II- THE KAPREKAR TRANSFORMATION AND ITS FIXED-POINT ATTRACTORS

We examine the Kaprekar transformation — “sort digits ↓, sort digits ↑, subtract” — as a discrete dynamical system on integers of arbitrary digit length and base. For the three- and four-digit systems in base 10, the unique attractors 495 and 6174 emerge. We present formal proofs using modular arithmetic, gap analysis, and weighted positional invariants, and generalize to n-digit and base-b settings. The framework reveals a family of deterministic attractors governed by base-dependent symmetry. Extensions include a contraction criterion predicting when unique fixed points arise, computational mapping of terminal cycles across (n, b), and exploration of the ‘gap polytope’ describing feasible equilibria. This study highlights how simple subtraction encodes deep numerical order, where finite arithmetic produces emergent self-symmetry. (Kaprekar, 1949; Hardy & Wright, 2008; Flajolet & Sedgewick, 2009; Milnor, 2011; Wolfram, 2002).