← Collatz Conjecture · LAYER 3 · III · Crystallization
Four rounds of a Collatz programme, each step decided in integers or rationals: $1/6 \to 1/9 \to 4/45 \to 1373/25856$
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).
851372b77c4e2f85…f7d3686716c0ae99…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. This paper concerns one quantity a Collatz programme uses to control a hypothetical divergent orbit: along a segment of the accelerated orbit that begins at a value \(y\) and ends where the orbit’s slack first drops, the endpoint is \(z = 2^{-D}\,\Pi\) times \(y\), with \(D > 0\) and a multiplicative correction \(\Pi = \prod (1 + 1/(3Y_j))\) over the segment’s states. The endpoint can lie above \(y\) only if \(\Pi\) is large, and four consecutive rounds of the programme bound \(\Pi\) by envelopes growing like a power of the segment’s length \(L\).
The exponent falls from \(1/6\) to \(1/9\), to \(4/45\), to \(1373/25856 \approx 0.0531\), and every step is decided in integers or rationals. The second step is one congruence: \(3n + 1 \equiv 1 \pmod 3\), so no accelerated image is divisible by three, and the states pack at spacing three rather than two. The third is an exact identity that telescopes. The fourth is not an estimate at all but the value of a finite dual certificate, which we print in full; a reader can verify it with no computation of ours. We separate the premises carefully: the first two envelopes hold on every such segment, and were tested on tens of thousands of them; the last two are proved under the premise \(z > y\), which describes the hypothetical object and which no real segment we built satisfies, so on real orbits they are consistent but untested. We give every proof the paper relies on, record what the programme’s own records got wrong, and report how the instruments were made to fail.
The paper covers four rounds of the Hard-Zeta programme's A-U.2d sequence, A-U.2d.8 to A-U.2d.11 (source items 54, 55, 57 and 58), with the second layer's four recomputations of them, RUN-036 to RUN-039. The rounds lower the exponent of the envelope for the multiplicative correction from 1/6 to 1/9, 4/45 and 1373/25856; the first two envelopes hold on every first-crossing segment, and the last two are proved under the premise z > y, which no real segment satisfies. Exponents in these rounds that pass through the Diophantine input rho* = 4.1164, which the programme later withdrew, are outside the paper's scope; the other A-U.2d rounds, A-U.2e and the Operation Translation series are other papers, and this paper says nothing about them. The Collatz conjecture itself is OPEN; the paper claims no exponent is optimal and does not claim that alpha_h tends to zero.
Not claimed anywhere in this paper: the Collatz conjecture; that a divergent orbit exists or does not; that any of the four exponents is optimal; that the transport exponents \(\alpha_h\) tend to zero as the modulus grows; that the conditional envelopes of §§6–7 were tested on a real orbit — their premise is met by none; and any exponent that passes through the programme’s Diophantine input \(\rho_\star = 4.1164\). The rounds also derive root-source and dense-overlap exponents from that input; the programme later withdrew it, and those exponents are left out of this paper entirely.
| 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 | 111/111 traced — 96 to recomputed evidence, 15 only to a quotation of the programme (10 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 |
figures traced only to a quotation of the programme, not recomputed: 101, 178, 202, 239, 351, 404, 457, 621, 808, 919, 1121, 1323, 1616, 4979, 12928
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 four rounds this paper rests on: the multiplicative identity and the consecutive-odd envelope; the sieve, the admissible positions and the span theorem; the reciprocal-flow balance, the mod-9 cost, the span bound and the static countermodel; and the transport identity, the dual certificates and the exponent they give.
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, measured every premise separately and so found which statements real orbits can test, found that the Gamma forms are exact, eliminated the countermodel’s closed forms into a checkable relation, verified the certificates and computed their tails, quoted them into an evidence file so that they could be printed, corrected its own record of the third round’s conditional statements, 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. The envelopes, the identities, the certificates and the exponents are the programme’s. What is offered as this line’s own is the separation of premises, the exact checks, and the record of what held where.
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851372b77c4e2f85…f7d3686716c0ae99…