The finite certificate frontier of the Collatz map: what a finite verification proves, and the quantifier it cannot cross

The Collatz conjecture is open, and nothing here bears on it. For \(T(n) = n/2\) or \((3n+1)/2\), let \(\sigma(n)\) be the first \(j \ge 1\) with \(T^j(n) < n\). The conjecture is equivalent to \(\sigma(n) < \infty\) for every \(n > 1\), and a verification that \(\sigma\) is finite on \([2, N]\) proves the conjecture on \([1, N]\) and nowhere else. This paper makes that precise in the terms of the local affine atlas. The integers that follow a parity word \(w\) and have not descended within its length are one residue class cut by one exact height, \(\Omega_w \cap [1, h(w)]\), with \(h(w)\) read off the word’s contracting prefixes; a verification on \([2, N]\) is exactly the extinction of a finite frontier of such cylinders, at depth \(K(N) = \max_{n \le N} \sigma(n)\); and \(\sigma(n) = \infty\) exactly when \(n\) anchors an infinite hard branch of residue classes, so that the conjecture is the absence of such a branch at any integer above 1.

Measured in these terms: \(K(2^{40}) = 550\), attained at \(n = 898696369947\), whose orbit climbs to 395806039507110357728, about \(4.4042 \times 10^8\) times the start, before first falling below it. And one sentence of the series is sharpened from evidence to proof. The series warns that the depth \(K(N)\) cannot be made uniform in \(N\), calling a uniform bound incompatible with observed behaviour; it is false outright, because \(2^m - 1\) climbs for \(m\) steps, and the same family is a nested chain of residue classes, hard at every depth, that is anchored at no integer at all. Every statement is checked against a referee that assumes none of them, and the equivalence, the absence of a uniform depth and the anchored-branch characterization are formalized.

Not claimed anywhere in this paper: the Collatz conjecture; any bound on \(K(N)\) beyond the one value measured, or on its growth; that a finite verification at any size says anything about the integers above it; that the proportion of integers needing explicit work tending to zero implies that none do; and novelty of the elementary statements, several of which are folklore of computational verification.

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 paper of the series this paper rests on: the finite certificate frontier, with its hard heights, its frontier extinction, its anchored branches and its quantifier boundary.

Neo.K set the series’ target and its method — that what can be made exact be made exact, and the one obligation that cannot be isolated by name — 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, measured the frontier function at \(2^{40}\) from this line’s own exhaustive run and walked its extremal start again, proved and checked that no uniform depth exists, formalized the equivalence, the absence of a uniform depth and the anchored-branch theorem 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 measurement of the frontier at one size, the refutation of the uniform reading by an explicit family, and the version of the framework that can be relied on: checked against an independent referee and formalized where it is universal.