BFD - Control Plane Independent (CPI) bit
Bidirectional Forwarding Detection (BFD) uses a series of control packets to allow BFD devices to communicate. Within these control packets, there are a series of flags. One of those flags is known as the C-bit, or the Control Plane Independent (CPI) bit.
The RFC 5880 that describes BFD has the following explanation for the use of this bit:
Control Plane Independent (C)
If set, the transmitting system's BFD implementation does not share fate with its control plane (in other words, BFD is implemented in the forwarding plane and can continue to function through disruptions in the control plane). If clear, the transmitting system's BFD implementation shares fate with its control plane. The use of this bit is application dependent and is outside the scope of this specification. See specific application specifications for details.
When it talks about the control plane here, it is primarily talking about the underlying routing protocol that BFD is supporting. So if you’re using OSPF for example, if the OSPF neighborship fails, and the C bit is set, the BFD session will continue to function (if it can) even if the OSPF adjacency has failed. If the C bit is not set, and the OSPF adjacency fails, the BFD session will also go down automatically.
Links:
https://forum.networklessons.com/t/bidirectional-forwarding-detection-bfd/1280/94?u=lagapidis
https://networklessons.com/cisco/ccie-routing-switching/bidirectional-forwarding-detection-bfd/