Registered in 2004 to "Indy Application Server" and essentially never seen since.
IANA assigns TCP and UDP 5963 to indy, "Indy Application Server", registered by Bjorn Lantz in November 2004. There is no RFC, no public protocol specification, and no widely deployed implementation attached to that name, so the registration is effectively all there is to say. In practice a listener on 5963 is either something private or — given the neighbourhood — a VNC server on display :63.
Expose with care
You cannot reason about this port from its number, because the registered service is unknown in practice. Identify the owning process first, then decide.
$ nmap -Pn -sV -p 5963 TARGETSeen this on your network? Beige Box scans your Wi-Fi and shows every open port on every device — and its Ports tool links straight back here.
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Corrections or a missing port? Reply @rimrocksystems.