4.4.3: Official Definition
4.X.X
[RFC3463] (Standards Track) from the The Internet Engineering Task Force (IETF) definition for codes that start with 4.X.X:
A persistent transient failure is one in which the message as sent is valid, but persistence of some temporary condition has caused abandonment or delay of attempts to send the message. If this code accompanies a delivery failure report, sending in the future may be successful.
X.X.3
[RFC3463] (Standards Track) from the The Internet Engineering Task Force (IETF) definition for codes that end with X.X.3:
The network system was unable to forward the message, because a directory server was unavailable. This is useful only as a persistent transient error. The inability to connect to an Internet DNS server is one example of the directory server failure error.