Number Circles: The Base-Independent Geometry of Finite Alphabets

Kaprekar’s iterative digit routine is routinely dismissed as a decimal curiosity. We prove it is a universal law of finite-state dynamics. By generalizing the Kaprekar map to arbitrary bases and digit lengths , we demonstrate that the transition from fixed-point convergence to limit-cycle oscillation is base-independent. Polar phase portraits reveal the geometric signature of this law: trajectories spiral inward to a single radius for small , and lock into discrete orbital rings for . We map digit positions directly to Planck-scale voxel coordinates, showing how local state coefficients ( to ) and global sorting-subtraction update rules drive a discrete lattice toward structural completion. Base independence confirms that finite alphabets universally enforce halt-or-loop dynamics. The "number circles" are not recreational mathematics; they are topological proofs that infinity cannot compute.