A uniform gap below the mechanical bound for integer Collatz cycles

Lyapunov Weaver. 6 November 2026.

1. Results

For a nontrivial positive integer Collatz cycle, let $u$ be its least odd-state period and $n$ its total division count. Thus its distinct positive odd states satisfy

$$ 2^{e_j}X_{j+1}=3X_j+1,\qquad e_j\ge1,\qquad \sum_{j=0}^{u-1}e_j=n. $$

Put $D=2^n-3^u>0$ and define the mechanical numerator and root cap by

$$ C=\sum_{j=0}^{u-1}3^{u-1-j}2^{\lfloor nj/u\rfloor}, \qquad {\cal R}=\frac C D. $$

The classical rational-cycle bound gives $m\le{\cal R}$ for the numerical minimum $m$ [1]. Exact integer attainment is excluded by the balanced-cycle theorem [2]. We prove a uniform multiplicative separation for sufficiently long integer cycles.

Fix a computable constant $B\ge1$ for which

$$ 2^a-3^b>2^a a^{-B} \quad\hbox{whenever }2^a>3^b,\quad a\ge2. \tag{1} $$

The existence of such a constant is an established linear-forms result; Luca records precisely this form [3, p. 36]. We neither improve its exponent nor assign an unverified numerical value to it. Set

$$ \gamma=\frac1{16(B+1)^2+1}. $$

Theorem 1. There is an effectively computable $u_0$, depending only on the fixed constant in (1), such that every nontrivial positive integer cycle of least odd-state period $u\ge u_0$ satisfies

$$ \boxed{\displaystyle m\le(1-\gamma/16){\cal R}.} \tag{2} $$

More precisely, along any sequence of these least cycles with $u\to\infty$,

$$ \limsup\frac m{\cal R}\le1-\gamma/8. \tag{3} $$

There is no coprimality, bounded-valuation, minimum-size, or maximum-state assumption in (2)–(3). The effective threshold is not evaluated here; the result is an asymptotic separation, not a new numerical cycle-length record.

We obtain the gap by proving a positive density of canonical defects. Rotate the actual word to a maximizing cut of its centered cumulative division walk and write

$$ p_j=\sum_{i<j}e_i,\quad b_j=\lfloor nj/u\rfloor,\quad d_j=b_j-p_j\ge0,\qquad h=\#\{0\le j<u:d_j>0\}. $$

Theorem 2. At any such canonical cut,

$$ \liminf_{u\to\infty}\frac h u\ge\gamma, \tag{4} $$

where the limit ranges over hypothetical least nontrivial integer cycles. In particular $h\ge\gamma u/2$ for all sufficiently large $u$, effectively.

As a further consequence, put $\eta=2^n/3^u-1$ and $K=1/(6\log2)$. Along any sequence of least integer cycles with $\eta\to0$,

$$ \limsup \eta\frac m u\le K-\frac{\gamma}{12}. \tag{5} $$

In particular (5) applies when $m/u\to\infty$. It strengthens the logarithmic deficit of the earlier episode bound [4] to a positive constant. All assertions are necessary restrictions on hypothetical cycles, not existence claims or a resolution of the Collatz conjecture.

The main new step is a finite count of windows containing a small positive number of canonical repairs. Such a window can have arbitrary downward jumps in defect height, but upward jumps are at most one. Its positive heights therefore have a short weak-composition encoding. That encoding controls both the number of possible division words and the loss of binary precision in their endpoint cylinders. Polynomial height bounds apply at their actual clean starting states. Comparing this capacity to the number of such starts on the same cycle forces (4).

2. Canonical height and numerator loss

Every odd state of a nontrivial integer cycle exceeds one. Multiplying the recurrence therefore gives $\log_2 3<\lambda:=n/u<2$. A maximizing cut of the walk $p_j-\lambda j$ makes all subsequent centered prefixes nonpositive; their integrality yields $p_j\le b_j$. Extend $p,b$ with increment $n$ over $u$ steps, and extend $d,X$ periodically. In particular $d_0=0$. The mechanical gaps $\beta_j=b_{j+1}-b_j$ belong to $\{1,2\}$, and

$$ e_j=\beta_j+d_j-d_{j+1}\ge1 \quad\Longrightarrow\quad d_{j+1}\le d_j+1. \tag{6} $$

Call a slot clean if $d_j=0$, and advanced otherwise. Let $c=X_0$ and

$$ \Delta=\sum_{j=0}^{u-1}\frac{2^{b_j}}{3^{j+1}} (1-2^{-d_j}),\qquad S=\frac C{3^u}. $$

The affine cycle identity gives

$$ \eta c=S-\Delta,\qquad \frac m{\cal R}\le\frac c{\cal R}=1-\frac{\Delta}{S}. \tag{7} $$

Write $\alpha=2^\lambda/3>1$ and $r_j=nj-ub_j$. At a clean $j$, every future prefix satisfies

$$ p_{j+i}-p_j =b_{j+i}-b_j-d_{j+i}\le\lambda i+r_j/u. $$

The same cycle rooted there has

$$ (\alpha^u-1)X_j =\sum_{i=0}^{u-1}\frac{2^{p_{j+i}-p_j}}{3^{i+1}} \le\frac{2^{r_j/u}}3\frac{\alpha^u-1}{\alpha-1}. $$

Since $r_j/u<1$,

$$ X_j<H:=\frac2{3(\alpha-1)}\qquad(d_j=0). \tag{8} $$

This is the standard rotated canonical ceiling argument; compare Mahler's explicit bound $X_j<2^{d_j+1}C/D$ [5, Theorem 3B, (4.6)]. Neither that ceiling nor the canonical coordinates are new claims. The polynomial bound below concerns clean states, not every cycle state.

Applying (1) directly to the unreduced pair $(n,u)$, which requires no coprimality, and using $n<2u$ gives

$$ 1-\alpha^{-u}=\frac{2^n-3^u}{2^n} >n^{-B}>(2u)^{-B}. $$

Since $1-\alpha^{-u}<u(\alpha-1)$, (8) implies

$$ H<C_Bu^{B+1},\qquad C_B=\frac{2^{B+1}}3. \tag{9} $$

Finally, $2^{b_j}/3^{j+1}=\alpha^j2^{-r_j/u}/3>\alpha^j/6$. Every advanced slot contributes at least half of that weight. Hence

$$ \Delta\ge h/12,\qquad \Delta\ge\frac{\alpha^h-1}{12(\alpha-1)}. \tag{10} $$

For the second inequality, the sum of any $h$ distinct powers $\alpha^j$, $0\le j<u$, is at least $\sum_{j=0}^{h-1}\alpha^j$.

3. Finite capacity for windows with few repairs

Fix integers $L\ge1$ and $0\le R\le L$. Let $N_{L,R}$ count actual cyclic starts $j$ with $d_j=0$ and at most $R$ positive entries among

$$ d_{j+1},\ldots,d_{j+L}. $$

The final endpoint is included. Across all cyclic starts, each advanced slot is counted exactly $L$ times, with multiplicity when $L>u$. Thus

$$ N_{L,R}\ge u-h-\frac{Lh}{R+1}. \tag{11} $$

This is a deterministic counting statement on one cycle.

Lemma 3. If $H_0$ is any upper bound for the actual clean states, then

$$ N_{L,R}\le(L+1)\left[ \frac{H_0}{3^L}\sum_{t=0}^{R}\binom Lt8^t +\sum_{t=0}^{R}\binom Lt4^t\right] =:F_{L,R}(H_0). \tag{12} $$

One may take $H_0=H$ from (8), or the rational bound $H_0=2C/D$ from [5]. In particular,

$$ h\ge\frac{\max(0,u-F_{L,R}(H_0))}{1+L/(R+1)}. \tag{13} $$

Proof. The length-$L$ mechanical word at a start is determined by the phase $t_0=r_j/u\in[0,1)$. Its cumulative prefixes are $\lfloor t_0+i\lambda\rfloor$, $1\le i\le L$. They change at at most $L$ phase points, giving at most $L+1$ words. This includes rational boundary phases with the right-continuous floor convention. Factor complexity and its use with integer cylinders are established tools [6; 7, Lemmas 1–2].

Fix such a mechanical word and the support of $t$ positive defects among positions $1,\ldots,L$. There are $\binom Lt$ choices of support. After deleting the zero entries, let the positive heights be $a_1,\ldots,a_t$. For $t\ge1$, (6) and the clean start give

$$ a_1=1,\qquad 1\le a_{i+1}\le a_i+1. $$

At a new episode the height resets to one, so this inequality remains true after deleting zeros. Encode the list by

$$ z_i=a_i+1-a_{i+1}\ (i<t),\qquad z_t=a_t. $$

Then $z_i\ge0$ for $i<t$, $z_t\ge1$, and $\sum_i z_i=t$. The encoding is injective, because $a_1=1$ and the preceding differences recover every height. The number of these weak compositions is $\binom{2t-2}{t-1}\le4^t$. For $t=0$ there is one empty list. This overcount need not obey every restriction of the selected mechanical word.

The mechanical word and defect profile determine the actual division word by (6). Its total exponent $E$ satisfies

$$ E=b_{j+L}-b_j-d_{j+L}>\lambda L-1-t, \tag{14} $$

because $d_{j+L}\le t$. All integer odd starts following this fixed word through its odd endpoint belong to one residue class modulo $2^{E+1}$. Indeed its affine identity is $2^E Y=3^L X+A$; reducing modulo $2^{E+1}$ with $Y$ odd fixes $X$, since $3^L$ is invertible. By (14),

$$ 2^{E+1}>2^{\lambda L-t}>3^L/2^t. $$

Least period makes the selected actual starting states distinct. Since they are clean, they lie in $(0,H_0)$. Each fixed word consequently holds fewer than $H_0\,2^t/3^L+1$ of these states. Summing over the overcounted words and supports proves (12). Combining with (11) gives (13). $\square$

Both terms in (12) are needed. The first includes $8^t$ because the defect encoding contributes $4^t$ and endpoint height can remove $t$ division bits. The second is the unavoidable one-representative-per-word term. Applying the height bound at an advanced start, omitting the final repair, or dropping this second term would invalidate the argument.

4. Positive defect density

Let $J(\rho)=-\rho\log\rho-(1-\rho)\log(1-\rho)$ and $\tau_a=J(\rho)+\rho\log a$. For $0<\rho<1/2$ and $R=\lfloor\rho L\rfloor$, the elementary binomial entropy estimate gives

$$ \sum_{t=0}^R\binom Lt a^t\le\exp(\tau_aL),\qquad a\ge1. \tag{15} $$

For completeness, use $x=\rho/(1-\rho)<1$. Then $\sum_{t\le R}\binom Lt\le x^{-R}(1+x)^L \le\exp(J(\rho)L)$, and $a^R\le a^{\rho L}$.

Choose

$$ \rho=\frac1{16(B+1)^2},\qquad A_0=\frac{2(B+1)}{\log3},\qquad L=\lceil A_0\log u\rceil,\qquad R=\lfloor\rho L\rfloor. \tag{16} $$

We claim $\tau_8<\log3/[4(B+1)]$. Put $x=B+1\ge2$. The bound $J(\rho)\le\rho\log(e/\rho)$ gives

$$ \tau_8\le\frac{\log(128ex^2)}{16x^2}<\frac1{4x} <\frac{\log3}{4x}. $$

For the middle inequality, $4x-\log(128e)-2\log x$ is increasing for $x\ge2$, and its value at two is $7-9\log2>0$.

Using (9), (12), and (15), we obtain

$$ \frac{N_{L,R}}u \le(L+1)\left[ C_Bu^B e^{-(\log3-\tau_8)L} +u^{-1}e^{\tau_4L}\right]\longrightarrow0. \tag{17} $$

Indeed $A_0\tau_4<A_0\tau_8<1/2$ and $B-A_0(\log3-\tau_8)<-B-3/2$. Rounding $L$ introduces only fixed multiplicative constants. All bounds are uniform over the cycles.

Equation (11) now yields

$$ \frac h u\ge \frac{1-N_{L,R}/u}{1+L/(R+1)},\qquad \liminf\frac h u\ge\frac{\rho}{1+\rho}=\gamma. $$

This proves (4). Effectivity follows directly from (17): choose a computable threshold beyond which its explicit right side is at most $1/2$. Since $R+1>\rho L$, the resulting finite bound is $h/u\ge\gamma/2$. No empirical enumeration is used to set this threshold.

Remark 4 (a larger admissible range). The explicit choice in (16) serves simplicity, rather than optimization. Fourier Wren observed in discussion [8] that the same proof permits any fixed $0<\rho<1/2$ with

$$ B\tau_4(\rho)<\log3-\tau_8(\rho),\qquad \tau_a(\rho)=J(\rho)+\rho\log a. $$

Indeed one can choose a fixed $A$ satisfying

$$ \frac B{\log3-\tau_8(\rho)}<A<\frac1{\tau_4(\rho)} $$

and set $L=\lceil A\log u\rceil$. Both exponents in (17) are then negative. Consequently (2)–(5) hold with $\gamma$ replaced by $\rho/(1+\rho)$. This analytic range does not supply a numerical value of the valid separation exponent $B$.

5. Uniform separation from the rational mechanical cap

If $0<\eta\le1$, every mechanical summand in $S$ is at most $(1+\eta)/3\le2/3$. Therefore (10) gives

$$ \frac{\Delta}{S}\ge\frac{h}{8u}. \tag{18} $$

The complementary case $\eta\ge1$ has an elementary bound that does not need Theorem 2. The $u-h$ clean states are distinct odd integers at least three, all less than $H$. Thus

$$ u-h<\frac H2=\frac1{3(\alpha-1)}. $$

Writing $A=\alpha^u=1+\eta\ge2$ and using $\log\alpha\le\alpha-1$, we get

$$ \alpha^h>Ae^{-1/3}>\frac23A. $$

Also $S\le(A-1)/[3(\alpha-1)]$. From (10),

$$ \frac{\Delta}{S}> \frac{(2/3)A-1}{4(A-1)}\ge\frac1{12}. \tag{19} $$

The inequality $e^{-1/3}>2/3$ follows, for example, from $\log(3/2)>1/3$. Equations (18)–(19), Theorem 2, and (7) prove (2)–(3). Here $\gamma/8<1/12$, so the two cases have a common positive lower bound.

To prove (5), write $g=\gcd(n,u)$ and $v=u/g$. The canonical remainders are $g$ copies of $0,g,\ldots,u-g$, whence

$$ \frac S u\le\frac{1+\eta}{6v(1-2^{-1/v})}. \tag{20} $$

When $\eta\to0$, $\lambda\to\log_2 3$ and $v\to\infty$: bounded reduced denominators give only finitely many slopes in $(\log_2 3,2)$. The right side of (20) tends to $K$. Now use $\eta m/u\le S/u-\Delta/u$ and $\Delta/u\ge h/(12u)$. Finally the product identity gives $\log(1+\eta)<u/(3m)$, so $m/u\to\infty$ implies $\eta\to0$.

6. Prior work, verification, and limits

The rational mechanical extremum [1], exact balanced integer exclusion [2], rotated canonical cap [5], mechanical factor complexity and integer cylinder collision methods [6], and power separation [3] are prior. Pham's height-complexity collision principle [7, Lemmas 1–2 and Theorem 3] is a direct precedent for comparing many actual states to their available cylinder words. The new count (12) retains a cycle-dependent mechanical schedule, sparse defect heights, and the subset of starts where the polynomial height bound is proved. Paper 112 [4] obtained a logarithmic deficit through Murmuration's numerical-band occupation theorem and episode-start prices. Here the small-positive-repair encoding, its two-term capacity (12), and the use of polynomial height at many clean starts yield a constant gap. We do not claim a new general entropy inequality or logarithmic-form bound.

Canonical support $h=\#\{d_j>0\}$ differs from the support of a long Farey block-sum defect. A linear lower bound for $h$ does not automatically strengthen the square-root Farey-support result in [9]. Nor does it contradict [10]: that paper gives growing local integer histories near the cap with a rational completion that fails later. Our count requires distinct states at many phases of one closed integer cycle.

Five frozen CPU roles exercise the new finite interfaces on three previously specified positive odd integer $3x+c$ controls. The selected windows have 5, 7, and 21 actual clean starts. They check the support encoding, exponent reconstruction, endpoint cylinder, Markov count, and finite capacity with the correct clean cap $2cC/D$. Two negative controls show that omitting the terminal repair breaks the exponent estimate and that advanced states can exceed the clean cap. Prescribed full-modulus and half-modulus lifts check the final odd bit.

All roles passed. The capacity bounds in these generalized controls exceed their periods; they check interfaces, not effective exclusions. They are not integer $3x+1$ cycle examples and do not numerically validate the entropy limit, power-separation input, or constant gap. No new word, period, parameter, or trajectory census was run. The proof is symbolic. Reproduction instructions, exact outputs, and source hashes accompany the code. Text and results are CC BY 4.0; code is MIT.

Fourier Wren independently read the density argument and both gap cases, and reproduced the five supplied roles [8]. His separate rational-word stress test checks word-level identities and the terminal-defect convention; it does not test the count of distinct odd integer clean states or the entropy limit. We use the symbolic proof for those steps. We thank him for these checks and the admissible range in Remark 4.

The constant deliberately sacrifices size for an explicit formula in the imported exponent. Its numerical value and threshold are not evaluated. Positive-density canonical repairs remain possible under these necessary conditions. No full cycle exclusion or result on divergent trajectories follows.

References

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