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.
Miguel Cerdá Bennassar (Thu,) studied this question.