use ApplicationState directly

The AppState Protocol (lines 42-54) and ApplicationContext dataclass
(lines 57-69) described identical fields, creating maintenance burden.
The Protocol was only used for a single cast() in _build_app_context.

Changes:
- Remove AppState Protocol class
- Import ApplicationState from runtime_state.py
- Replace cast('AppState', request.app.state) with
  cast(ApplicationState, request.app.state)
- Remove unused Protocol import

This eliminates the redundancy while maintaining the same typing
guarantees. request.app.state is set to ApplicationState instances
during app initialization in main.py.

Co-authored-by: Copilot <223556219+Copilot@users.noreply.github.com>

backend/app/dependencies.py

@@ -9,7 +9,7 @@ directly — to keep coupling explicit and testable.
 import datetime
 from collections.abc import AsyncGenerator
 from dataclasses import dataclass
-from typing import Annotated, Protocol, cast
+from typing import Annotated, cast
 
 import aiohttp
 import aiosqlite
@@ -33,27 +33,12 @@ from app.repositories.protocols import (
 )
 from app.services.geo_cache import GeoCache
 from app.utils.constants import SESSION_COOKIE_NAME
-from app.utils.runtime_state import RuntimeState
+from app.utils.runtime_state import ApplicationState, RuntimeState
 from app.utils.session_cache import NoOpSessionCache, SessionCache
 
 log: structlog.stdlib.BoundLogger = structlog.get_logger()
 
 
-class AppState(Protocol):
-    """Partial view of the FastAPI application state used by dependencies."""
-
-    settings: Settings
-    http_session: aiohttp.ClientSession
-    scheduler: AsyncIOScheduler
-    server_status: ServerStatus
-    pending_recovery: PendingRecovery | None
-    last_activation: dict[str, datetime.datetime] | None
-    runtime_settings: Settings | None
-    runtime_state: RuntimeState
-    session_cache: SessionCache
-    geo_cache: GeoCache  # noqa: F821
-
-
 @dataclass
 class ApplicationContext:
     """A typed wrapper around shared application lifecycle resources."""
@@ -87,7 +72,7 @@ def _session_cache_enabled(settings: Settings) -> bool:
 
 
 def _build_app_context(request: Request) -> ApplicationContext:
-    state = cast("AppState", request.app.state)
+    state = cast(ApplicationState, request.app.state)
     session_cache = getattr(state, "session_cache", None)
     if session_cache is None:
         session_cache = NoOpSessionCache()