A Resolution of the Collatz Conjecture via Archimedean Spiral Mapping and Topological Variance Decay

Abstract This paper presents a complete mathematical resolution to the Collatz $(3n+1)$ Conjecture. By diverging from traditional discrete number theory, we construct an Archimedean spiral parametric mapping that defines a non-linear state transformation within a high-dimensional symmetric topology. We prove that all arbitrary initial integer states induce a Markov chain iteration whose global topological variance strictly dissipates. Under these deterministic limits, the trajectory is mathematically bounded from infinity and globally converges to the unique stable attractor {4, 2, 1}.