Add granular DB error types with retry logic

New exceptions: DatabaseBusyError, DatabasePermissionDeniedError,
DatabasePathInvalidError, DatabaseCorruptedError, DatabaseUnavailableError.

open_db creates parent directory if missing. Catches all aiosqlite errors
and maps to specific exception types.

get_db retries up to 3x on locked database with backoff.
Propagates specific exceptions instead of generic HTTPException.

Tests for all new error types and retry behavior.

backend/app/dependencies.py

@@ -165,22 +165,61 @@ async def get_db(
 
     Yields:
         An open :class:`aiosqlite.Connection` for the request.
+
+    Raises:
+        DatabaseBusyError: After 3 retries when database is locked by concurrent writers.
+        DatabasePermissionDeniedError: When the database file cannot be accessed.
+        DatabasePathInvalidError: When the database path is invalid or directory missing.
+        DatabaseCorruptedError: When the database file is corrupted.
+        DatabaseUnavailableError: For any other unexpected database error.
     """
     from app.db import open_db  # noqa: PLC0415
+    from app.exceptions import (
+        DatabaseBusyError,
+        DatabaseCorruptedError,
+        DatabasePathInvalidError,
+        DatabasePermissionDeniedError,
+        DatabaseUnavailableError,
+    )
 
-    try:
-        db = await open_db(settings.database_path)
-    except Exception as exc:
-        log.error("database_open_failed", error=str(exc))
-        raise HTTPException(
-            status_code=status.HTTP_503_SERVICE_UNAVAILABLE,
-            detail="Database is not available.",
-        ) from exc
+    db = None
+    retries = 3
+    retry_delay = 0.1
+    last_exc = None
+
+    for attempt in range(1, retries + 1):
+        try:
+            db = await open_db(settings.database_path)
+            break
+        except DatabaseBusyError:
+            raise
+        except (DatabasePermissionDeniedError, DatabasePathInvalidError, DatabaseCorruptedError):
+            raise
+        except DatabaseUnavailableError as exc:
+            error_str = str(exc).lower()
+            if "database is locked" in error_str or "busy" in error_str:
+                last_exc = exc
+                if attempt < retries:
+                    log.warning(
+                        "database_open_retry",
+                        attempt=attempt,
+                        max_retries=retries,
+                        database_path=settings.database_path,
+                    )
+                    import asyncio
+                    await asyncio.sleep(retry_delay * attempt)
+                    continue
+                raise DatabaseBusyError(settings.database_path, retries) from exc
+            raise
+
+    if last_exc is not None and db is None:
+        raise DatabaseBusyError(settings.database_path, retries)
 
     try:
         yield db
     finally:
-        await db.close()
+        if db is not None:
+            await db.close()
 
 
 async def get_http_session(