The Tangential of a point X with respect to a cubic is the 3rd intersection point of the tangent at X to the cubic, assuming that the 1st and 2nd intersection points are both the point of tangency.
7P-s-P3 is the Tangential of 7P-s-P2 with respect to the cubic 7P-s-Cu1.
Special about 7P-s-P3 is that it is the CB-conjugate (7P-s-Tf1) of 7P-s-P2.
This special property was noted by Eckart Schmidt. See Ref-66, QPG-message #652.