All orbits of the collatz map attain the trivial cycle 4 2 1: A proof via the theory of huts and 2-adic classification classes

Abstract

We present a complete proof of the Collatz conjecture. We transport Syracuse orbits by multiplying every odd term by three. The evolution then depends only on the local 2-adic valuations of the two neighboring even multiples of six. To formalize this we introduce huts, equivalence classes of odd multiples of three defined by the pair of valuations of their borders. For the factor three only three families exist: the base hut, huts of type one-alpha, and huts of type alpha-one. Ordered by the sum of minimal representatives, the set of huts is well-ordered, allowing strong induction. We classify the Collatz transformation on huts. A hut of type one-alpha always moves to the strictly smaller hut one-(alpha minus one). A hut of type alpha-one moves either to a hut of type one-beta or to a hut of type beta-one where beta is strictly smaller than alpha in the second case, proved by a parity obstruction. When beta is larger, finite iteration of the first rule forces a return to a smaller hut. Thus every hut enters a strictly smaller hut after finitely many steps. By induction every hut converges to three. Since every odd multiple of three belongs to a unique hut, every Syracuse orbit reaches one, so every Collatz orbit attains the trivial cycle four, two, one. We also explain why factor three is singular and why induction fails for qx plus one with q at least five and for three x plus r.