The rotation envelope of a Collatz first crossing, and the integers underneath it

The Collatz conjecture is open, and nothing here bears on it directly. A Collatz programme studies the stretch of an accelerated orbit from a value \(y\) to the first point where the orbit’s slack drops — its first crossing — and the additive correction \(B\) by which the endpoint exceeds the pure multiplicative prediction, \(2^Q z = 3^L y + B\). Five consecutive rounds of the programme prove statements about this correction and about the nested chains of such crossings. This paper separates the statements that hold on every real orbit from those that describe only the hypothetical divergent one, and decides the first kind in integers.

The correction obeys an exact cap \(B/3^L \le U_\beta(L)\), where \(U_\beta(L)\) looks irrational and is a rational number, since \(2^{-\{\beta j\}} = 2^{\lfloor \beta j \rfloor}/3^j\); the cap is attained exactly when the orbit’s valuation code is mechanical, an equivalence exercised in both directions on 9999 first crossings. A valuation code selects one source class modulo \(2^{Q+1}\) and one endpoint class modulo \(3^k\), checked in both directions; every crossing source of length at least two is \(3 \pmod 4\); the number of codes under a line of rational slope is a rational Catalan number, checked against exhaustive enumeration; and along a nested chain the crossing slope changes by a ratio of integers, \(\xi_{i+1} - \xi_i = J_i/(L_iL_{i+1})\), because \(\beta\) cancels — verified on 144295 renewal edges. The rounds’ depth caps rest on survival premises that one real chain in 29468 meets, and one of their theorems is false if lifted out of its round without a bound stated a section earlier.

Not claimed anywhere in this paper: the Collatz conjecture; that the rounds’ depth caps, survival gates or non-attainment gap hold — their premises are met by essentially no real orbit, and they were not tested; that the Denjoy–Koksma inequality is sharp at \(\log_2 3\); the rounds’ density and overlap exponents, which pass through a Diophantine input the programme later withdrew and are left out; 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 five rounds this paper rests on: the rotation envelope and its Denjoy–Koksma step, the saturation equivalence and the non-attainment gap, the exponent-code classes and the anchor residue, the rational Catalan count, and the jump law with its plateau and reset determinants, together with the survival caps built on them.

Neo.K set the programme’s target and its method, and wrote its documents with Aletheia.

墜衡 (Zhuì Héng) — the independent verification line; model: Claude, Opus series — rechecked each round with independently written code in exact arithmetic, found that the envelope is rational and that every sign needed is an integer comparison, checked each equivalence and class statement in both directions, enumerated the count against its closed form, measured every survival premise and so separated what real orbits test from what they cannot, found the missing hypothesis of the high-source theorem, 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 premises, the exact checks, and the record of what held where.