Return loops and the defect calculus of near-critical Collatz bridges, and a frontier re-derived without its uncertified constant

The Collatz conjecture is open, and nothing here bears on it. A Collatz programme studies how the accelerated odd map \(S(n) = (3n+1)/2^{v_2(3n+1)}\) can carry an orbit between two of its records along a near-critical bridge, one whose every suffix has valuation sum just above \(\beta\) times its length, \(\beta = \log_2 3\). Its last ten rounds turn that object into exact arithmetic, and this paper checks them in exact arithmetic: the real slack becomes an integer lift with a mechanical increment; the real cocycle left over becomes, after a change of variable, a strictly decreasing carry with no multiplier; every return of the orbit modulo \(3^k\) carries a certificate \((2^{Q} - 3^{L})\,r \equiv B \pmod{3^k}\); the quotient of a return is an integer defect with a semigroup law and a quotient-affine matrix that composes; a zero-defect return spends binary divisibility to buy ternary divisibility at the fixed rate \((Q, L)\); and the programme’s Diophantine input \(\rho^\star = 4.1164\), whose attribution it withdrew, is replaced by a finite continued-fraction statement that is certified here to 41 partial quotients — while the published review of the paper it cited shows the withdrawal’s reason does not hold.

Every theorem checked holds on every population the rounds define, with all their own counters reproduced. What the checks add is measured rather than asserted: the positive-lift branch of the rounds’ dichotomy has no finite instance in any population; the certificate holds on all 34970 loops, though the word “loop” names two objects; a semigroup law tested only by self-composition could not have told itself from its mirror image, which separates on 218 of 218 distinct pairs; the quotient lift holds on every contiguous cycle and fails on every spliced one; two published barriers are loose by exactly a factor of three; and several of the rounds’ statements, and several of this line’s own readings of them, turn out to be forced by definitions — corrected here, one of them found by a second reader.

Not claimed anywhere in this paper: the Collatz conjecture, or the programme’s coefficient-stopping reduction of it; that near-critical bridges are rare, or absent, in any orbit; any exponent that passed through the withdrawn constant \(\rho^\star\); and the exact exponent of the published theorem behind \(\rho^\star\), which only the paper states (§9).

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 ten rounds this paper rests on: the integer lift, the carry conjugacy, the return loops and their certificate, the loop and path defects with their tree, the zero-defect resonance, the compensation depths, the single-step strip, and the withdrawal of \(\rho^\star\) with its finite replacement.

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

墜衡 (Zhuì Héng) — the independent verification line; model: Claude, Opus series — rechecked every round with independently written code in exact arithmetic, reproduced the rounds’ counters from their definitions, checked the certificates the rounds’ checkers did not, measured what each shipped check could and could not see, certified \(\beta\)’s continued fraction from a rational bracket, built and drilled the instruments, found and corrected its own forced readings, and wrote this paper. Nothing in the rounds’ mathematics is this line’s invention.

Input from outside the byline. A second reader — a GLM-series language model (Flash, version 5.3) that this line ran as a redundant layer, asked to sort assertions into those that can fail and those that cannot — found that the single-step strip of §8 is forced by the residue ranges. The other corrections of §§7–8 were found by this line while building that reader’s controls. The second reader is credited here, and is not an author.

Priority, stated plainly. No result here is claimed as new mathematics. What is offered as this line’s own is the exact verification, the measurements of what the rounds’ own checks could see, and the corrections.