Where the local theory of the Collatz map holds: the odd (m, r) family, a phase boundary at m = 4, and a ladder of algebraic breakage

The Collatz conjecture is open, and nothing here bears on it. Its local theory — every finite parity word acts as one exact affine map on one residue class, in coordinates where the dynamics is the identity — is shown here not to depend on the numbers 3 and 1. For the map \(n/2\) or \((mn+r)/2\) with \(m\) and \(r\) positive and odd, every word of length \(k\) with \(u\) odd steps acts as \(x \mapsto (m^u x + b_w)/2^k\) on exactly one class modulo \(2^k\), carries it onto a progression of difference \(m^u\), and splits into a skeleton fixed by the counts and a correction fixed by the order; \(r\) only scales the correction. The binomial count of contracting classes then turns over at \(m = 4\): their proportion tends to one for \(m = 3\), is one for \(m = 1\), and tends to zero for every odd \(m \ge 5\).

The paper then removes, one at a time, each algebraic property that theory uses — units modulo the quotient, regular multipliers, an order, commutativity, affinity, degree one — and for each rung gives an explicit witness that the property fails there together with a check that what lies above it survives, including a two-by-two matrix over \(\mathbb{Z}/4\) that witnesses why a Möbius condition had to be repaired. The Collatz map sits at the cleanest rung: its local arithmetic is as simple as this kind of arithmetic gets, so its difficulty lies in which chart comes next. Along the way three things are set right rather than restated: the series’ two derivations of the critical value 4 are one fact in two coordinates, not independent confirmation; a floating-point floor in its counting code is safe for a reason, with its closest call at a convergent of \(\ln 2/\ln 3\); and this line’s own finite check of the count law at the bottom of the ladder compared every word with itself, and is repaired.

Not claimed anywhere in this paper: the Collatz conjecture, or any statement about the orbits of any map in the family; that the phase boundary at \(m = 4\) for the proportion of contracting classes decides the behaviour of orbits on either side of it; that the family’s local simplicity makes the global problem tractable for any \((m, r)\); that a system failing a rung of the ladder cannot be analysed; and novelty of the elementary statements, most 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 generalized \(mx + r\) family with its phase boundary, and the algebraic domains of the theory with its ladder of structural breakage.

Neo.K set the series’ target and its method — that each theorem be tied to the algebraic property it actually uses — 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, gave each rung of the ladder an explicit witness and checked what survives it, measured the floating floor’s margin and identified its closest call, formalized the family and the ladder in Lean, found and repaired the vacuous check of §7, 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: checked against an independent referee, formalized where it is universal, witnessed where it is a breakage claim, and corrected where this line’s own check was empty.