Structural Analysis of Collatz Sequences Under Systematic Initial Value Doubling: A Rigorous Introduction to Parametric Family Dynamics

We present a rigorous framework for analyzing Collatz sequences through systematic initial value doubling. For any positive odd integer a₁, we investigate the parametric family {aₙ = 2ⁿ⁻¹a₁ : n ∈ ℕ} and establish complete structural characterizations of the corresponding Collatz sequences C*(aₙ). Under the assumption that a₁ converges to 1 (i.e., the Collatz conjecture holds for a₁), we prove that: (1) The first n-1 iterations consist exclusively of deterministic halvings; (2) The total stopping time satisfies the exact additive formula t(aₙ) = (n-1) + t(a₁); (3) Several key properties including odd element count and 3n+1 operation frequency remain invariant across the entire parametric family. These results, verified computationally for over 100,000 test cases, reveal previously unexplored connections between geometric progressions and Collatz dynamics. While this work does not resolve the Collatz conjecture itself, it provides rigorous analytical tools for studying infinite families simultaneously and establishes foundations for parametric approaches to iterative number-theoretic functions. Mathematics Subject Classification (MSC2020): Primária: 11B83 — Sequências especiais e polinômios Secundária: 37B20 — Noções de recorrência 05A15 — Enumeração exata 11A07 — Congruências