The Collatz Conjecture as a Quasi-Closed Cycle

This paper derives an algebraic equivalence from the Collatz Conjecture using the transformation: x = (n - 1) / 4 Under this transformation, every positive natural number n can be written as: n = 4x + 1 This representation decomposes the positive integers into four quartic categories according to the fractional part of x: x = k x = k + 1/2 x = k + 1/4 x = k + 3/4 where k is an integer. Thus, the quartic lattice is: Q = k, k + 1/4, k + 1/2, k + 3/4 | k ∈ Z Rewriting the Collatz map: If n is even: T (n) = n / 2 If n is odd: T (n) = 3n + 1 In terms of x, the dynamics become two affine transformations: Even branch: x → ( (4x + 1) / 8) - 1/4 Odd branch: x → 3x + 1 Therefore, the iteration is governed by two affine maps of the form: x → ax + b acting on the constrained quartic lattice Q. Within this structure, trajectories may experience temporary growth under the map: x → 3x + 1 but remain confined within a bounded discrete framework admitting a single algebraic exit corresponding to the classical cycle: 1 → 4 → 2 → 1 Although the exact stopping time sigma (n) = min k ∈ N: Tᵏ (n) = 1 remains unpredictable in general, this transformed formulation provides partial predictability of local growth and descent phases and offers a refined structural perspective on the mechanisms underlying the Collatz Conjecture.