---
title: "The rotation envelope of a Collatz first crossing, and the integers underneath it"
subtitle: "An exact cap and when it is attained, exponent codes and their residue classes, a rational Catalan count, and a slope that moves by whole numbers — five rounds of a Collatz programme, decided without a logarithm"
author:
  - "Aletheia — the programme's documents and rounds; model: ChatGPT, GPT series"
  - "Neo.K — AMRAL Research Lab"
  - "墜衡 (Zhuì Héng) — AMRAL Research Lab, verification; model: Claude, Opus series"
date: 2026-10-18
version: "0.1"
standard: "1"
headline_problem: "Collatz conjecture"
evidence:
  branch: agent/collatz-verification-zhuiheng
  commit: "a2c25aa757c0704bc110458684ef3d459cb3d9e0"
  dir: collatz-verification-zhuiheng
  sources: "data/gate-logs/src45-*.json, data/gate-logs/src46-*.json, data/gate-logs/src51-*.json, data/gate-logs/src52-*.json, data/gate-logs/src53-*.json, data/external/au2d1-literature-check.json, data/external/au2d2-literature-check.json, data/gate-logs/suite-totals.json, data/results.v1.json"
abstract: |
  The Collatz conjecture is open, and nothing here bears on it directly. A Collatz programme
  studies the stretch of an accelerated orbit from a value $y$ to the first point where the orbit's
  slack drops — its first crossing — and the additive correction $B$ by which the endpoint exceeds
  the pure multiplicative prediction, $2^Q z = 3^L y + B$. Five consecutive rounds of the programme
  prove statements about this correction and about the nested chains of such crossings. This paper
  separates the statements that hold on every real orbit from those that describe only the
  hypothetical divergent one, and decides the first kind in integers.

  The correction obeys an exact cap $B/3^L \le U_\beta(L)$, where $U_\beta(L)$ looks irrational and is
  a rational number, since $2^{-\{\beta j\}} = 2^{\lfloor \beta j \rfloor}/3^j$; the cap is attained
  exactly when the orbit's valuation code is mechanical, an equivalence exercised in both directions
  on 9999 first crossings. A valuation code selects one source class modulo $2^{Q+1}$ and one endpoint
  class modulo $3^k$, checked in both directions; every crossing source of length at least two is
  $3 \pmod 4$; the number of codes under a line of rational slope is a rational Catalan number,
  checked against exhaustive enumeration; and along a nested chain the crossing slope changes by a
  ratio of integers, $\xi_{i+1} - \xi_i = J_i/(L_iL_{i+1})$, because $\beta$ cancels — verified on 144295
  renewal edges. The rounds' depth caps rest on survival premises that one real chain in 29468 meets,
  and one of their theorems is false if lifted out of its round without a bound stated a section
  earlier.
keywords: [Collatz conjecture, first crossings, Sturmian words, rational Catalan numbers, verification]
---

> **Status of this document.** Preprint, version 0.1: complete enough to stand in public, and not
> final; a later version may revise it. Not prepared for journal submission. Section 9 records who
> did which part, including the parts this line did not do.

# 0. What is claimed, and what is not

Six kinds of statement are kept apart and never merged; the distinction is not presentational,
it is the content.

| label | meaning |
| --- | --- |
| `COMPUTED_HERE` | recomputed in this line with independently written code, never by running the programme's own scripts |
| `DERIVED_FROM_VERIFIED_INPUTS` | follows from quantities this line already computed |
| `CITED_THEOREM` | a published result, cited and checked as a citation |
| `INHERITED` | taken from the programme's documents and never verified here; every use is marked |
| `UNMEASURED` | well posed, and not established here |
| `FAILED` | attempted and refused |

In particular `PASS` and `PASS by cited theorem` are **different verdicts** and are never written
as one. No verdict in this paper rests on a cited theorem; two classical facts — Weyl's
equidistribution and the cycle lemma — are cited for an asymptotic constant and a closed form whose
finite cases are checked here exhaustively.

**Not claimed anywhere in this paper:** the Collatz conjecture; that the rounds' depth caps, survival
gates or non-attainment gap hold — their premises are met by essentially no real orbit, and they were
not tested; that the Denjoy–Koksma inequality is sharp at $\log_2 3$; the rounds' density and overlap
exponents, which pass through a Diophantine input the programme later withdrew and are left out; and
novelty of the elementary statements, several of which are classical in other notation.

**Inherited:** nothing. Every statement below is proved in the text or computed here; where a round's
statement is conditional, its premise is named and measured.

| round | second-layer record | what this paper takes from it |
| --- | --- | --- |
| A-U.2d.1 | RUN-027 | the rotation envelope and its rationality |
| A-U.2d.2 | RUN-028 | the saturation equivalence |
| A-U.2d.5 | RUN-033 | exponent codes and their classes; sources are $3 \pmod 4$ |
| A-U.2d.6 | RUN-034 | the rational Catalan count; the extra bit for anchor-to-anchor codes |
| A-U.2d.7 | RUN-035 | the jump law of the crossing slope |

# 1. The setting

For odd $n$ let $q(n) = v_2(3n+1)$ and $\mathrm{Syr}(n) = (3n+1)/2^{q(n)}$. An orbit is
$Y_0 = n$, $Y_{j+1} = \mathrm{Syr}(Y_j)$ with valuations $q_j$ and $K_u = q_0 + \cdots + q_{u-1}$. Write
$\beta = \log_2 3$ and call $\delta_u = u\beta - K_u$ the **slack**. The **first crossing** of a step $s$
is $e(s) = \min\{u > s : \delta_u < \delta_s\}$. The segment from source $y = Y_s$ to endpoint
$z = Y_{e(s)}$ has length $L = e(s) - s$, partial valuations $Q_j = K_{s+j} - K_s$ and total
$Q = Q_L$, and
$$2^Q z = 3^L y + B, \qquad B = \sum_{j<L} 3^{L-1-j}\, 2^{Q_j}, \qquad \frac{B}{3^L} = \frac13 \sum_{j<L} \frac{2^{Q_j}}{3^j}.$$
For $0 < j < L$ the slack has not yet dropped: $Q_j < \beta j$, the integer statement $2^{Q_j} < 3^j$.
Every quantity of the form $c\beta + k$ with integers $c, k$ has the sign of $3^c 2^k - 1$, so every
sign this paper needs is decided by comparing two integers; no logarithm is evaluated anywhere.

# 2. The rotation envelope

**Proposition 2.1 (the cap, and when it is attained).** For every first-crossing segment,
$$\frac{B}{3^L} \le U_\beta(L) := \frac13 \sum_{j<L} \frac{2^{\lfloor \beta j \rfloor}}{3^j},$$
with equality if and only if $Q_j = \lfloor \beta j \rfloor$ for every $j < L$.

*Proof.* $Q_0 = 0$, and for $0 < j < L$ the integer $Q_j$ lies strictly below $\beta j$, so
$Q_j \le \lfloor \beta j \rfloor$ term by term. The sum of an increasing function of each term is
maximal exactly when every term is. $\square$

The first half is the first round's cap; the second is the next round's saturation equivalence,
which it states unconditionally — it is the one statement of that round that needs no hypothetical
object. Both are inequalities between rationals: $2^{-\{\beta j\}} = 2^{\lfloor \beta j\rfloor - \beta j}
= 2^{\lfloor \beta j \rfloor}/3^j$ because $2^{\beta j} = 3^j$, and $\lfloor \beta j \rfloor$ is the bit
length of $3^j$ minus one. So **$U_\beta(L)$ is a rational number**, although the round computes it
in 80-digit floating point. Since $\beta j \bmod 1$ is equidistributed (Weyl),
$U_\beta(L)/L \to \tfrac13\int_0^1 2^{-x}\,dx = 1/(6\ln 2)$, against the universal cap $L/3$ that the
programme had used before; the ratio $U_\beta(L)/(L/3)$ is $0.787654$ at $L = 5$, $0.756048$ at
$L = 10$ and $0.733817$ at $L = 34$, on its way to $1/(2\ln 2)$.

**Checked.** At the first crossing of each of the first 9999 odd starting values, from 3 up, the
cap and the termwise inequality hold with no violation, as exact rationals. The cap is attained, so
it is the best bound of its form:

| | mechanical prefix | not mechanical |
| --- | --- | --- |
| **cap attained** | 7137 | **0** |
| **not attained** | **0** | 2862 |

Both off-diagonal cells are empty and both diagonal cells inhabited, so the equivalence is exercised
in both directions, not only where it is easy. Attainment is a short-crossing phenomenon: the longest
crossing that attains the cap has length 9, and on average real crossings use 0.917624 of it. The
round's non-attainment gap $G(y, L)$ belongs to its survival setting, but it makes a prediction about
orbits that exist — wherever $G > 0$ the cap is not attained — and that prediction holds: 75 real
crossings have $G > 0$, the shortest of them of length 19, and none attains the cap.

**The Denjoy–Koksma step.** The round bounds $|\sum_{j<q} 2^{-\{\beta j\}}/3 - q/(6\ln 2)|$ at the
convergent denominators $q$ of $\beta$ by the variation of the summand, as Denjoy–Koksma allows at
every irrational. That variation is $1/3$, not $1/6$: the periodic extension of $\tfrac13 2^{-x}$
descends by $1/6$ on each period and then jumps back by $1/6$. (This line reached for $1/6$ first, and
the drill showed the data could not have caught it: the largest deviation at the round's convergents
is below $1/6$ too. The variation is now derived from its definition, not from the data.) At the
twelve convergent denominators up to 190537, recomputed exactly by mediant descent, the bound holds in
both directions and is used to at most 0.4161 of its allowance. It is not tight here, and a sharpness
result that holds for almost every rotation number, Frühwirth and Hauke's, says nothing about this one.

# 3. Exponent codes and their residue classes

A **code** is a word $w = (a_1, \ldots, a_k)$ of positive valuations, with partial sums
$A_0 = 0, A_j = a_1 + \cdots + a_j$ and total $Q = A_k$; put $B_w = \sum_{j<k} 3^{k-1-j}2^{A_j}$.

**Proposition 3.1.** If an odd $x$ has the code $w$ as its next $k$ valuations, with endpoint $z$,
then $2^Q z = 3^k x + B_w$. The sources of $w$ form exactly one class modulo $2^{Q+1}$, the class
$3^k x \equiv 2^Q - B_w \pmod{2^{Q+1}}$, and their endpoints lie in one class modulo $3^k$; so two
sources of one code differ by a multiple of $2^{Q+1}$, and their endpoints by a multiple of
$2 \cdot 3^k$.

*Proof.* The identity is the affine form of $k$ steps, and $z$ odd means $3^k x + B_w$ is divisible by
$2^Q$ exactly — the stated congruence, which is one class since 3 is invertible. Conversely, in terms of
the map $n \mapsto n/2$ or $(3n+1)/2$, having the code $w$ is following one fixed parity word of length
$Q + 1$: each valuation $a$ is an odd step followed by $a - 1$ halvings, and the endpoint's oddness is
one more letter. The integers that follow a parity word of length $m$ form one class modulo $2^m$, so
the sources of $w$ fill a whole class modulo $2^{Q+1}$; it lies inside the congruence class above, and
so equals it. Finally $2^Q z \equiv B_w \pmod{3^k}$ fixes $z$ modulo $3^k$, and two odd endpoints in one
class modulo $3^k$ differ by an even multiple of it. $\square$

Checked both ways, because "the sources lie in one class" is the easy half and cannot tell one class
from some class: on 400 random codes the identity, both class statements and the separation hold; and
1200 members of the claimed classes at which the code had never been observed were each required to
realize it, and did. On the 400 repeated-code pairs the smallest source gap is exactly one unit of
$2^{Q+1}$, so the separation is attained.

**Proposition 3.2.** A first-crossing source with $L \ge 2$ is $3 \pmod 4$.

*Proof.* $L \ge 2$ means the first step did not cross, $q_s < \beta < 2$, so $q_s = 1$, which on odd
integers is $y \equiv 3 \pmod 4$. $\square$

It held on all 27556 such sources among 54825 first crossings of the 2000 orbits of the odd starts from 3
to 4001. The next round sharpens
the separation by one bit for codes between two such sources — anchors, both $3 \pmod 4$ — to the
modulus $2^{Q+2}$, and the extra bit is exactly the size of the claim, so it was checked the same way:
on 376 codes whose destination is $3 \pmod 4$, and on 1128 class members never observed, each required
to realize the code and land on a $3 \pmod 4$ destination.

**A cap that no real orbit tests.** The same section of the round derives a depth cap for nested chains
from a source corridor $y_r - y_1 < U_\beta(L)$, which is a survival property. Of 10214 real chains with
distinct increasing sources, none lies in the corridor; the cap is untested, and is not called wrong.
A first version of the check that applied it without its premise flagged every one of them, which
measured only the check.

# 4. A count that needs no tolerance

For coprime $p, g$ with $p/g < \beta$, let $W_{p,g}$ be the set of codes of length $g$ and total $p$
whose partial sums stay on or below the line of slope $p/g$, $A_j \le jp/g$. The round states
$$\#W_{p,g} = \frac{1}{g}\binom{p-1}{g-1} = \frac{1}{p}\binom{p}{g},$$
a rational Catalan number. It follows from the cycle lemma: the $g$ cyclic rotations of a composition
of $p$ into $g$ parts are distinct when $\gcd(p, g) = 1$, and exactly one of them stays under the line.
A closed form against a set that can simply be enumerated is the most decisive kind of check there is —
no tolerance, no sampling, no shared method to share a bug. On the 34 coprime pairs $(p, g)$ with $2 \le g \le 13$, $g < p \le 22$ and $p/g < \beta$, up to $(20, 13)$ with
3876 members, enumeration and both closed forms agree exactly, and the round's five capacity examples
recompute, including one of more than a hundred digits.

The same round's binary bridge and concatenation law for $B_w$ hold exactly on 800 random codes.

# 5. A slope that moves by whole numbers

A **chain** is a set of first-crossing intervals $[s_1, e_1) \supset [s_2, e_2) \supset \cdots$ that all
cover one common point — the intervals open at that point, which the first crossings' nesting
totally orders. Concretely, at each position $t$ the chain is the set of starts $\{s \le t : e(s) > t\}$;
each distinct such set is taken once, and only when it has at least two members. For interval $i$ write $L_i$, $Q_i$ and the slack drop $D_i = Q_i - \beta L_i > 0$, and
call $\xi_i = D_i/L_i = Q_i/L_i - \beta$ its **crossing slope**. Between consecutive intervals put the
start gap $g_i = s_{i+1} - s_i$ with valuations $p_i$, the end gap $h_i = e_i - e_{i+1} \ge 0$ with
valuations $r_i$, and the slack gained and lost there, $A_i = \beta g_i - p_i > 0$ and
$E_i = r_i - \beta h_i$. Since $L_i = g_i + L_{i+1} + h_i$ and $Q_i = p_i + Q_{i+1} + r_i$,
$$A_i + D_i = D_{i+1} + E_i$$
identically, and as a pair of integer coefficients of $\beta$ and of 1 it is exact.

**Proposition 5.1 (jump law).** $\xi_{i+1} - \xi_i = J_i/(L_iL_{i+1})$ with
$J_i = Q_{i+1}L_i - Q_iL_{i+1} \in \mathbf Z$. So a slope that moves at all moves by at least
$1/(L_iL_{i+1})$.

*Proof.* $\beta$ appears in both slopes and cancels in the difference. $\square$

**Proposition 5.2 (the two kinds of edge).** On a **plateau** edge ($h_i = 0$),
$J_i = \Pi_i := Q_{i+1}g_i - p_iL_{i+1} = g_iD_{i+1} + L_{i+1}A_i \ge 1$, so the slope strictly
increases. On a **strict** edge ($h_i > 0$), $\Delta_i := r_ig_i - p_ih_i = g_iE_i + h_iA_i \ge 1$ and
$J_i = (g_i + h_i)D_{i+1} - L_{i+1}(E_i - A_i)$. A **genuine reset** — the slope falling — is exactly
$J_i < 0$, which happens only on strict edges, and then $E_i - A_i > 1/L_{i+1}$.

*Proof.* In each identity the $\beta$-terms cancel, so a positive real quantity equals an integer, which
is therefore at least 1. If $J_i < 0$ then $L_{i+1}(E_i - A_i) \ge 1 + (g_i + h_i)D_{i+1} > 1$. $\square$

These were checked on 29468 chains built from the 1499 orbits of the odd starts from 3 — the deepest of depth 17 — with 144295
renewal edges, 107592 of them plateau and 36703 strict: no violation of the jump law, of the
quantization, of either determinant or of either of its two forms, of the renewal identity, or of the
reset bound on the 21991 genuine resets, each decided as $2^N$ against $3^D$. The plateau mass also
stays under its bound, $\sum g_iL_{i+1} < L^2/2$, on every chain. And where a strict edge has
$\Delta_i = 1$, its two slopes $p_i/g_i < \beta < r_i/h_i$ are Farey neighbours — on all 6365 such edges
— with the orientation the round predicts from where the mediant falls.

**How the chains had to be built.** A first chain builder took the records of $\delta$ instead: start at
a source and take each later index whose slack exceeds the current one. That satisfies the increasing
slacks as written, but two record intervals can be disjoint, which makes $h_i$ negative and the
determinant meaningless; it reported a large fraction of edges as violating a boxed theorem that holds.
The chain is the set of intervals covering one point, and rebuilt that way the violations vanished. A
violation rate that large is a statement about the checker first.

# 6. What rests on survival

The round's caps on strict drops and plateau mass, the depth inequality they combine into, its
inversion and its high-source no-go all descend from three inputs its first section declares: an
origin-slack budget, an endpoint budget, and an overshoot bound on each crossing slope. Those are
**survival properties** of the hypothetical divergent orbit; a real orbit owes them nothing. Measured
on the 29468 chains: the overshoot bound holds on 4898 of 173763 nested intervals, and on every
interval of only one chain; the origin-slack budget on one chain; the endpoint budget on 9637; and all
three at once on one. So those theorems were applied to one chain, which settles nothing, and are not
claimed. Their derivations are arithmetic even where their hypotheses are unavailable, and were checked
on 378 grid points with no violation. One threshold in them, printed as $7.7712$, is in fact algebraic:
the $\ln 2$ in its numerator and denominator cancel, leaving $(12 + 8\sqrt2)/3$, and the inequality it
must satisfy is decidable exactly.

**A theorem that is false out of context.** The round's high-source theorem assumes $y_1 > L^2/(3\ln 2)$
and concludes, among other things, that all crossing slopes of the chain are equal and its depth at most
4. The hypothesis is common — 13433 of the 29468 chains meet it — and on them the conclusion fails: only
569 have equal slopes, and depths reach 16. That is not a counterexample. The proof also uses the
overshoot bound $\xi_i < 1/(3y_1\ln 2)$ — the round declares it a section earlier per interval, as
$\xi_i < 1/(3y_i\ln 2)$, which gives this form because sources increase along a chain — and with the
hypothesis that gives $\xi_i < 1/L^2$ and leaves room for one slope; none of the 13433 chains meets it.
A reader who lifts the theorem out of its round, with only its stated hypothesis, gets a false statement.
The arithmetic half of its lemma — two distinct rationals with denominators at most $L$ differ by at
least $1/L^2$ — holds on every chain with more than one slope.

# 7. What the records got wrong

None of the following changes a result.

- **A published file the published script does not produce.** In the first two rounds the constants
  file's keys differ from those the shipped script writes, so re-running the program would not
  reproduce the file; every value is correct.
- **Decimals beyond the computation.** The first round prints its envelope to as many as 79 decimals
  from an 80-digit floating sum, and the last two or three are wrong at $L \ge 300$ — the digits that
  summing $L$ terms at fixed precision costs. Recomputed as exact rationals, every leading digit is
  right.
- **A floating residual for an identity that is exact.** The renewal identity and both determinants
  have integer coefficients, so their residual is the pair $(0, 0)$; the shipped checker reports
  $1.93 \times 10^{-12}$.
- **A withdrawn reference, cited three times.** The literature notes of the round before these, A-U.2d,
  and of two of them, A-U.2d.1 and A-U.2d.2, cite a preprint withdrawn before the first of those notes as
  a primary reference, the third time under a heading saying the references were rechecked. Every claim attributed to it is present in it, and nothing rests on it.
- **A ledger that under-reports its own paper.** The last round's new machine-readable theorem ledger
  lists 16 internally proved results where the paper's own list has 19, and 6 no-go boundaries where
  it has 7; the inherited and external lists agree exactly.

# 8. The instrument, and how it was made to fail

Each round's recheck has its own gate and its own mutation drill: defects are planted one at a time and
each must be caught by the check **named** for it, while null controls must disturb nothing. The five
drills caught 16 of 16, 12 of 12, 18 of 18, 19 of 19 and 27 of 27 planted defects. The line's whole suite at the pinned commit stands at 57 drills and 1474 planted defects: 1472 caught by their named check, one detected only by a crash and one only by a time-out, and 128 controls undisturbed. Three things the drills found are worth stating:

- **A guard that could not fail.** The report emitters carried a check that replaced one character of a
  string and asked whether the result equalled the original — false in every possible case — and printed
  the count of digits as if it were a count of digits checked. It was replaced by one that perturbs the
  logs and requires the reported figures to move, and a demonstration that freezes a figure and watches
  the emitter refuse.
- **A total that had refused, unread.** The suite's defect total was built to refuse anything it could not
  interpret, and after two drills renamed their tally keys it refused on every run — while the archived
  figure it fed kept its old value, because a refusal writes nothing. A check that refuses is half a
  control; the other half is somebody reading the refusal.
- **A bracket the wrong way round.** A hard-coded lower bound for $\beta$ was in fact above it. The
  assertion certifying the bracket stopped it before any result used it — which is the case for
  certifying an instrument one could as easily have trusted — and in the end almost nothing needed a
  bracket at all, since every comparison is between two integers.

# 9. Contributions

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 five rounds this paper rests on: the
rotation envelope and its Denjoy–Koksma step, the saturation equivalence and the non-attainment gap, the
exponent-code classes and the anchor residue, the rational Catalan count, and the jump law with its
plateau and reset determinants, together with the survival caps built on them.

**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, found that the envelope is rational and that
every sign needed is an integer comparison, checked each equivalence and class statement in both
directions, enumerated the count against its closed form, measured every survival premise and so
separated what real orbits test from what they cannot, found the missing hypothesis of the high-source
theorem, 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. What is offered as this line's
own is the separation of premises, the exact checks, and the record of what held where.

# 10. What is open

1. Whether any segment of any orbit meets the survival premises of §6; on the orbits built here, one
   chain does.
2. Whether the Denjoy–Koksma bound is ever close to tight at $\beta$; at the convergents checked it is
   used to less than half.
3. The rounds' results not covered here, which their own lists name: the upper-wing envelopes, the
   Christoffel deficits, and the local continued-fraction bound (`UNMEASURED`).
4. The density and overlap exponents that pass through the Diophantine input. A later round of this line
   found that the theorem behind the input does bound $\beta$, by its published review, at an exponent
   that does not quite reach the programme's, and that the reason the programme gave for withdrawing it
   does not hold; it re-evaluated those exponents there. They are reported with the rounds that carry
   them, not here.

# Acknowledgements

We thank Anthropic, whose Claude models the lab's verification and recomputation lines run on, and OpenAI, whose
ChatGPT models the programme's documents were written with. Not every AI model or instance that took part in this work
is named in it — among them instances that the named authors ran as helpers; that an AI is not listed here does not
mean that none took part.

# References

For the cycle lemma, see Dvoretzky and Motzkin, and Raney; for the Denjoy–Koksma inequality, Koksma and Herman,
and for its sharpness at almost every rotation number, Frühwirth and Hauke. The withdrawn preprint the rounds cite is Niu's. The theorem behind the Diophantine input in open item 4 is Wu and Wang's, and its published review is zbMATH Open's.

1. A. Dvoretzky and Th. Motzkin, *A problem of arrangements*, Duke Math. J. 14 (1947) 305–313, doi:10.1215/S0012-7094-47-01423-3.
2. Lorenz Frühwirth and Manuel Hauke, *On the metric upper density of Birkhoff sums for irrational rotations*, Nonlinearity 36 (2023) 7065–7104, doi:10.1088/1361-6544/ad086f; arXiv:2303.15992.
3. Michael-Robert Herman, *Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations*, Publ. Math. IHÉS 49 (1979) 5–233, doi:10.1007/bf02684798.
4. J. F. Koksma, *Ein mengentheoretischer Satz über die Gleichverteilung modulo Eins*, Compositio Math. 2 (1935) 250–258.
5. Tong Niu, *Parity vectors and paradoxical sequences in the accelerated Collatz map*, arXiv:2605.13886 (2026); withdrawn in its second version.
6. G. N. Raney, *Functional composition patterns and power series reversion*, Trans. Amer. Math. Soc. 94 (1960) 441–451.
7. Hermann Weyl, *Über die Gleichverteilung von Zahlen mod. Eins*, Math. Ann. 77 (1916) 313–352, doi:10.1007/BF01475864.
8. Qiang Wu and Lihong Wang, *On the irrationality measure of log 3*, J. Number Theory 142 (2014) 264–273, doi:10.1016/j.jnt.2014.03.007; reviewed in zbMATH Open as Zbl 1303.11081.

## Data and software

- Python Software Foundation, *Python* (CPython), version 3.14.5, https://www.python.org/ — the language of every gate; the version is the one the line's tree records.
- The mpmath development team, *mpmath: a Python library for arbitrary-precision floating-point arithmetic* (version 1.3.0), https://mpmath.org/ (2023).

# A. Reproduction

The gates and the drills are Python with arbitrary-precision integers and exact rationals, standard
library only, no network. Each gate writes a JSON log, and the literature checks are archived records.
The line's own tree is not public. Its public projection is, and it is what this paper's figures can be
checked against without trusting the authors:

- **the run-by-run record**, rendered from the tree with a privacy gate on each file:
  <https://amral.evemisslab.com/collatz/recompute/> (and `/en/`);
- **the secondary-verification ledger**, whose numbers are rendered from the line's own results file and
  read back against it before the site ships:
  <https://verify.evemisslab.com/collatz-verification-zhuiheng/>.

The evidence is pinned, not copied: this paper's front matter names the research branch, the commit and
the directory its figures were drawn from, and the checker reads them exactly as they stood at that
commit. The crystallization standard and its checker live in the release repository beside the paper
(`STANDARD.md`, `tools/check_standard.py`).