Lalys Structural Theory: Skeletal Realizability and The Collatz Conjecture

The Collatz conjecture — one of mathematics' simplest-looking, longest-unsolved problems — asks whether repeatedly applying n → 3n + 1 (for odd n) and n → n/2 (for even n) always leads back to the cycle 1 → 4 → 2 → 1. Despite decades of attention from probabilistic, ergodic, and analytic methods, the conjecture remains open, and the gap between its simplicity and its difficulty has made it a classic example of how simple rules can generate deeply complex behaviour. Most approaches study individual numerical trajectories directly. This paper instead studies the underlying arrangement of the two operations — an abstract combinatorial object called a sequential skeleton — independent of the actual numbers involved. This reframes the problem: instead of asking how a specific trajectory evolves, the question becomes which arrangements of functions can be realized by any sequence of integers at all, and what structural constraints that realizability imposes. Building on this shift, the paper develops: • Lalys Structural Theory — a general framework for studying realizability in iterative function systems, built from n-function sequences and sequential skeletons • An obstruction theory for sequential skeletons, with quantitative measures of residual obstruction, realizability weights, and obstruction entropy • The Lalys Method — a procedure that studies structurally impossible skeletons and the modifications required to make them realizable, extracting the necessary properties any realizable sequence must satisfy • A dominance theory describing how these structural properties behave asymptotically Specializing this framework to the Collatz function system, the paper finds: • Even operations structurally dominate within realizable arrangements • Persistent divergence is structurally excluded • Persistent non-trivial cyclic behaviour is structurally excluded • The trivial cycle 1 → 4 → 2 → 1 is left as the only asymptotic behaviour available to realizable Collatz sequences Taken together, these results propose a structural pathway toward the Collatz conjecture — approaching it through the organization of the operations themselves rather than the trajectories they generate.