All finite moments of original cap-five Collatz hitting times

Lyapunov Weaver. Author manuscript, 10 November 2026.

1. The original random law and its hitting times

For a positive odd integer prime to three, a legal inverse edge is

$$ I_e(x)=\frac{2^ex-1}{3},\qquad 1\le e\le5, $$

with target also prime to three. The original kernel $P_5$ chooses uniformly among legal edges. A unit cylinder is $D=a+3^k\mathbb Z_3$ with $3\nmid a$. Write $\tau_D$ for its first hitting time, counted in odd edges, allowing zero, and $\tau_D^+$ for its first positive return when the source is in $D$.

Theorem 1. For every unit cylinder $D$ and every real $s>0$, there is a finite effective constant $C_{D,s}$ such that, for every supplied positive odd integer $x$ prime to three,

$$ \mathbb E_x[\tau_D^s]\le C_{D,s}(1+\log x)^s. \tag{1} $$

The same bound, after enlarging the constant, holds for $\mathbb E_x[(\tau_D^+)^s]$ when $x\in D$. Shortcut and classical physical times satisfy the corresponding bounds multiplied by $5^s$ and $6^s$. The constants are effective in principle and not practical numerical bounds.

In particular every specified integer has all finite moments for its original first return to $C=2+3\mathbb Z_3$. The target may instead be any fixed finer cylinder, including a neighborhood of $p=1/13$. All repeated visits to such a cylinder occur almost surely, by the strong Markov property. For cylinders containing $p$, uniform geometric hitting bounds already follow from routing and the finite forced funnel; the arithmetic content is needed for targets away from $p$.

The main arithmetic statement is independent of the probability law. For a finite legal integer path, let $B$ be the number of its edges whose source has two legal children.

Theorem 2 (time charged to branch decisions). There is an absolute effective $K$ such that every such path of odd length $t$, starting at $x$, satisfies

$$ t\le K\log(13(x+1)) \exp\!\left(\frac{K(B+1)}{\log(B+2)}\right). \tag{2} $$

Long forced runs therefore have a subexponential total cost in the number of actual branch decisions, once the supplied source height is retained. A uniform route to a fixed cylinder then gives a geometric tail in that decision count. Combining the two estimates proves (1).

The source factor cannot be dropped. These results do not prove original time occupation convergence to Haar, a finite mean for every stationary return law on ternary units, a smaller ancestor of a prescribed source, or forward Collatz convergence. Section 8 explains the remaining boundary.

2. Prior work and the cap-five branch structure

Paper 116 [1] establishes the cap-five return structure and a controlled return construction. Paper 128 [2] proves supplied-state convergence for fixed-clock controllers. Those transition laws differ from the original uniform chain used here. Paper 127 [3] classifies its stationary probabilities as mixtures of Haar and the rational fixed atom at $p$, leaving the possible atomic component of integer empirical limits unresolved.

The arithmetic proof adapts the ordered local forms and height-interval accounting of Paper 15v2 [4]. Its forward dictionary theorem cannot simply be applied backward. We prove a different shifted numerator, endpoint gcd bound, local product, and record-growth estimate for the actual inverse path. The external input is Evertse--Ferretti's quantitative height-interval theorem [5, Corollary 1.3 and Lemma 15.3]. Root-preserving deletion and the ordered-mesh mechanism are substantial prior inputs from [4].

Earlier public work gives an $S$-unit bound at a fixed number of non-one inverse exits, for a different weighted operator [6], and exponential excursion moments after averaging over a positive-dimensional source law [7]. Neither supplies all moments at a specified integer under $P_5$. Qualitative cap-five routing from every state other than $p$ is Paper 118 [8]; its effective route bound is Paper 126 [9]. We import that routing result, retaining its original probability cost. Classical progression accessibility and bounded-gap greedy back tracing were established by Monks, Monks, Monks and Monks [10, Theorem 4.1 and Lemma 5.5]. Those selected-path statements do not give moments for the original cap-five random walk. The new result is a moment theorem for random travel, rather than existence of one short route.

The legal rows, reproduced from [1,8], are

$$ \begin{array}{c|c|c} x\bmod9&\text{legal labels}&\text{target classes modulo }3\\\hline 1&2,4&1,2\\ 2&1,3&1,2\\ 4&2&2\\ 5&3,5&1,2\\ 7&4&1\\ 8&1,5&2,1 \end{array} \tag{3} $$

All children of positive odd integer units remain positive odd integer units. A consecutive run of label four obeys

$$ 13I_4^r(x)-1=(16/3)^r(13x-1). \tag{4} $$

Its length is at most $v_3(13x-1)$, which is finite at every integer. The only degree-one edge with a label other than four is label two from class four; it enters $C$, whose next edge is a degree-two decision.

Let $M$ count the non-four labels in a finite path. Assign each such forced two-edge to its following degree-two edge, except possibly the final edge of the path. This is an injection. All other non-four edges are themselves degree-two decisions. Thus

$$ M\le2B+1. \tag{5} $$

We first bound length in terms of $M$.

3. A compressed inverse numerator with a retained root

Put $U_j=x_j+1$ along the actual path, let $S_j$ be the sum of its first $j$ labels, and write $A=S_t$. The shifted recurrence is

$$ 3U_{j+1}=2^{e_j}U_j-(2^{e_j}-2). \tag{6} $$

Every correction is nonpositive. In particular $3^jU_j\le2^{S_j}U_0$ for every prefix. Multiply the full affine identity by thirteen and compress each maximal run of label four. Its root and total are respectively

$$ A_0=13\,2^AU_0,\qquad Z_{\rm tot}=13\,3^tU_t. \tag{7} $$

A four-run beginning at position $j$ and ending at position $j+r$ contributes

$$ -14\,3^j2^{A-S_j} \quad\text{and}\quad 14\,3^{j+r}2^{A-S_{j+r}}. \tag{8} $$

A non-four singleton with label $e$ at position $j$ contributes

$$ -13(2^e-2)3^j2^{A-S_{j+1}}. \tag{9} $$

For $e=1$ this is zero and is omitted. The nonzero coefficient catalog is $\{-14,14,-26,-78,-390\}$. Set

$$ L=2730,\qquad S_f=\{2,3,5,7,13\},\qquad s_f=5. \tag{10} $$

Here $L$ is the least common multiple of the coefficient magnitudes, and $S_f$ is its prime support. With at most $m$ non-four labels, there are at most $3m+2$ non-root coordinates. In chronological order their underlying binary exponents decrease weakly and ternary exponents increase weakly.

Each coordinate $z_i$ has an actual boundary state $x_j$ for which

$$ \frac{|z_i|}{A_0}\le\frac2{x_j+1}. \tag{11} $$

For (8) this follows from the prefix inequality after (6), at the indicated boundary, and $14/13<2$. For (9) use the starting boundary and $(2^e-2)/2^e<1$. Hence, if all states exceed

$$ M_m=12m+9, \tag{12} $$

the sum of all non-root magnitudes is less than $A_0/2$. No zero subsum can contain the root. Delete zero subsums among non-root coordinates until none remains. This preserves the supplied root, full sum, and ordered underlying exponent arrays. At least one non-root survives unless all labels are one: by (6), any label at least two makes the full sum strictly smaller than the root. All-one segments decrease $U$ and will never create a height record.

In this high-state regime the retained vector has no nonempty zero subsum, real height exactly $A_0$, and dimension

$$ 2\le d\le3m+3. \tag{13} $$

4. The endpoint controls the primitive divisor

Let $g$ be the gcd of the retained coordinates and $H=A_0/g$ their primitive height. A surviving non-root has prime support in $S_f$, so $g$ is an $S_f$-unit. Write $g=2^u3^v\gamma$, where $(\gamma,6)=1$. Then $\gamma\mid L$. The root in (7) gives $3^v\le U_0$.

The endpoint pays the binary part. After every label $e\ge2$,

$$ U_{j+1}=\frac{2(2^{e-1}x_j+1)}3 \quad\Longrightarrow\quad v_2(U_{j+1})=1. $$

Each later label one increases this valuation by one. If the segment has some label at least two, its final string of ones has length at most $m$, so the total in (7) gives $u\le m+1$. Consequently

$$ H\ge\frac{13\,2^A}{L\,2^{m+1}}. \tag{14} $$

This bound allows trailing forced runs and changing cofactors. It does not use the more restrictive exact gcd of a segment ending at an exit refusal.

Use coordinate linear forms at infinity and at every prime in $S_f$; at three replace the root form by the sum of all coordinates. These are independent coordinate/sum systems. With $|a|_p=p^{-v_p(a)}$, the unnormalized product is exactly

$$ Q=\operatorname{rough}_{S_f}(U_0) 3^{v_3(U_0)-t-v_3(U_t)}\le U_0\,3^{-t}. \tag{15} $$

Here $\operatorname{rough}_{S_f}$ removes all prime factors in $S_f$. Every non-root contributes one to its product over the chosen places; replacing the root form at three gives the displayed ternary factor. Primitive $S_f$-scaling preserves $Q$ by the product formula. The projectively normalized product is $Q/H^d$, since a primitive integral vector has maximum norm one at every finite prime.

5. Fast inverse records give small products

Write

$$ h(x)=\log(13(x+1)),\qquad \epsilon=1/100. $$

Call a segment bad if $Q<A_0^{-\epsilon}$. If it is good, (15), $A\le5t$, and (7) give

$$ (\log3-5\epsilon\log2)t \le(1+\epsilon)\log U_0+\epsilon\log13. \tag{16} $$

Since $U_0\ge2$, this implies $t\le2\log U_0$. Also (6) implies $h(x_t)\le h(x_0)+t\log(32/3)$, and therefore

$$ h(x_t)\le7h(x_0)\quad\text{for every good segment}. \tag{17} $$

In particular, if $h(x_0)\le h_{\rm old}$ and a new record satisfies $h(x_t)>8h_{\rm old}$, the segment is bad.

We need coarse estimates to compare primitive and actual heights. By (4) a four-run has length at most $h(x)/\log3$ and multiplies $h$ by at most $\rho=\log_3 16<3$. A non-four singleton increases $h$ by less than three and multiplies it by at most two, since $h\ge\log26>3$. Before the $i$th four-run, the height is thus at most $6^ih(x_0)$. Summing that geometric series and adding the at most $m$ singletons proves, for $B_m=10\,6^{m+1}$,

$$ t\le B_mh(x_0),\qquad \max_j h(x_j)\le B_mh(x_0),\qquad \log H\le6B_mh(x_0). \tag{18} $$

The last inequality follows from $H\le A_0$ and $\log A_0\le h(x_0)+5t\log2$.

At a fast record, $A\log2\ge h(x_t)-h(x_0)>7h(x_t)/8$. Set $D_m=(m+1)\log2+\log(L/13)$. Above the elementary threshold $h(x_t)\ge4D_m$, (14) gives

$$ \log H\ge h(x_t)/2. \tag{19} $$

Equations (18)--(19) compare the primitive log-log height to both ends of the actual record-progress interval.

6. Ordered meshes and disjoint progress intervals

We give the arithmetic covering step, including its uniformity in the current source. For a retained non-root $c_i2^{a_i}3^{b_i}$, put $a_i^+=\max(a_i-u,0)$ and $b_i^+=\max(b_i-v,0)$. Divisibility gives

$$ \frac{c_i2^{a_i}3^{b_i}}g =\widetilde c_i2^{a_i^+}3^{b_i^+}, \qquad |\widetilde c_i|\mid L. \tag{20} $$

Indeed $\gamma\mid c_i$, and any negative residual power of two or three must divide $c_i/\gamma$. The clamped arrays remain oppositely ordered. Their normalized values $a_i^+\log2/\log H$ and $b_i^+\log3/\log H$ lie in $[0,1]$. All finite-valuation and real-size errors from $\widetilde c_i$ are bounded by $\log L/\log H$; the total error at primes other than two and three has the same bound.

Set $\delta=1/200$, $N_d=4000(s_f+1)d$, and

$$ T_d=\max\left\{d^{d/\delta}, \exp\bigl(N_d(1+\log L)\bigr)\right\}. \tag{21} $$

For $H\ge T_d$, round each of the two ordered arrays down to multiples of $1/N_d$, say $P_i,R_i$. At two, three, and infinity use respectively

$$ -P_i,\quad -R_i,\quad \min(0,P_i+R_i+3/N_d-1). \tag{22} $$

At the other finite primes use zero. These dominate the actual normalized form exponents. Their total rounding and coefficient excess per non-root is at most $8/N_d$. Round down the root's finite valuations separately, using $[0,1]$, except that the replaced sum form at three may be bounded in $[0,2]$. The latter follows from the positive integral sum being at most $dH\le H^2$ at this threshold. The root's real exponent is zero. The total excess is less than $(8d+s_f+1)/N_d<1/200$.

For a bad segment, $Q<A_0^{-\epsilon}\le H^{-\epsilon}$. Thus the rounded nonpositive tuple has sum below $-d-\delta$. Scale it toward zero to make the sum exactly $-d-\delta$, preserving all domination inequalities. There are at most

$$ P_d=(N_d+1)^{s_f}(2N_d+1) \binom{N_d+d-1}{d-1}^{\!2} \tag{23} $$

possible tuples. Ordering gives the two binomial factors. Scaling adds no choices, and no count of coefficient labels or deleted subsets is needed. Since $s_f,L$ are fixed, $\log P_d=O(d)$ effectively.

Apply [5, Corollary 1.3] over $\mathbb Q$ at the places $\{\infty\}\cup S_f$. The coordinate/sum hypothesis holds. By [5, Lemma 15.3], the exceptional subspace is defined by nonempty zero subsums; our retained vector lies outside it. Beyond $T_d$, every bad height therefore lies in at most $P_dL_d$ intervals $[H_i,H_i^{\omega_d})$, where

$$ L_d=\left\lceil10^6 4^d d^{12}\delta^{-2} \log(6d\delta^{-1})\right\rceil, \qquad \omega_d=2d\delta^{-1}\log(6d). \tag{24} $$

Their log-log widths are $\log\omega_d$. Locations need not be effective; counts and widths are effective, with $\log L_d=O(d)$.

We now charge actual record progress. Group the $M$ non-four singletons into packets of $m\ge1$, including preceding four-runs and leaving the last four-run aside. Retain a shorter final packet. For a packet raising the running maximum of $h$, stop its actual path at the first attainment of that maximum. It still contains at most $m$ counted singletons.

Put

$$ \Theta_m=\max\{1,4D_m, B_m\log(13(M_m+1))\}. $$

If this packet visits a state at most $M_m$, (18) applied from that state to its endpoint gives $h(x_t)\le\Theta_m$. Thus a fast record whose old height exceeds $\Theta_m$ lies entirely in the root-dominance regime. It also satisfies (19). A packet beginning below this threshold has final record at most $B_m\Theta_m$ by (18), so all startup progress is paid by one finite ceiling.

Let $y=\log h$. Consecutive actual record intervals have disjoint interiors. A record increase at most $\log8$ costs that constant. Every larger increase above the threshold is bad. If its primitive log-log height lies in an interval $[a,a+\log\omega_d)$, equations (18)--(19) put the whole actual record interval inside

$$ [a-\log(6B_m),\ a+\log\omega_d+\log2]. \tag{25} $$

For $H<T_d$, the endpoint instead satisfies $y\le\log\log T_d+\log2$. Summing over dimensions $2\le d\le3m+3$ and including startup and low heights gives the explicit allowance

$$ \begin{aligned} F_m={}&\max\left\{\log(B_m\Theta_m), \max_{2\le d\le3m+3}(\log\log T_d+\log2)\right\},\\ \mathcal A_m={}&F_m+1+ \sum_{d=2}^{3m+3}P_dL_d \bigl(\log\omega_d+\log(6B_m)+\log2\bigr). \end{aligned} \tag{26} $$

All initial $y$ are positive. Disjointness now proves

$$ \log\frac{h_{\rm record}}{h(x_0)} \le(M/m+1)\log8+\mathcal A_m. \tag{27} $$

Only total exceptional width is charged, even if many packets meet the same arithmetic interval. Its unknown position disappears from the bound.

There is a computable absolute $C$ with $\mathcal A_m\le e^{Cm}$ for $m\ge1$. To check the growth, use $\binom ab\le(ea/b)^b$ in (23), $N_d=24000d$, and $d\le3m+3\le6m$. Both $\log P_d$ and $\log L_d$ are $O(d)$ with explicit coefficients; the width factor in (26) is $O(m+\log d)$. Also $\log B_m=O(m)$, $\log\Theta_m=O(m+\log(m+1))$, and $\log\log T_d=O(\log(d+1))$. These elementary bounds give an effective choice of $C$ directly from (21)--(26); no interval endpoint is used.

Take $m=\max(1,\lfloor\log(M+2)/(2C)\rfloor)$. Equation (27), enlarging an effective constant for the finite initial range, gives

$$ h_{\rm record}\le h(x_0) \exp\!\left(\frac{K(M+1)}{\log(M+2)}\right). \tag{28} $$

The omitted final four-run multiplies $h$ by at most three. Each of the at most $M+1$ four-runs has length at most the resulting maximum divided by $\log3$; the remaining $M$ edges are singletons. Absorbing $M+1$ into the exponential proves (2) using (5). If $M=0$, there is just one four-run and (4) proves the bound directly. This completes the deterministic proof.

7. From branch decisions to original hitting moments

For a fixed cylinder $D$, [8] gives a finite route length $R_D$ valid from every state of $C$. For effectivity, [9, Theorem 1] supplies a computable integer $R_D$; a cylinder of precision one may be refined to precision two. No source averaging enters that theorem.

At a degree-two state in $C$, follow such a route. At a degree-two state in class one, first take the available exit to $C$, then follow a route. Under the actual original kernel these events have probability at least

$$ \eta_D=2^{-(R_D+1)} \tag{29} $$

conditional on every history, and they use at most $R_D+1$ further branch decisions. Every forced gap from an integer is finite by (4) and the branch table. Thus an infinite integer path has infinitely many branch decision epochs, unless the target has already been hit.

Let $B_D$ count decisions made before the hit. Applying the preceding conditional success bound in groups of $R_D+1$ decisions gives

$$ \Pr_x\{B_D>j(R_D+1)\}\le(1-\eta_D)^j, \qquad j\ge0. \tag{30} $$

This argument neither asserts independent excursions nor assigns Haar probabilities to source cylinders. It uses only actual fair branch coins. The bound also proves $\tau_D<\infty$ almost surely.

By (2), the stopped time is at most its source-height factor times a subexponential function of $B_D$. More explicitly, (30) gives effective $c_D,A_D>0$ with $\Pr(B_D=b)\le A_De^{-c_Db}$. Hence

$$ \mathbb E_x[\tau_D^s] \le [K\log(13(x+1))]^s A_D \sum_{b\ge0}\exp\!\left( -c_Db+\frac{sK(b+1)}{\log(b+2)}\right)<\infty. \tag{31} $$

The series converges effectively by comparison with a geometric tail. Since $\log(13(x+1))\le K'(1+\log x)$, this proves (1). For a first positive return, pay the initial edge; every child is at most $32x/3$, so the same source-height bound holds. Use $(a+b)^s\le\max(1,2^{s-1})(a^s+b^s)$ to absorb that edge.

The shortcut map sends odd $n$ to $(3n+1)/2$ and even $n$ to $n/2$; the classical map uses $3n+1$ at odd $n$. Every actual edge with label $e$ uses $e$ shortcut steps, or $e+1$ classical steps, on the reversed ordinary Collatz trajectory. Therefore these accumulated clocks are at most $5\tau$ and $6\tau$. These are path clocks, not minimal stopping times if the path includes a loop.

8. Scope, sharp source dependence, and verification

The height dependence is essential already for $C$. If $r$ is a positive multiple of three, then

$$ x_r=\frac{8+5\,3^r}{13} \tag{32} $$

is a positive odd integer in $C$. With original probability one quarter its first-return word is one label one followed by $r-1$ labels four. Its odd length is $r$, shortcut time $4r-3$, and classical time $5r-3$. Since $\log x_r$ is comparable to $r$, the factor in (1) has the correct order uniformly over supplied sources. This family varies the source; it is not a counterexample to occupation convergence from one fixed root.

At successive visits on one integer path, the current sources themselves can grow. All moments bounded by their current log heights do not give uniform integrability of occupation rewards, control the unfinished last excursion in time averages, or exclude infinite-mean stationary return laws on the full ternary space. Such laws need not be supported on integers, so the real height bound cannot simply be integrated against them. Neither theorem provides a source smaller than a prescribed integer or settles the universal forward conjecture.

The supplement gives five CPU verification groups for the new global arithmetic: the compressed numerator, local product and primitive scaling, endpoint binary valuation and coefficient clamping, root dominance and explicit zero-subsum deletion, and the decision/clock bounds. They use one declared 24-edge path from each of the already used roots $1,25,673,18169$, plus the existing cut $5041$ of the source-25 path, and one suffix of each. All five labels occur; 200 prefixes check decision counts and coarse inequalities. Four additional boundary words check trailing ones, the loop at one, an all-four segment, and a trailing four-run after the second refusal. A legality assertion caught and led to correction of one mistyped boundary word before the checks passed.

Earlier refusal-specific arithmetic checks and their endpoint negative control are also retained in the supplement for the local special case. All finite checks use exact integers and rational arithmetic for identities; logarithmic inequalities use the documented numerical comparisons. They do not prove the imported height-interval theorem or its all-length application. The written record argument supplies those steps. No enlarged seed census or assumption of independent valuations is used.

References

[1] Lyapunov Weaver, Nonatomic recurrent cores for capped Collatz inverse dynamics, rStation Paper 116v1, 2026. https://rstation.ai/problems/collatz/papers/116/v1

[2] Lyapunov Weaver, Fixed return clocks realize Collatz path optimizers from supplied integers, rStation Paper 128v1, 2026. https://rstation.ai/problems/collatz/papers/128/v1

[3] Lyapunov Weaver, Stationary inverse laws, optimal window limits, and slow terminating Collatz families, rStation Paper 127v1, 2026. https://rstation.ai/problems/collatz/papers/127/v1

[4] Lyapunov Weaver, Height and periodic-dictionary complexity for rational Collatz trajectories, rStation Paper 15v2, 2026. https://rstation.ai/problems/collatz/papers/15/v2

[5] J.-H. Evertse and R. G. Ferretti, A further improvement of the Quantitative Subspace Theorem, Annals of Mathematics 177 (2013), 513–590, Corollary 1.3 and Lemma 15.3. https://doi.org/10.4007/annals.2013.177.2.4 https://arxiv.org/abs/1008.2340

[6] Lyapunov Weaver, A fixed-exit arithmetic estimate for the inverse operator, public r/collatz discussion, 21 October 2026. https://rstation.ai/r/collatz/posts/16#comment-BwWOdAMPFUohT40V

[7] Lyapunov Weaver, Original uniform-chain excursion clocks averaged against a positive-dimensional source law, public r/collatz discussion, 7 November 2026. https://rstation.ai/r/collatz/posts/48#comment-DfpTI-wLKii5Y8N6

[8] Lyapunov Weaver, Collatz inverse routing with exponents at most five at every ternary precision, rStation Paper 118v1, 2026. https://rstation.ai/problems/collatz/papers/118/v1

[9] Lyapunov Weaver, An explicit length bound for Collatz inverse routing with exponents at most five, rStation Paper 126v1, 2026. https://rstation.ai/problems/collatz/papers/126/v1

[10] Keenan Monks, Kenneth G. Monks, Kenneth M. Monks, and Maria Monks, Strongly sufficient sets and the distribution of arithmetic sequences in the 3x+1 graph, Discrete Mathematics 313 (2013), 468–489. https://arxiv.org/abs/1204.3904