Why is the traction vector dependent on just normal vector instead of the geometry of the entire cuting surface?

Given a continuum G with surface forces on its surface S being cut arbitrarily (not necessarily by a plane) it is natural that the traction vector at any point along the cut depends on the cut itself. If the forces are represented as tractions T across the surface of the object, Let the cutting surface be C, the surface of the first half of the cut object as S1, and the surface of the second half S2. In the image, the light blue surface would be C, the dark blue would be S1 on the left and S2 on the right. Then

$$oint_{Ccup S_1} vec{T}dS = m vec{a}_{cg}$$

where m is the mass of the whole object and cg is the center of gravity for the whole object. Thus it is natural that the traction field on C depends on the geometry of C. But I don’t understand how to show that the traction at a point is strictly dependent on the normal at that point, rather than the entire cut.