Note_Kaprekar_Bounds

Abstract We investigate the convergence properties of the four-digit Kaprekar process across infinite families of numerical bases. Building upon previously established results in the mathematical literature, we derive general bounds on convergence probabilities using analytical methods and exact combinatorial counting techniques. Our findings provide mathematical evidence challenging previously published asymptotic conjectures and reveal structural properties of the underlying discrete dynamical system. The complete manuscript, including the explicit formulas, proofs, numerical verification, and detailed results, is preserved in this dated, restricted-access version pending independent scientific evaluation. The originality of these findings relative to subsequent research remains to be independently assessed.