← Collatz Conjecture · LAYER 3 · III · Crystallization
Countermodels for the critical invariant limit, a lift cocycle that separates every integer by step 11, and source tails that freeze to zero for every positive integer
I · Precursor (the nine-paper Operation Translation series and the Hard-Zeta research programme) → II · Computation and Reproduction (73 source items rechecked, 52 reports; the bounded verification of [3, 240]) → III · Crystallization (this page, the formal paper).
7501c6a21ac34910…0bd74ae089d1dd5d…The PDF is built from the Markdown source by tools/build.py of the crystallization repository, amral-crystallization, at commit 86f12815a964f62926154a6c50b851784963f457. XeLaTeX draws random font-subset tags and a document ID on every build, so a rebuild matches in content, not byte for byte; the SHA-256 above identifies the file released here.
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.
The paper covers three rounds of the Hard-Zeta programme — A-U.1, A-U.2a and A-U.2d (source items 31, 32 and 44) — with the second layer's recomputations of them, RUN-013, RUN-014 and RUN-026: the critical invariant limit and its two countermodels, the source-lift cocycle measured, the source's binary blocks and the decoupling, the compact coordinates that send every integer anchor to one point, and source freeze with the standalone no-go for source-only observables. The limit theorems about infinite orbits are the programme's proofs (INHERITED); the paper verifies the exact algebra they rest on and measures the datum they say is discarded. The Collatz conjecture itself is OPEN; the paper does not decide whether the mechanical point is an ordinary positive integer.
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.
| Item | Result | Detail |
|---|---|---|
| 1 · claim vocabulary | PASS | all six labels, and PASS kept apart from PASS by cited theorem |
| 2 · not-claimed list | PASS | present, and it names the open problem itself |
| 3 · object specified | review item (not machine-checked) | review: can a reader recompute every figure from what is stated? |
| 4 · numbers trace to logs | PASS | 52/52 traced — 52 to recomputed evidence, 0 only to a quotation of the programme (9 small integers set aside) |
| 5 · instrument can fail | PASS | 1,474 defects planted, 1,472 caught by the check named for each; 128 controls; 57 checks |
| 6 · reproduction | PASS | section present; running it is a review step |
| 7 · contributions | PASS | section present; 3 author(s), listed alphabetically, each named in it |
| 8 · attachments | PASS | 165 gate logs; 1 input manifest(s) with SHA-256; public projection linked |
| rule · 4 nothing local | PASS | nothing machine-local or credential-shaped |
Evidence pinned at agent/collatz-verification-zhuiheng @ a2c25aa757c0704bc110458684ef3d459cb3d9e0 · collatz-verification-zhuiheng/
This paper pins its evidence at research-tree commit a2c25aa (RUN-056, 2026-10-24), from which the second layer's pages are regenerated, so every figure in it can be seen there, on the pages of the runs it cites; the corrections this version carries are recorded on RUN-056's page.
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.
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7501c6a21ac34910…0bd74ae089d1dd5d…