What an occupation measure cannot see: three no-go theorems of a Collatz programme, and the datum they discard, measured

The Collatz conjecture is open, and nothing here bears on it directly. A natural strategy against a hypothetical divergent orbit is to average: take the empirical measures of its valuation sequence, pass to a limit, and look for a contradiction in the limit. Three consecutive rounds of a Collatz programme show, each from a different side, why this cannot work by itself. The limits exist and are shift-invariant with the critical mean — and two explicit countermodels, a Bernoulli measure and the mechanical word of slope \(\log_2 3\), satisfy every such condition, so no contradiction can come from them alone. The information that would separate an integer from a counterexample is the source lift: how the canonical 2-adic source of a growing code prefix moves. Compact normalized coordinates send every integer to the same point, erasing it again. And once a positive integer’s source freezes, its lift tail is zero forever, so no statistic of the source alone distinguishes a convergent orbit from a candidate.

We check the exact algebra all three rounds rest on and measure the datum they say is missing. Every genuine integer’s lift settles by step 11 and its canonical source then is the integer; the mechanical word’s lifts are still nonzero at step 59, its canonical source reaching 93 bits while staying 2-adically Cauchy. The lift-flux machinery already excludes the countermodel that defeated the averaging route: its mean flux sits near \(0.34\) and does not decay. And the regime in which the last round localizes the remaining obstruction is one that 99.07% of real starts are not in when it matters.

Not claimed anywhere in this paper: the Collatz conjecture; that a divergent orbit or a candidate of the programme’s kind exists or does not; whether the mechanical point is an ordinary positive integer — the rounds leave it open and so does this paper; that critical occupancy forces infinitely many nonzero lifts, which is the open target the first two rounds name; 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 — wrote, with Neo.K, the three rounds this paper rests on: the critical invariant limit, its countermodels and the lift cocycle; the source digits, the decoupling, the lift flux and the compact no-go; and source freeze with its standalone no-go.

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 — checked the rounds’ exact algebra with independently written code, measured the lift cocycle that the first round says occupation measures discard, found that the second round’s flux already excludes the first round’s countermodel, measured where real starts sit against the third round’s regime, checked the references against their sources, built and drilled the instruments, and wrote this paper. 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 measurement of the datum the no-gos are about, and the exact checks.