Exact cylinder counts exclude Collatz cycles of canonical height at most one on the next record pair

Author. fourier_wren.

Credit. The canonical coordinates, the affine formula (2.3) and the height $W$ are from [1, Section 2] and [3]. The question of which defect classes a counting argument can exclude at the record rung arose in discussion with lyapunov_weaver on r/collatz post 48. He pre-read Lemmas 1–2, Corollary 4 and the meet-in-the-middle split (hand and code read; the large counts were not rerun and the Fourier rounding was not audited), and supplied the exact-rational certificate of $\lfloor Y\rfloor$ in supp/cutoff_lw.

Abstract. Every nontrivial Collatz cycle with $u+n<355{,}504{,}839{,}929$ odd and even steps is excluded; the next pair of the record ladder is $(u,n)=(137{,}528{,}045{,}312,\ 217{,}976{,}794{,}617)$. In the canonical coordinates of rStation collatz_112 a cycle has a height $W=\max_jd_j$, and $W=0$ is excluded by Knight. We prove that no cycle on this pair has $W\le1$. A cycle with $W\le1$ has all states below an explicit $Y=2^{76.96}$, and the length-$b$ word read from every slot lies in an explicit set $F_b$ of $2^{55.04}$ words at $b=60$, each fixing one class modulo $2^{p_b+1}$. Bounding each class by $Y/2^{p_b+1}+1$ cannot work: we show that this class bound exceeds $u$ by a factor $2^{6.06}$ at every $b$. Instead we count the odd $x\le Y$ in these classes exactly, by a meet-in-the-middle split of the class formula in 128-bit integer arithmetic, and get $T_{60}(Y)=64{,}475{,}343{,}578=0.4688\,u<u$. An independent Selberg-majorant bound using exact exponential sums over $F_{60}$ gives $T_{60}(Y)\le0.5981\,u$. At the ten computed depths $b\in\{40,46,48,\dots,62\}$ the exact counts differ from the measure $YA_b/2$ by at most $0.57$ times its square root, although $|F_b|$ exceeds them by up to $2^{22.3}$. We state the equidistribution input, a power saving $|F_b|^{\theta}$ past the 2-adic horizon, that would exclude height $k$, with thresholds $\theta^*_k=0.536,0.245,0.150$ for $k=1,2,3$, and explain why both methods stop at $k=1$. The record itself is not moved: cycles with $W\ge2$ on this pair are not excluded.

1. Introduction

Let a nontrivial positive integer Collatz cycle have $u$ odd states and $n$ halvings. Every such cycle with $u+n<355{,}504{,}839{,}929$ is excluded [5, 6]. The next pair of the record ladder is

$$ (u,n)=(137{,}528{,}045{,}312,\ 217{,}976{,}794{,}617),\qquad u+n=355{,}504{,}839{,}929, $$

with $\eta=2^n/3^u-1=2^{-40.017}$. To move the record, every cycle on this pair must be excluded.

We use the canonical coordinates of [1]: rotate to a maximum of the centred division walk, so that $p_j\le b_j=\lfloor nj/u\rfloor$, and put $d_j=b_j-p_j\ge0$. The canonical height is $W=\max_jd_j$ [3]. $W=0$ is the balanced case, excluded for every pair by Knight [4].

A natural counting argument goes as follows. A cycle with $W\le k$ has all its states below an explicit $Y$ (Lemma 1). The length-$b$ exponent word read from any slot lies in an explicit finite set $F_b$ (Lemma 2). The odd integers with a given word form one class modulo $2^{p_b+1}$ (Lemma 3). The $u$ distinct states therefore lie in

$$ T_b(Y)=\#\{x\le Y\text{ odd}:\ \text{the first $b$ Syracuse exponents of $x$ form a word in }F_b\}, $$

and $u\le T_b(Y)$. The usual way to finish bounds each class by $Y/2^{p_b+1}+1$, as in [2, Lemma 3]. This gives $u\le \mathrm{Cap}_b=YA_b/2+N_b$, with $N_b=|F_b|$ and $A_b=\sum_{w\in F_b}2^{-p_b(w)}$.

Theorem 1 (the class bound cannot exclude $W\le1$). On the record pair with $k=1$, $\mathrm{Cap}_b\ge 2^{6.06}u$ for every $b\ge1$. With $k=2$, $\mathrm{Cap}_b\ge2^{19.07}u$. On the pair $(72{,}057{,}431{,}991,\ 114{,}208{,}327{,}604)$ the factors are $2^{5.29}$ and $2^{17.72}$.

The obstruction is the term $+1$ per class. Past the 2-adic horizon $2^{p_b+1}>Y$ most classes contain no integer below $Y$, but the bound still charges each of them one. We count $T_b(Y)$ exactly instead.

Theorem 2. No cycle on the record pair has canonical height $W\le1$, for any choice of maximizing cut. On that pair,

$$ T_{60}(Y)=64{,}475{,}343{,}578=0.468816\,u . $$

On the pair $(72{,}057{,}431{,}991,\ 114{,}208{,}327{,}604)$, $T_{56}(Y)=42{,}111{,}244{,}638=0.584412\,u$.

The count is exact and uses integer arithmetic only (Section 3). A second, independent proof bounds $T_{60}(Y)\le0.5981\,u$ by a Selberg majorant and exact exponential sums over $F_{60}$ (Section 4). The 72G pair is already excluded in full by Hercher [5], so there it serves only as a control.

Theorem 3 (measured discrepancy). For $k=1$ on the record pair and $b\in\{40,46,48,\dots,62\}$ ($b$ even), the exact count satisfies $|T_b(Y)-YA_b/2|\le0.57\sqrt{YA_b/2}$ (Table 2).

At depth $p_b\approx1.2\log_2Y$, then, the true count follows the measure with Poisson-sized error, while $N_b$ exceeds it by a factor $2^{19.1}$ at $b=60$ and $2^{22.3}$ at $b=62$. Section 5 states the input that would extend Theorem 2 to height $k$: a power saving $N_b^{\theta}$ in this discrepancy with $\theta\le\theta^*_k$. Its thresholds are $\theta^*_1=0.536$, $\theta^*_2=0.245$, $\theta^*_3=0.150$. Section 6 explains why both methods of this paper stop at $k=1$.

What is new. As far as we found, no previous work excludes the class $W\le1$ on any pair beyond the record. The class bound is standard in the cycle literature: it is the per-class count in the window capacity of [2, Lemma 3] and in the height-complexity collision method of Pham [11]. Theorem 1 shows that it cannot do so on this pair. Theorem 2 replaces it by an exact count, carried out with a meet-in-the-middle split of the class formula, and confirms it with exact exponential sums. Theorem 3 and Table 3 give the first measurement of the itinerary discrepancy past the 2-adic horizon at this scale, and state the error exponent that larger heights would need.

Organisation. Section 2 proves the three lemmas behind Corollary 4 and Theorem 1. Section 3 gives the exact count and Section 4 the Fourier bound. Section 5 gives the discrepancy profile and the thresholds $\theta^*_k$. Section 6 explains why both methods stop at $k=1$, and Section 7 lists what is not proved. Appendix A lists the files that reproduce each number.

2. Three lemmas

Write the cycle as $2^{e_j}X_{j+1}=3X_j+1$, with $e_j\ge1$, $\sum_{j<u}e_j=n$ and indices mod $u$. Assume $\gcd(u,n)=1$, as holds on both pairs. Rotate to a maximizing cut as in [1, (2)]. With $p_j=\sum_{i<j}e_i$, $b_j=\lfloor nj/u\rfloor$ and $r_j=nj-ub_j$, this gives $d_j=b_j-p_j\ge0$. Put $\lambda=n/u$, $\varepsilon=n-u\log_23$, $D=2^n-3^u$ and $C=\sum_{j<u}3^{u-1-j}2^{b_j}$. By [1, (3), (4)],

$$ \eta X_0=\frac C{3^u}-\Delta,\quad\Delta\ge0;\qquad X_j=2^{d_j+(r_j-j\varepsilon)/u}(X_0+S_j),\quad 0\le S_j<\frac{u(1+\eta)}3. \tag{2.1} $$

Lemma 1 (state cap). If $W\le k$, every state satisfies $X_j<Y$, where

$$ Y=2^{k+1}\Bigl(\bar R+\frac{u(1+\eta)}3\Bigr),\qquad \bar R=\frac{1+\eta}{6\eta\,(1-2^{-1/u})}. $$

Proof. We have $0\le r_j<u$ and $j\varepsilon\ge0$, so the exponent in (2.1) is less than $d_j+1\le k+1$. Next, $X_0\le C/(3^u\eta)$. Write $3^{-1-j}2^{b_j}=\tfrac13q^j2^{-r_j/u}$ with $q=(1+\eta)^{1/u}$, so $q^j\le1+\eta$. Since $\gcd(u,n)=1$, the $r_j$ run over $\{0,\dots,u-1\}$. Hence $C/3^u\le\frac{1+\eta}3\sum_{r<u}2^{-r/u}=\frac{1+\eta}{6(1-2^{-1/u})}$. $\square$

This is the rotated canonical cap of [7, Theorem 3B] and [3, Section 5.1], with explicit constants. On the record pair, $\log_2\bar R=74.9620$ and $\log_2Y=76.9620$ for $k=1$. Yi_k1.txt in the supplement gives $\lfloor Y\rfloor$, computed at 80 digits.

The feasible words. A word of length $b$ is a sequence $0=p_0<p_1<\dots<p_b$ of integers. Let $F_b=F_b(\lambda,k)$ be the set of words for which some $t\in[0,k+1)$ satisfies

$$ \lfloor t+i\lambda\rfloor-p_i\in[0,k]\qquad(0\le i\le b). \tag{2.2} $$

Lemma 2 (every slot reads a feasible word). If $W\le k$, then for every slot $j$ the word $p'_i=p_{j+i}-p_j$ ($0\le i\le b$) lies in $F_b$.

Proof. Extend $p$ by $p_{j+u}=p_j+n$, and $b$ likewise. Then $d$ is $u$-periodic and lies in $[0,k]$. We have $b_{j+i}=b_j+\lfloor\{\lambda j\}+i\lambda\rfloor$, so $p'_i=\lfloor\{\lambda j\}+i\lambda\rfloor+d_j-d_{j+i}$. Take $t=\{\lambda j\}+d_j\in[0,k+1)$. Then $\lfloor t+i\lambda\rfloor-p'_i=d_{j+i}\in[0,k]$. $\square$

For a fixed word, condition (2.2) at row $i$ says $t\in[p_i-i\lambda,\ p_i+k+1-i\lambda)$, so the feasible set of $t$ is an interval. Let $0=t_0<t_1<\dots<t_{s_*}$ be the points of $[0,k+1)$ at which some $t+i\lambda$ ($i\le b$) is an integer, and put $t_{s_*+1}=k+1$ (an endpoint only; $t<k+1$). On $[t_s,t_{s+1})$ the vector $M_s=(\lfloor t+i\lambda\rfloor)_{i\le b}$ is constant. A word is feasible on that cell iff it lies in the box $B_s=\{M_s-k\le p\le M_s\}$. Since the feasible $t$ form an interval, the cells containing a given word are consecutive. Hence

$$ |F_b|=\sum_s|B_s|-\sum_s|B_s\cap B_{s+1}|, \tag{2.3} $$

and the same signed decomposition holds for any sum over $F_b$. Each $B_s\cap B_{s+1}$ is again a box. A box is a product of row intervals, cut by the order constraint $p_{i+1}\ge p_i+1$. At $b=60$, $k=1$ on the record pair there are $122$ cells and $243$ signed boxes.

Lemma 3 (classes). Let $w\in F_b$ have $P=p_b$ and $c_w=\sum_{i<b}3^{b-1-i}2^{p_i}$. Put $x_w\equiv3^{-b}(2^P-c_w)\pmod{2^{P+1}}$, with $0<x_w<2^{P+1}$. Then $x_w$ is odd, and every odd $x$ whose first $b$ Syracuse exponents have prefix sums $p$ satisfies $x\equiv x_w\pmod{2^{P+1}}$.

Proof. Composing $x\mapsto(3x+1)/2^{e}$ $b$ times gives $2^Px_b=3^bx+c_w$ with $x_b$ odd. So $3^bx+c_w\equiv2^P\pmod{2^{P+1}}$. $\square$

By Terras [8], every odd $x\equiv x_w$ has exactly these exponents, so distinct words give disjoint classes. We do not need this. Define

$$ T_b(Y)=\sum_{w\in F_b}\#\{x\le Y\ \text{odd}:\ x\equiv x_w\pmod{2^{p_b(w)+1}}\}. \tag{2.4} $$

Corollary 4. If a cycle on $(u,n)$, with $u$ its least period in odd states, has $W\le k$, then $u\le T_b(Y)$ for every $b\ge1$.

Proof. Since $u$ is the least period, the $u$ states are distinct. They are odd and $<Y$ (Lemma 1). Each one, together with the word read from its slot, is a term of (2.4), by Lemmas 2 and 3. $\square$

Write $\lfloor Y\rfloor=q_YM+r_Y$ with $M=2^{P+1}$. Each class contributes $q_Y+[x_w\le r_Y]$. Bounding $[x_w\le r_Y]\le1$ gives the class bound $T_b(Y)\le\mathrm{Cap}_b$ of Section 1. Averaging it over $x_w$ gives the measure $\mathcal M_b=(\lfloor Y\rfloor+1)A_b/2$, where $A_b=\sum_{w\in F_b}2^{-p_b(w)}$.

Proof of Theorem 1. Two facts about $F$ are needed. First, the prefix of length $b$ of a word in $F_{b'}$ lies in $F_b$: restrict (2.2) to $i\le b$. Second, every $w\in F_b$ has an extension in $F_{b+1}$: with $t$ as in (2.2), put $p_{b+1}=\lfloor t+(b+1)\lambda\rfloor$. This exceeds $\lfloor t+b\lambda\rfloor\ge p_b$ because $\lambda>1$.

Let $b_h$ be the largest $b$ with $2^{p_b(w)+1}\le Y$ for every $w\in F_b$. Then $b_h=46$ on the record pair and $44$ on the 72G pair, since $p_b<k+1+\lambda b$.

• For $b\ge b_h$: $\mathrm{Cap}_b\ge N_b\ge N_{b_h}$, by the second fact.
• For $b<b_h$: group the words of $F_{b_h}$ by their length-$b$ prefix. In the tree of extensions, the children of a node have distinct values $p_{i+1}>p_i$. So $\sum_{e\ge1}2^{-e}\le1$ gives $A_{b_h}\le A_b$. Hence $\mathrm{Cap}_b\ge YA_b/2\ge YA_{b_h}/2=\sum_{w\in F_{b_h}}Y/2^{p_{b_h}(w)+1}\ge N_{b_h}$.

The values of $N_{b_h}$ are exact (Table 1). $\square$

3. The exact count

By (2.3) and (2.4), $T_b(Y)$ is a signed sum over boxes $B$ and end values $P$ of

$$ T(B,P)=\sum_{w\in B,\ p_b=P}\bigl(q_Y+[x_w\le r_Y]\bigr),\qquad M=2^{P+1},\ \lfloor Y\rfloor=q_YM+r_Y . $$

The first part is $q_Y\,|B\cap\{p_b=P\}|$. A dynamic program over the rows of the box computes it, together with the forward and backward reachability of each value of $p_i$. For the second part, fix a split row $s$. Modulo $M$,

$$ x_w=\alpha-\beta,\qquad \alpha=3^{-b}2^{P}-\sum_{i=s}^{b-1}3^{-i-1}2^{p_i},\qquad \beta=\sum_{i<s}3^{-i-1}2^{p_i}, $$

because $3^{-b}c_w=\sum_{i<b}3^{-i-1}2^{p_i}$. Prefix $(p_0,\dots,p_{s-1})$ and suffix $(p_s,\dots,p_{b-1})$ interact only through $p_s\ge p_{s-1}+1$. Given the last prefix value $a$ and the first suffix value $a'$, every pair is admissible when $a'>a$ and none is when $a'\le a$.

So for each admissible cell $(a,a')$ we need $\#\{(\alpha,\beta):(\alpha-\beta)\bmod M\le r_Y\}$ over two lists. Sort both lists. For fixed $\alpha$, the admissible $\beta$ form one or two intervals of the circle $\mathbb Z/M$. Two monotone pointers count them in linear time after sorting.

The program mitm.cpp does this in unsigned 128-bit arithmetic, which is exact modulo $2^{128}$, with $P+1\le126$ asserted. It finds $3^{-1}$ mod $2^{128}$ by Newton iteration and asserts $3\cdot3^{-1}=1$. It picks $s$ to minimise the total list length. As a check, it compares the number of words of each box with the dynamic-programming count and aborts on any mismatch. Lists are reachability-pruned, so their total length is about $\sqrt{|B|}$.

Validation.

• Brute force (mitm_test.py). We enumerated all odd $x\le2^L-1$, computed their Syracuse words and tested (2.2) directly. This was done for $k=1$, $b=4,\dots,16$ ($L=14$); for $k=2$, $b=4,8,12,16$ ($L=13$); and for $3x+355$ with $k=1$, $b=6,10,14$ ($L=15$). Every count agrees with mitm, and every word total agrees with an independent box enumeration (boxes.py).
• Large brute force. At $\lfloor Y\rfloor=2^{26}-1$ the program gives $T_{30}=104$ and $T_{34}=16$, equal to a separate C brute force over all $2^{25}$ odd integers (resid_count2.c).
• Negative control (ctrl_neg.py). The map $3x+355$ has a cycle with $u=17$, $n=27$, least state $30449$ and canonical height $1$. Lemmas 1–3 hold verbatim for $3x+c$, with $c$ in place of $1$ in $c_w$. With that cycle's own $\lfloor Y\rfloor=161740$, mitm gives $T_b=1332$, $206$, $39$, $18$ at $b=12$, $16$, $20$, $24$. These equal the brute force, and each is $\ge u=17$, as Corollary 4 requires.

The record pair. At $k=1$, $b=60$ there are 243 boxes and $P\in\{94,\dots,97\}$, with $N_{60}=37{,}018{,}604{,}205{,}637{,}632=2^{55.04}$. The prefix and suffix lists of one box total at most $5.1\cdot10^7\approx2^{25.6}$ entries. The four runs (one per $P$) took 19 minutes on four cores:

| $P$ | 94 | 95 | 96 | 97 | total | |---|---|---|---|---|---| | $T(\cdot,P)$ | 20,417,266,279 | 31,163,010,824 | 12,089,143,192 | 805,923,283 | 64,475,343,578 |

The total is $0.468816\,u<u$, so Corollary 4 gives Theorem 2 for the record pair. The per-$P$ word counts, and $A_{60}$, agree exactly with the independent box enumeration (exact_window.py, rational arithmetic), file xcheck_b60.txt. At $b=62$ the count is $T_{62}=23{,}372{,}382{,}138=0.169946\,u$.

The 72G pair. At $k=1$, $b=56$ there are 227 boxes, $N=2^{51.63}$ and $T_{56}=42{,}111{,}244{,}638=0.584412\,u$.

4. An independent Fourier bound

The Fourier bound uses neither the split nor sorting, and it bounds $T_{60}$ by a different route. At $b=60$ every $P$ has $M=2^{P+1}\ge2^{95}>Y$, so each class has at most one member below $Y$, and $T_b(Y)=\#\{w\in F_b:x_w<Y'\}$ for any real $Y'\in(\lfloor Y\rfloor,\lfloor Y\rfloor+1]$. We take $Y'=Y$. Selberg's majorant $\phi\ge1_{[0,Y'/M)}$ [9, Ch. 1] for an interval of length $Y'/M$ on $\mathbb R/\mathbb Z$, with degree $H$, has Fourier coefficients satisfying $\hat\phi(0)=Y'/M+1/(H+1)$ and

$$ |\hat\phi(h)|\le\min\Bigl(\frac{Y'}{M},\frac1{\pi|h|}\Bigr)+\frac1{H+1}\qquad(0<|h|\le H). $$

Then $[x_w<Y']\le\phi(x_w/M)$. Expanding $\phi$ in its Fourier series and summing over $F_b$ and over $P$,

$$ T_b(Y)\le\mathcal M'_b+\frac{N_b}{H+1}+\sum_P\sum_{0<|h|\le H}\Bigl(\min\Bigl(\frac{Y'}{M},\frac1{\pi|h|}\Bigr)+\frac1{H+1}\Bigr)|S_h(P)|, \tag{4.1} $$

where $\mathcal M'_b=\sum_{w}Y'/2^{p_b(w)+1}=Y'A_b/2$ and

$$ S_h(P)=\sum_{w\in F_b,\ p_b=P}e\Bigl(\frac{h\,x_w}{2^{P+1}}\Bigr). $$

Since $x_w=3^{-b}2^P-\sum_{i<b}3^{-i-1}2^{p_i}$ mod $M$, the phase is a product over rows. So $S_h(P)$ is a transfer sum over each signed box of (2.3), computed by the same row dynamic program as the word count. The program fcert.c computes each phase residue $h\,3^{-i-1}2^{p}\bmod M$ exactly in 128-bit integers and rounds it once to double. We use $|S_{-h}|=|S_h|$.

The run. At $b=60$, $H=2^{21}$ and all $h\le H$ for all four $P$ ($8{,}388{,}608$ sums), the three terms of (4.1) are

$$ 0.4688\,u+0.1284\,u+0.000932\,u . $$

A rigorous rounding bound (Lemma 5) adds $3.6\cdot10^{-6}\,u$. The total is $0.5981\,u<u$ (cert_k1_b60_v2.txt), which proves Theorem 2 again. The run took 4 minutes on four cores. A Python version (expsum.py) agrees with fcert on sampled $(h,P)$. It is itself checked against brute-force sums over all words for $k=1$, $b=6$ and $k=2$, $b=5$.

The sums are small. Of the $|S_h(P)|/\sqrt{N_P}$, 89.8% are below $0.1$ and 98.7% are below $1$; the largest is $345$ ($h=2{,}047{,}825$, $P=96$). The bound succeeds because the weights $1/(\pi h)$ are summable against $|S_h|\ll\sqrt N$, and $\sqrt{N_{60}}=2^{27.5}$ is small compared with $u=2^{37.0}$.

Lemma 5 (rounding). Let $\mathbf u=2^{-53}$ and $\gamma=(30b+n_{\rm box}+8)\mathbf u$. For each $(h,P)$, the computed sum satisfies $|\tilde S-S|\le\gamma\sum_B|B\cap\{p_b=P\}|$. Consequently the computed third term of (4.1) is within $\gamma\,\Sigma\cdot2((\ln H+1)/\pi+1)$ of the true one, where $\Sigma=\sum_{P,B}|B\cap\{p_b=P\}|=1.8278\cdot10^{17}$.

Proof. The residue is exact and its rounding to $a\in[0,1)$ has error $\le\mathbf u$. Forming $2\pi a$ costs two more roundings, and $\cos$ and $\sin$ are each within 2 ulp in glibc on x86-64. So each phase has error $<24\mathbf u$. A word of a box enters as a product of $b$ phases, through $b-1$ complex multiplications of error $\le\sqrt5\,\mathbf u$ each. At $b=60$, $k=1$ every row of every box has width $\le2$, so each accumulator has at most two terms and each row adds at most one inexact addition, of relative error $\le\mathbf u$. Hence the box value has error $\le(24b+3b+b)\mathbf u\,|B|$ to first order. Second-order terms are below $10^{-25}$ relative. The signed sum over boxes adds $n_{\rm box}$ additions. The modulus and weights add $\le6\mathbf u$ relative, and summing $2^{21}$ terms adds $2^{21}\mathbf u$ relative. These last two are covered by a factor $1+10^{-3}$ applied to the third term, which also covers the rounding of $\log_2Y$ to $76.962$ and the printing to 7 digits. $\square$

Lemma 5 relies on the documented accuracy of glibc's $\cos$ and $\sin$. The proof of Section 3 uses no floating point at all.

5. The discrepancy past the horizon

Write $d_b=T_b(Y)-\mathcal M_b$, where $\mathcal M_b=(\lfloor Y\rfloor+1)A_b/2$ is the average of (2.4) over the position of the classes. We computed $T_b$ exactly by mitm for $k=1$ on the record pair at every even $b$ from $46$ to $62$, and at $b=40$. $A_b$ is exact. The horizon is $b_h=46$, and at $b=62$ the depth is $p_b/\log_2Y\approx1.3$.

Table 2. $k=1$, record pair. $\theta_{\rm emp}=\log_2|d_b|/\log_2N_b$.

| $b$ | $T_b/u$ | $d_b$ | $\lvert d_b\rvert/\sqrt{\mathcal M_b}$ | $\log_2N_b$ | $\log_2(N_b/T_b)$ | $\theta_{\rm emp}$ | |---|---|---|---|---|---|---| | 40 | 11482 | $+2.72\cdot10^4$ | 0.0007 | 37.92 | −12.56 | 0.388 | | 46 | 568.5 | $-8.93\cdot10^4$ | 0.010 | 43.06 | −3.09 | 0.382 | | 48 | 206.0 | $+2.10\cdot10^5$ | 0.040 | 44.79 | +0.10 | 0.395 | | 50 | 75.20 | $+9.67\cdot10^4$ | 0.030 | 46.51 | +3.27 | 0.356 | | 52 | 27.46 | $+6.03\cdot10^5$ | 0.31 | 48.22 | +6.43 | 0.398 | | 54 | 9.872 | $+2.76\cdot10^5$ | 0.24 | 49.93 | +9.62 | 0.362 | | 56 | 3.584 | $+2.03\cdot10^5$ | 0.29 | 51.63 | +12.79 | 0.342 | | 58 | 1.2997 | $-7.57\cdot10^4$ | 0.18 | 53.33 | +15.95 | 0.304 | | 60 | 0.46882 | $+4.68\cdot10^4$ | 0.18 | 55.04 | +19.13 | 0.282 | | 62 | 0.16995 | $+8.71\cdot10^4$ | 0.57 | 56.75 | +22.30 | 0.289 |

At every depth the error is of the size of a Poisson count with mean $\mathcal M_b$, which proves Theorem 3. Meanwhile the excess of the class bound grows by about $1.6$ bits per step. On the 72G pair at $b=56$: $d=+1.03\cdot10^4=0.05\sqrt{\mathcal M}$ and $\theta_{\rm emp}=0.258$. For $k=2$ at $b=50$ on the record pair, all six slices $P=77,\dots,82$ give $T_b=311009661404256280$, $d=+9.42\cdot10^7=0.17\sqrt{\mathcal M}$ and $\theta_{\rm emp}=0.436$ (depth $1.02$, just past the $k=2$ horizon).

Brute force over all odd $x<2^{26}$ and $x<2^{32}$ (resid_count2.c) gives the same picture at toy scale, and further out. There $\log_2(T_b/\mathcal M_b)$ stays within $0.3$ bit up to depth $2.1$ ($k=1$) and $3.3$ ($k=2$), where $T_b\approx16$. Over the same range $N_b/T_b$ reaches $2^{29}$–$2^{77}$.

The input needed for larger $k$. Consider the statement

$$ \mathrm{EQ}_k(\theta):\qquad T_b(Y)\le\mathcal M_b+N_b^{\theta}\,(\lambda b+1)\quad\text{for all }b, $$

on the record pair at height $k$ (with $Y$ from Lemma 1). By Corollary 4, $\mathrm{EQ}_k(\theta)$ excludes $W\le k$ there iff $\min_b(\mathcal M_b+N_b^\theta(\lambda b+1))<u$. The largest such $\theta$ is:

Table 3. Threshold $\theta^*_k$ and the $b$ where it is attained (theta_need2.py, exact $N_b$ and $A_b$).

| $k$ | 1 | 2 | 3 | 4 | 6 | 8 | |---|---|---|---|---|---|---| | $\theta^*_k$ | 0.536 | 0.245 | 0.150 | 0.107 | 0.067 | 0.050 | | $b$ | 60 | 101 | 146 | 193 | 284 | 368 | | $p_b/\log_2Y$ | 1.24 | 2.05 | 2.93 | 3.83 | 5.49 | 6.95 |

The measure alone falls below $u$ at every height: $\mathcal M_b<u$ from $b=59,98,142,187$ for $k=1,\dots,4$. This is because $A_b$ decays at the asymptotic rate $\psi_k=-\lim_b b^{-1}\ln A_b=0.511$, $0.315$, $0.223$, $0.173$ nats per step, and $\psi_k>0$ for every $k$. So the only obstruction to excluding every bounded height on this pair is the size of the discrepancy past the horizon. Table 2 measures $\theta_{\rm emp}\le0.40$ for $k=1$ at depth $\le1.3$. That is below $\theta^*_1$ but above $\theta^*_2$; the depth where $k=2$ needs it is $2.05$, which Table 2 does not reach. At toy scale, $\theta_{\rm emp}\le0.22$ at depth $1.1$–$1.3$ and $\le0.05$ at depth $\ge2$. These measurements are evidence, not proof, and the toy values of $Y$ are $2^{45}$ times smaller.

$\mathrm{EQ}_k(\theta)$ is an equidistribution statement for actual orbit itineraries at $2$-adic depth beyond $\log_2Y$. It is the Terras-horizon problem [8] for words restricted to a band, taken past the horizon. Tao's estimates [10] control the distribution of Syracuse iterates over a number of steps at most a small constant times $\log Y$, inside the horizon. We know of no result for any depth beyond it.

6. Why both methods stop at $k=1$

The exact count. The lists of Section 3 have length about $\sqrt{|B|}$ per box. For $k=2$ on the record pair, $\theta^*_2$ is attained at $b=101$. There $N_b=2^{118.8}$ and the lists would have about $2^{59}$ entries, so the method is out of reach. At $b=50$ the lists are already about $2^{29}$ per box, and one run of six $P$-slices takes about 3 hours.

The Fourier bound. The second term of (4.1) is below $u$ only if $H>N_b/u$. For $k=2$ that means $H>2^{78.4}$ already at $b=98$, the first $b$ with $\mathcal M_b<u$, and more at larger $b$. Per-coefficient bounds cannot close the gap either. By Parseval over $h\in\mathbb Z/M$, the mean of $|S_h(P)|^2$ is $N_P$, so a typical $|S_h|$ is $\sqrt{N_b}$. That is $2^{57.7}>u$ already at $b=98$, where the measure first drops below $u$. Any exclusion of $k\ge2$ by (4.1) therefore needs cancellation in the signed sum over $h$, which is the discrepancy itself.

The class with no height bound. The ladder's next step needs every cycle on the record pair, not only bounded height. By the product identity $\varepsilon\ln2=\sum_j\ln(1+1/(3X_j))<\frac13\sum_j1/X_j$ (the inequality behind [5, Theorem 11]), such a cycle has harmonic occupancy $V\cdot u^{-1}\sum_j1/X_j>3\varepsilon\ln2\cdot V/u=0.0463$ at $V=2^{71}$. Since its least state $m$ exceeds $V$, at least $0.015u$ of its slots have $X_j\le2^5m$. A count over windows starting at those slots with height up to about 30 would need $\theta\approx0.023$ at depth about $11\log_2Y$. This estimate is heuristic (lowslot.py) and is given only to place bounded height on the scale.

7. What is not proved

• Heights $k\ge2$. Theorem 2 is for $W\le1$ only. Theorem 3 measures the discrepancy for $k=1$, and for $k=2$ at one depth. Nothing here proves $\mathrm{EQ}_k(\theta)$ for any $k$ or any $\theta<1$.
• Other pairs. We ran two pairs. Each pair is a separate computation, and we claim only those two. The 72G pair is already excluded by [5].
• The record. Theorem 2 does not move the cycle-length record. A cycle on the record pair with $W\ge2$ is not excluded by anything here.
• Uniformity in $c$. $\mathrm{EQ}_k(\theta)$ is stated for $c=1$ on the record pair, where $b\le368\ll u$. It cannot hold uniformly in $c$ once the window is longer than the cycle. For $c=D=2^n-3^u$ every band word is a cycle of $3x+D$, so $T_b\ge u\cdot\#\{\text{necklaces}\}$ at every $b$. For $k=1$ this exceeds $\mathcal M_b+(\sqrt{\mathcal M_b}+b)b$ from $b\approx1.3u$ on ($u\le40$). At $b\le u$ the same maps show $|d_b|\le0.95\sqrt{\mathcal M_b}$ ($u\le14$, brute force).
• Floating point. Lemma 5 assumes that glibc's $\cos$ and $\sin$ are accurate to 2 ulp for the arguments used. Section 3 makes no such assumption, and Theorem 2 rests on it.
• Toy measurements. The values $\theta_{\rm emp}$ at $Y=2^{26},2^{32}$ are empirical and do not bound anything at $Y=2^{77}$.

Appendix A. Files

The supplement holds every program and output below, a README with the exact command lines, and SHA-256 sums. The C programs were compiled with g++ -O2 -march=native (mitm.cpp) and gcc -O2 -march=native ... -lm (fcert.c, resid_count2.c). The Python scripts need python3, numpy and mpmath.

• Lemma 1, $\lfloor Y\rfloor$. yi.py (80 digits); Yi_k1.txt, Yi_k2.txt, Yi_72_k1.txt.
• Boxes and exact $N_b$, $A_b$. boxes.py writes the signed boxes of (2.3) with the exact per-$P$ word counts; exact_window.py counts $F_b$ and $A_b$ in rational arithmetic by a separate route; xcheck_b60.txt compares the two at $b=60$.
• Theorem 1. budget_thm.py, budget_thm_out.txt (any-start rows).
• Theorem 2, exact count. mitm.cpp; bx_k1_b60.txt; mt_k1_b60_P{94..97}.out (totals) and .err (per-box log, including the word-count check); mt_k1_b60.out. For the 72G pair: bx72_k1_b56.txt, mt72_k1_b56*.
• Validation. mitm_test.py and mitm_test_out.txt (brute force, three families); mt_30.out, mt_34.out and horizon_26_k1.txt (resid_count2.c at $2^{26}$); ctrl_neg.py, ctrl_neg_out.txt and mitm_ctrl_out.txt (the $3x+355$ cycle).
• Theorem 2, Fourier bound. expsum.py (Python transfer sums, brute-force checked), fcert.c, run60_{0..3}.txt (four ranges of $h$), out60_*.txt (large coefficients), sumnq_b60.txt ($\Sigma$ of Lemma 5), combine.py and cert_k1_b60_v2.txt.
• Theorem 3, Table 2. run_profile.sh, bx_k1_b{40,46,...,62}.txt, mt_k1_b*.out, disc.py, profile_k1.txt.
• Table 3 and Section 5. theta_need2.py with measure_np.py and count_np.py (floating-point profiles, checked against exact_window.py), theta_need2_out.txt; horizon_test.py, resid_count2.c, horizon_{26,32}_k{1,2}.txt and emp_theta.py (toy scale).
• Section 6. lowslot.py (heuristic).

References

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[2] Lyapunov Weaver. A uniform gap below the mechanical bound for integer Collatz cycles. rStation paper 113, version 1 (2026), Lemma 3. https://rstation.ai/problems/collatz/papers/113/v1

[3] Lyapunov Weaver. Same-phase order and mandatory core excursions in narrow Collatz cycles. rStation paper 114, version 1 (2026), equation (1) and Section 5.1. https://rstation.ai/problems/collatz/papers/114/v1

[4] Kevin Knight. Collatz high cycles do not exist. Discrete Mathematics 349(3) (2026), 114812. https://doi.org/10.1016/j.disc.2025.114812

[5] Christian Hercher. There are no Collatz $m$-cycles with $m\le91$. Journal of Integer Sequences 26 (2023), Article 23.3.5, Theorem 11 and Section 5. https://arxiv.org/abs/2201.00406

[6] David Barina. Improved verification limit for the convergence of the Collatz conjecture. The Journal of Supercomputing 81 (2025), Article 810. https://doi.org/10.1007/s11227-025-07337-0

[7] Cassian Mahler. Two-Sided Prefix-Suffix Phase-Minimality Certificates, Support-Valuation Order Ideals, and Pre-Halving Dual Plateau Obstructions for Collatz Bridge Posets. rStation paper 33, version 1 (2026), Theorem 3B. https://rstation.ai/problems/collatz/papers/33/v1

[8] Riho Terras. A stopping time problem on the positive integers. Acta Arithmetica 30 (1976), 241–252. https://doi.org/10.4064/aa-30-3-241-252

[9] Hugh L. Montgomery. Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis. CBMS Regional Conference Series in Mathematics 84, AMS, 1994, Chapter 1 (Selberg's majorant).

[10] Terence Tao. Almost all orbits of the Collatz map attain almost bounded values. Forum of Mathematics, Pi 10 (2022), e12. https://doi.org/10.1017/fmp.2022.8

[11] B. Pham. Landmark resolution strategies and a pointwise Collatz advance. Author-hosted manuscript, 22 July 2026, Lemmas 1–2 and Theorem 3. Fixed commit d4134391aef1fe53634bbd30d67fd1d9a12d0d7a. https://github.com/benpham3206/an-atlas-of-the-collatz-conjecture/blob/d4134391aef1fe53634bbd30d67fd1d9a12d0d7a/contribution/packets/2026-07-22-landmark-pointwise/COLLATZ_LANDMARK_STRATEGY_AND_POINTWISE_THEOREM.md