The 3-Adic Arithmetic of the 6174 Kaprekar Family: Valuations, Regular Roots, an Explicit Form, a Digit Transducer and Transcendence

Kaprekar's routine, which subtracts the smallest rearrangement of the digits of a number from the largest, has been studied since the 1940s, and its fixed points are known for many digit lengths and bases. In base ten there is a well-known infinite family of fixed points in the even digit lengths $2k+4$, beginning with $6174$ and $631764$ (OEIS A214556). The family and the classification of fixed points are not new; what was missing is a structural description of how its members are related. Their Archimedean distances grow like $10^{2n}$, and the existing analysis treats each digit length separately. A remark on terminology is necessary: in the literature on Kaprekar constants the phrases ``$2$-adic'' and ``$3$-adic'' refer to numeral systems with base $2$ or $3$, whereas here the term is used in the sense of non-Archimedean analysis. We study the family in the field $\mathbb{Q}_3$ with its $3$-adic valuation, a choice dictated by the problem itself, since the base ten exceeds one by nine. The novelty of the paper is the construction of this genuinely $3$-adic theory. Its main results are: (i) an exact formula for the valuation of the difference of two members, with a static and a dynamic component, and a classification of the pairs of indices into four classes of supports (two rigid and two regular) with a finer hierarchy of sub-supports; (ii) the existence, uniqueness and geometry of the regular roots, the $3$-adic numbers that govern the dynamic component, which form a Cantor set of Haar measure $1/24$; (iii) an exact distance law between roots, which implies injectivity up to a reflection and an exact stabilization of the calibration constants; (iv) a description of the closure of the family in $\mathbb{Z}_3$, with explicit limit points and speeds of convergence; (v) the explicit form of the roots, an analytic isometry $\Theta$ of $\mathbb{Z}_3$ and its global Schr"oder linearization with an explicit small-divisor estimate; (vi) the structure of the reductions modulo $3^m$, including a semigroup of root operators, a lifting monoid of order $5\cdot3^{m-1}$, and an $S_3$-valued monodromy cocycle that records, but does not compute, the carries of an infinite-state digit transducer; (vii) the contractive dynamics of the root map; (viii) the transcendence of the roots. The results are confirmed by direct computation for pairs of Kaprekar numbers with thousands of digits, and they open the way to the $3$-adic study of other families of Kaprekar fixed points.