This work introduces a new mathematical interpretation of the solution space for the
Traveling Salesperson Problem (TSP); which is a combinatorial optimization problem that has applications in many fields, from logistics and manufacturing to design and configuration of products. The proposed interpretation shows that all solutions
are contained on the edge of a sliced a spheroid. Thus,  modeling the solution space from a novel perspective. The contribution of this paper is fourfold: First the development  of  a linear programming problem which objective function value leads to an upper bound on the Euclidean distance from a current feasible TSP solution to the optimal one; second, if the current feasible solution is optimal the solution of the later model will provide information that might aid in a further effort of confirmation;  third once the bound on the distance  is found the computer effort to achieve TSP optimally  is realized; and finally  a  set of heuristics which are  highly suitable for parallel graphical processing unit (GPU) computing are proposed. Numerical experimentation shows that the heuristics and the GPU implementation can efficiently find solutions with an average of 3.5% of the lower bound solutions for symmetric TSP, and an average of 4.15% for asymmetric best known benchmark solutions. In addition, the computer effort to achieve optimality is also shown for these instances.