Since these fields are discrete, an algorithm examines each possible permutation of Pn, with the form Z2,...,Z2, where Z2 = ±1, P = 2, and n = 3, 6, 12 and verifies that all the axioms and the theorems are valid within these groups and various subgroups. First, each element is multiplied by itself to verify theorem 4.1.  Next, each pair of element  a,b of the groups, Gf8, Gf64, and Gf4096, are multiplied together to check that the result ab and ba are equal, and are also in these groups. This proves axioms A-1 and A-8. Continuing in this manner, the axioms, A-2, A-3, A-4, A-5, A-6, and A-7, are also verified for these groups and semi-groups, thus proving these axioms. Again, using the result ab as a pole point, P, the algorithm finds the subset of all the orthogonal point pairs over the hyper-complex field, which are perpendicular to each other and form the group Gf'64XGf64. The property of points being perpendicular is very important because we define the property of handedness with three perpendicular base vectors. Two of these perpendicular vectors are identified, and their product defines a third perpendicular vector, the pole point, P. This orthogonal vector triplet forms the complex that models handedness. With this vector triplet, a geodesic line is also defined. By evaluating specific bits in the pole's array, (0,1,2,3,4,5), the direction of the geodesic line is determined by the sense of its pole. When the product is negative, a counterclockwise rotation is determined, which models left-handedness. When the product of bits two and five are positive, a clockwise rotation is determined, which models right-handedness. RETURN to Hilbert's 14-th Problem