VI. Structural reduction of the Collatz conjecture: segments, portals and 2-adic survival sets

This paper develops a structural analysis of the Collatz conjecture by decomposing orbits into finite building blocks called segments, each governed by exact algebraic rules. A key notion is introduced — the portal — a special class of odd numbers that act as the decisive transition points of the dynamics: every segment closes at a portal, and at every portal a structural functional measuring the size of the orbit decreases unconditionally.
Working within this framework, the paper proves that no non-trivial cycle can remain entirely within the class of portals, establishes an exact measure-theoretic result showing that the set of initial conditions compatible with a non-convergent orbit has measure zero, and then proves unconditionally — using a discrete Lyapunov function defined pointwise on ordinary integers — that no positive integer can sustain an orbit that stays inside this class forever without converging. The result reduces the full Collatz conjecture to a single precisely identified open question: whether orbits that alternate between two complementary dynamical classes can grow without bound. The paper quantifies the structural constraints on such growth and locates the remaining difficulty in the theory of linear forms in logarithms.

Version 1.1: Maintenance update including minor revisions and the resolution of issues identified in the previous version.

This paper is part of a series of six works on the Collatz conjecture. In reading order:

I. 2-adic structure of tails and survival sets in Collatz dynamics https://doi.org/10.5281/zenodo.18831439

II. Cylinder collision, bit non-reusage, and effective non-degeneration in 2-adic Collatz dynamics https://doi.org/10.5281/zenodo.18831527

III. Arithmetic obstruction to indefinite survival in 2-adic Collatz dynamics https://doi.org/10.5281/zenodo.18831690

IV. Arithmetic obstruction to mixed orbits in 2-adic Collatz dynamics https://doi.org/10.5281/zenodo.18831791

V. The ϕ function and the extension of the 2-adic budget argument to arbitrary k0 in Collatz dynamics https://doi.org/10.5281/zenodo.18831874

VI. Structural reduction of the Collatz conjecture: stretches, portals, and 2-adic survival sets https://doi.org/10.5281/zenodo.18831607