Part 4/10:
Starting with the number 1 and repeatedly doubling it produces a sequence that, when processed through vortex sums or mod 9 operations, cycles through the same set of values: 1, 2, 4, 8, 7, 5. This cycle repeats indefinitely, illustrating a stable repeating vortex pattern.
For example, doubling 1 yields 2; the vortex sum of 2 is 2. Doubling 2 results in 4; the vortex sum remains 4. Continuing this process, 4 doubles to 8, then 8 doubles to 16, which reduces to 7 (since 1 + 6 = 7), and so on, forming a perfect cyclic loop. This pattern (124875) demonstrates the robustness of vortex sums under doubling and hints at deeper mathematical symmetries.