On the Inverse Problem for the Evaluation of Regular Roots of Kaprekar Polynomials: \\ Profinite Injectivity Thresholds and Methodological Frameworks for Open Challenges in $p$-adic Analysis

In this work, the Inverse Problem for the locally analytical regular Hensel root mapping of generalized dynamical Kaprekar systems over the local field $\mathbb{Q}_p$ is formalized and investigated for the first time. Based on the multiplicative structure of profinite groups and the $p$-adic theory of measures, we substantiate the conditions for the existence and uniqueness of an exact algebraic preimage for given finite core projections. The algebraic formulation of the problem is established as the reconstruction of the step's square equivalence class from a fixed calibration constant. Furthermore, we determine the exact boundaries of metric information loss caused by the action of the Kaprekar base filtration co-factor and the algebraic degeneration arising from the inversion of the quadratic operator within higher-order zero balls.\end{abstract}