A research program that does not prove the Collatz conjecture. Its purpose is to reduce the conjecture, by honest, one-directional, individually-tracked implications, to a single analytic statement, and then to name that statement and certify the boundary around it. "The way is shut. It was made by those who are Dead, and the Dead keep it, until the time comes. The way is shut. " — J. R. R. Tolkien, The Return of the King No claim of a proof of Collatz is made anywhere in this work. The endpoint of the program, in one paragraphCollatz (modulo the elimination of nontrivial cycles) is equivalent to a bounded-deficit statement: along every odd orbit the cumulative 2-adic valuation tracks the equilibrium slope, Dₘ = m·log₂3 + O (1). The program reduces the conjecture to this, and identifies the residual obstruction precisely: it is the quenched (single-orbit) equidistribution / deterministic normality of the accelerated map's valuation cocycle against the log₂3 rotation. Under the Haar (annealed) measure this is elementary; the gap is its realization on one deterministic integer orbit. We show why this is genuinely irreducible: it has the form of dynamical Borel–Cantelli, measure rigidity, and homogeneous nondivergence, and the hypotheses of none, the cocycle has ×2-entropy but is rank-one (rigid) in the ×3 direction. It is one sharply-pinned instance of the deterministic-versus- random (pseudorandomness / normality) barrier.
John Janik (Mon,) studied this question.