The indeterminate channel problem arises from uncertainty in finding a closure relation for alluvial channels created by self-organizing erosional and depositional processes. Extremal hypotheses have been proposed as one potential approach to closing the system of governing equations for alluvial channels. Many different extremal hypotheses have been presented, but no substantive evidence has been developed to select which hypothesis may be most appropriate for natural alluvial river systems. This paper evaluates the ability of ten extremal hypotheses to identify dynamic equilibrium across a geomorphic gradient in the remote and undeveloped mid-latitude watershed of Rio Murta, Chile. This study (a) introduces extremal hypotheses, (b) describes the field site and geomorphic conditions, and (c) examines which extremal hypotheses are supported by the field data in identifying the evolutionary trend toward dynamic-equilibrium. The extremal hypotheses that identified dynamic equilibrium within the geomorphic gradient in the field are: (1) minimum kinetic energy, (2) minimum specific stream power, (3) maximum friction factor, and (4) maximum total friction factor, which collectively support minimizing kinetic energy of the system.