First crossings of the Collatz map: one correction inequality, a dichotomy that is a partition, and a sparsity argument that fails at log 3 / log 2

The Collatz conjecture is open, and nothing here bears on it directly. A Collatz programme studies how an accelerated orbit’s slack \(\delta_m = m\log_2 3 - K_m\) falls and recovers, and what a divergent orbit would have to look like if it existed. Five consecutive rounds prove statements of two kinds: identities and bounds that hold on every orbit, and theorems about a surviving first crossing — one where the slack has dropped and the value has not — which would itself refute Terras’s coefficient-stopping conjecture. On the first crossings of 99999 odd starting values, there is none. This paper checks the first kind exactly and the second as algebra, and reports three structural facts about the rounds.

The round’s two correction inequalities are one: substituting the affine form, the starting value cancels identically, and the slack of the second is exactly three times that of the first. Its “duration or Diophantine” dichotomy is a partition, each side forced by a different argument, so neither branch can fail on its own side. And one inference does not follow: a Farey-locked bracket’s next denominator is at least the sum of its two — true at every step — but that gives Fibonacci growth only when the bracket alternates, and along the Stern–Brocot descent to \(\log_2 3\) itself the lower endpoint \(1054/665\) stays frozen for 23 steps while the denominators grow by 665 each time. Logarithmic sparsity of Farey-locked record updates therefore holds exactly when the partial quotients of \(\log_2 3\) sum to \(O(\log q_k)\) — which bounded partial quotients would give, and which is itself open.

Not claimed anywhere in this paper: the Collatz conjecture; Terras’s coefficient-stopping conjecture beyond the range measured; that a surviving crossing or reset exists or does not; that any theorem conditional on survival holds of an orbit — those were checked as algebra on synthetic configurations satisfying their hypotheses, never as statements about the Collatz map; that the partial quotients of \(\log_2 3\) are bounded or unbounded; and novelty of the elementary statements.

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 — co-wrote two of the five rounds this paper rests on with Neo.K — the mechanical word and the mismatch barrier, and the renewal determinant with the Farey lock and its sparsity argument — and is credited in the other three as their collaborating editor: the reset-block arithmetic and the anchor, the correction caps with survival and the dichotomy, and the infinite-support degeneration with its corrigendum.

Neo.K set the programme’s target and its method; three of these rounds are under Neo.K’s name, with Aletheia as collaborating editor, and two are joint.

墜衡 (Zhuì Héng) — the independent verification line; model: Claude, Opus series — rechecked each round with independently written code in exact arithmetic, separated what real orbits test from what only a counterexample could, found that the two correction inequalities are one and that the dichotomy is a partition, measured the two crossing indices the corrigendum distinguishes, found that the sparsity inference fails at \(\log_2 3\) and what is true instead, built and drilled the instruments, and wrote this paper, including the proofs as written out here. Nothing in the rounds’ mathematics 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 separation of what holds on orbits from what is conditional, the exact checks, and the one counterexample to an inference.