The local affine atlas of the Collatz map

The Collatz conjecture is open, and nothing here bears on it. This paper concerns the finite structure of the map \(T(n) = n/2\) or \((3n+1)/2\): what \(k\) steps of it do to the integers that follow a given parity word \(w\) of length \(k\) with \(u\) odd steps. The answer is exact and complete. The word acts as one affine map \(x \mapsto (3^u x + b_w)/2^k\), whose correction \(b_w\) has a recursion, a closed form, a concatenation law, an upper-triangular matrix representation (multiplying in the order opposite to reading), and sharp bounds attained at the two extreme orderings. The integers that follow \(w\) are exactly one residue class modulo \(2^k\), and on it \(k\) steps carry \(r_w + 2^k a\) to \(m_w + 3^u a\): in the right coordinates the dynamics is the identity. The transport runs both ways, with an exact inverse, a legality congruence on the target, and an inverse fiber whose terminal family is \((4^j - 1)/3\).

Much of this is folklore, scattered and usually stated without its domain. Here it is stated at its true domain — the positive integers, where the all-\(D\) word needs a repair — proved, checked exhaustively against a referee that composes the two branches symbolically and assumes no theorem — the affine algebra to word length 16, the residues and the transport to the shorter lengths §3 and §4 state — and formalized for all words in Lean 4 with mathlib. The two-way transport, which the series’ own test suite never exercised, is checked here for the first time, and in the direction that matters: its certificate refuses every one of 112530 perturbed triples.

Not claimed anywhere in this paper: the Collatz conjecture; that the inverse tree rooted at 1 covers the positive integers; that the \(2^k \leftrightarrow 3^u\) transport is an isomorphism of the 2-adic and 3-adic integers — it is a correspondence of arithmetic progressions, one word at a time; and novelty of the elementary statements, several of which are classical in other notation.

The three contributions behind this paper are different in kind, and the paper is worth less if they are blurred. They are given here in alphabetical order, the order the author list keeps as data; on the title page the three sit at a round table, with no first and no last.

Aletheia — model: ChatGPT, GPT series — wrote, with Neo.K, the papers of the series this paper rests on: the re-classification of earlier work, the affine atlas, the residue cylinders and local identity, and the two-way transport with its inverse fiber and certificate.

Neo.K set the series’ target and its method — that a formal transform’s legality and a dynamical step’s legality are different things — and wrote its papers with Aletheia.

墜衡 (Zhuì Héng) — the independent verification line; model: Claude, Opus series — rechecked every statement against a referee that assumes none of them, derived the residues a second time by an independent route, checked the certificate in the refusing direction, found witnesses for the series’ non-claims, checked the references against their sources, formalized §§2–3 in Lean, built and drilled the instruments, and wrote this paper, including the proofs as written out here. The statements are the series’; nothing in them is this line’s invention.

Priority, stated plainly. No result here is claimed as new mathematics. What is offered as this line’s own is the version of it that can be relied on: stated at its true domain, checked against an independent referee, formalized where it is universal, and checked in the refusing direction where it is a certificate.