The exceptional weight of the Collatz stopping time

The Collatz conjecture is open, and nothing here bears on it. This paper concerns one reformulation of it and the first proof obligation that reformulation names. Let \(\sigma(n)\) be the stopping time of the shortcut map \(T\) and \(Z_k(s) = \sum_{\sigma(n) > k} n^{-s}\) the weight of the integers not yet settled at depth \(k\). For a monotone certificate system on a countable domain, holding everywhere is equivalent to the exceptional weight tending to zero; so the conjecture is equivalent to \(Z_k(s) \to 0\) for any fixed \(s > 1\), and the hypothesis of monotonicity cannot be dropped.

We measure \(Z_k(s)\) for the first time, on all 4,294,967,294 integers of \([2, 2^{32})\) for \(k \le 160\), as a rigorous two-sided bracket, and compute it exactly by a refinement algebra over parity charts; the exact values land inside the independently measured bracket at every depth checked. The natural uniform proof obligation \(Z_{k+1} \le q\,Z_k\) with \(q < 1\) is false, by a three-line argument: \(\sigma(n) = 3\) is impossible for every \(n\), so \(Z_2 = Z_3\) exactly. More generally the per-step hazard vanishes at every depth that is not an admissible stopping time, so no uniform positive lower bound on it exists, while the cumulative form survives; a single small integer that stays unsettled caps what every level in its window can contribute together. The Terras coefficient-stopping conjecture, in the programme’s finite-word form, holds on all 81,119 first-crossing words of length at most 24, the worst at 49% of its bound.

Not claimed anywhere in this paper: the Collatz conjecture; the Terras coefficient-stopping conjecture beyond word length 24; that \(Z_k(s) \to 0\); that the hazard series diverges; that any uniform decay with \(L \ge 2\) holds; and novelty of the bridge itself, which is the programme’s.

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 programme documents this paper rests on: the general bridge and its Collatz instance, the uniform and cumulative proof obligations, the first round’s chart algebra, conservation law and hazard process, and the second round’s quotient thresholds, mass-weighted hazard, two compartments and finite-word form of the Terras conjecture.

Neo.K set the programme’s target and its method — compressing the global quantifier into a single family of sums — and wrote its documents with Aletheia.

墜衡 (Zhuì Héng) — the independent verification line; model: Claude, Opus series — measured \(Z_k(s)\) and its bracket, rendered the algebra executable and confronted it with iteration, proved and checked that \(\sigma(n) = 3\) is impossible and drew the consequences for \(L\) and for the hazard, derived the hazard budget from the round’s own identity, enumerated the first-crossing words, exhibited the monotonicity counterexample, built and drilled the instruments, and wrote this paper. The bridge and the algebra are the programme’s; nothing here should be read as the verification line proposing a route.

Priority, stated plainly. No result here is claimed as new mathematics beyond what is stated as this line’s: the measurement, the impossibility of \(\sigma(n) = 3\) and its consequences, the hazard budget, and the enumeration.