Continuous Cap Replacement requires the computer program know how to assign a single point on the equatorial circle EDP from Cap Replacement at the point P. The computer program puts the map around a special point S_P in the center of the screen. Here the map awaits the user to scroll along the equator circle E_P. The assignment of S_P must be a continuous function of P; the scroll point moves just a little if P moves only slightly.
To scroll continuously, the Cap Replacement program continually replaces the original cap with a new cap as the user moves the point P on the map image. To each continuous selection of P, the program makes a cap replacement and removal as it chooses a scroll point S_P on E_P. As we saw previously, the right-hand rule lets us choose a particular unit tangent direction \bv_P on E_P at S_P. Now, here is the most important point. While \bv_P is a tangent direction at S_P, using P and S_P, we can assign a unit tangent vector at P. The process to do this is parallel translation of the vector \bv_P from S_P to P. Here is how it works.
There is a unique great circle on the sphere through P and S_P. This is because P and S_P can never be a pair of antipodal points. Further, we can recognize the short arc from P to S_P. The short arc goes exactly one-quarter of the way around the great circle. The vector \bv_P is a unit vector in a direction perpendicular to the great circle through P and S_P. There is a unique way to slide along this great circle from S_P to P with your back along the great circle and your right hand pointing in the direction of \bv_P when you are at S_P. Suppose during the slide, you keep your right hand perpendicular to the great circle. Then, when you arrive at P, your right hand points in a direction \bw_P, tangent to the sphere, and perpendicular to the great circle through P and S_P.
Thus, assuming the computer program knows how to continuously select a scroll point S_P, to each point P, this provides a unit tangent vector to P. This is for each point P on the sphere. So, the computer program seems to know how to create a continuous unit tangent vector field on the Earth. This is a function assigning a unit tangent vector to each point on the sphere. H. Poincare and L.E Brouwer considered this problem almost a century ago. They proved this topology theorem which states the sphere has no continuously varying unit tangent vector field.
The Poincare--Brouwer "Hairy Ball Theorem" implies there can be no Cap Replacement program. There is no continuous unit tangent vector assignment to every point on the surface of a sphere. Cap replacement is not a viable method of representing the spherical Earth.