The p-Adic Classification and Local Structural Invariants of Block Sequences Under Base Filtration b = 1 + p2

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 + p², where p ≥ 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 Zₚ. By evaluating the analytical framework of the unnormalized rational generating operator Fₚ(t) = μ t² + λ t + ν/p ∈ Qₚ[t], we induce a complete, field-independent topologico-algebraic factorization of the infinite discrete parameter space onto the two-dimensional discrete torus Fₚ × Fₚ. 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 Δ = n - k operators, isolating the regular isometric supports Xreg from rigidly blocked singular domains where the dynamic kernel valuation remains frozen at the uniform metric barrier νₚ(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 Aff(Fₚ) Fₚ Fₚ^×. Furthermore, the branching laws of the underlying Hensel lifting trajectories are investigated, demonstrating that the cardinality of active calibration constants sₚ(m-2) at an arbitrary accuracy layer m ≥ 2 is a topological invariant perfectly coinciding with the quadratic residue spectrum of the finite ring Z/pⁿZ 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^*(Δ) ∈ Zₚ 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 B₂ₖ, completely bypassing multi-layer iterative lifting cascades and establishing an exact root-finding routine of single-step computational complexity 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.