The p-Adic Classification and Local Structural Invariants of Block Sequences Under Base Filtration b = 1 + p2
Year: 2026
Type: preprint
Abstract: In this work, a comprehensive non-Archimedean dynamical framework is constructed to formalize the local $p$-adic geometry, metric stability, and structural invariants of infinite block sequences that replicate and generalize the profinite carry-propagation dynamics of parametric Kaprekar families. Under the strict criteria of the structural base filtration $b = 1 + p^2$, where $p \ge 3$ is an arbitrary prime, we transcend the empirical limits of finite combinatorial digit-sorting endomorphisms on the real line, establishing their rigorous continuous embedding into the compact ring of $3$-adic and $p$-adic integers $\mathbb{Z}_p$. By evaluating the analytical framework of the unnormalized rational generating operator $F_p(t) = \frac{\mu t^2 + \lambda t + \nu}{p} \in \mathbb{Q}_p[t]$, we induce a complete, field-independent topologico-algebraic factorization of the infinite discrete parameter space onto the two-dimensional discrete torus $\mathbb{F}_p \times \mathbb{F}_p$. This geometric projection exhaustively segregates the total dynamical territory into four mutually exclusive fundamental residue classes governed by the joint divisibility of the sum $S = n + k$ and mapping step $\Delta = n - k$ operators, isolating the regular isometric supports $\mathcal{X}_{\mathrm{reg}}$ from rigidly blocked singular domains where the dynamic kernel valuation remains frozen at the uniform metric barrier $\nu_p(Y) = 2$. We formalize the non-Archimedean kinetics of these systems by proving that the bitwise carry routing is driven by an infinite-state digit transducer whose algebraic transformations induce an exact, non-commutative representation of the affine group $\mathrm{Aff}(\mathbb{F}_p) \cong \mathbb{F}_p \rtimes \mathbb{F}_p^\times$. Furthermore, the branching laws of the underlying Hensel lifting trajectories are investigated, demonstrating that the cardinality of active calibration constants $s_p(m-2)$ at an arbitrary accuracy layer $m \ge 2$ is a topological invariant perfectly coinciding with the quadratic residue spectrum of the finite ring $\mathbb{Z}/p^{n}\mathbb{Z}$ via shifted Stangl--Bidassi index filtrations. Crucially, this structural deconvolution provides the explicit discrete foundations required to execute an exact analytical evaluation of the unique regular root $S^*(\Delta) \in \mathbb{Z}_p$ through the integration of the formal power series of the $p$-adic hyperbolic tangent. This constructs a strictly convergent power series expansion in even powers of the step variable, whose global numerical skeleton is governed by the rational Bernoulli numbers $B_{2k}$, completely bypassing multi-layer iterative lifting cascades and establishing an exact root-finding routine of single-step computational complexity $\mathcal{O}(1)$. Finally, the generalized theoretical model is validated through explicit implementations and structural mappings over known expanding multi-digit architectures, confirming the absolute, field-independent stability of the discovered algebraic invariants.
Author: Lunio, Nataliia
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225662339
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