← Collatz Conjecture · LAYER 3 · III · Crystallization
Ten rounds of a Collatz programme, checked in exact arithmetic: an integer slack profile, a carry with no multiplier, a certificate on every return loop, a defect that composes, zero-defect returns that trade binary for ternary divisibility at a fixed rate, and a disputed Diophantine input replaced by a certified continued fraction
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).
2ba5338d20b3f7c1…4f2047a95d71412a…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. 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.
The paper covers the last ten rounds of the Hard-Zeta programme's A-U.2d sequence, A-U.2d.17 to A-U.2d.26 (source items 64 to 73), with the second layer's ten recomputations of them, RUN-045 to RUN-054: the integer lift profile, the carry conjugacy that removes the multiplier, the certificate on every return loop, the loop and path defects with their quotient-affine matrix and interval tree, the parity refinement of zero-defect returns, the compensation depths, the single-step strip, and the finite continued-fraction route of beta that replaces the withdrawn rho* (certified here from a rational bracket to 41 partial quotients). The asymptotic halves of the rounds' statements, with their o(1) and 'sufficiently large', are checked only in their finite content; the limits are the programme's (INHERITED). Which number the theorem behind rho* bounds is read from its published review: its log 3 bound follows from a linear form in 1, log 2 and log 3 that bounds beta directly, so the withdrawal's reason does not hold for the general result (RUN-055); the paper's own exact exponent is not read (UNMEASURED). Exponents that passed through rho* are outside this paper. The Collatz conjecture itself is OPEN.
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).
| 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 | 85/85 traced — 85 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 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.
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2ba5338d20b3f7c1…4f2047a95d71412a…