We formulate an edge-complete contact-power theory for spin-gradient continua. The internal mechanical power contains a term in which a second-order hyperstress \(\varvec{G}\) is conjugate to \(\operatorname {grad}\operatorname {curl}\varvec{v}\) , where \(\varvec{v}\) is the velocity field. For an arbitrary piecewise-smooth part, the associated contact power must be distributed on the boundary strata of that part and must be expressed in terms of kinematical fields that can be prescribed independently on those strata. On each smooth face, the velocity and its normal derivative may be prescribed independently, whereas the velocity and its curl may not. We establish and use a face-wise surface-curl identity to rewrite the moment-type component of the contact power in terms of the independent face pair. After performing the face-wise integrations by parts, we obtain a boundary-power representation containing curvature-dependent face contributions and contributions distributed along edges. By comparing that representation with the edge-complete external power, we identify the associated face traction, face hypertraction, and line traction on each edge. The corresponding weak problem is formulated in terms of admissible face and edge data. Face separability and edge completeness are distinct requirements: the first concerns independently assignable face data, whereas the second concerns the retention of line tractions on edges of piecewise-smooth parts. On that basis, we classify contact-power formulations according to face separability, order of the internal-gradient expenditure, curvature dependence, and edge completeness.
