Coarse-Grained Drift Fields and Attractor-Basin Entropy in Kaprekar's Routine.

Area of Science:

• Number Theory
• Dynamical Systems
• Information Theory

Background:

• Kaprekar's routine involves sorting digits to reveal fixed attractors like 495 (D=3) and 6174 (D=4).
• The global information-theoretic dynamics and digit length dependence remain underexplored.

Purpose of the Study:

• To exhaustively analyze the information-theoretic structure of Kaprekar's routine for digit lengths D=3, 4, 5, and 6.
• To investigate attractor convergence, entropy decay, and state space dynamics.

Main Methods:

• Enumerating all states and computing the transition structure for each digit length.
• Constructing "entropy funnels" from attractor distributions.
• Reducing state space using permutation symmetry and digit-gap features.
• Empirically estimating a first-order Markov approximation and computing drift fields and stationary distributions.

Main Results:

• Average convergence distances remain small despite combinatorial state space growth.
• Entropy decays rapidly, followed by a slow tail, indicating predictable dynamics.
• A Markov approximation effectively describes the projected dynamics on the reduced gap space.

Conclusions:

• Kaprekar's routine exhibits complex yet rapidly converging dynamics.
• Information-theoretic analysis reveals underlying structure independent of closed-form solutions.
• The study provides numerical summaries of projected dynamics for varying digit lengths.