Thales Alenia Space Orange University of Aberdeen Private Octopus Inc. This document specifies careful convergence of Congestion Control (CC), providing a cautious method that enables fast startup for a wide range of connections or reconnections. The method reuses a set of computed CC parameters that are based on the previously observed path characteristics between the same pair of transport endpoints, such as the bottleneck bandwidth, available capacity, or the RTT. These parameters are stored, allowing them to be later used to modify the CC behavior of a subsequent connection. The document also discusses assumptions and defines requirements around how a sender utilizes these parameters to provide opportunities for a connection to more quickly get up to speed (i.e. utilize the available capacity). It discusses how these changes impact the capacity at a shared network bottleneck and the safe response that is needed after any indication that the new rate is inappropriate. The method is expected to be appropriate to IETF transports.