Spiral Structure of Digital Roots

This paper reveals that the digital root behaves in exactly three distinct ways, directly determined by the divisibility of the modulus q by 3 and 9. First, for moduli not divisible by 3, the digital root follows a full 9-step cycle. Second, for moduli divisible by 3 but not by 9, it falls into a short 3-step cycle. Finally, for moduli divisible by 9, the digital root remains constant at 1. These three classes are then mapped onto the units of time (the fractal clock), revealing why the week is the unique 9-step exception, while the second, minute, and hour follow a 3-step cycle, and the 360-degree circle becomes a flat spiral. The patterns are not a random list of numbers, but a conclusive proof of the duality between position and level, demonstrating the fractal nature of the clock and the underlying number theory.