# Synchronization of growing Collatz blocks and selected references Lyapunov Weaver. ## Result and prior work Let $$ T(n)= \begin{cases} n/2,&n\text{ even},\\ (3n+1)/2,&n\text{ odd}, \end{cases} $$ and let $C$ be the ordinary Collatz map. Two trajectories have *counted-fused* at shortcut time $t$ when their values and accumulated odd-step counts agree. Counted fusion persists. For Haar $x\in\mathbb Z_2$, start $x,x+1,\ldots,x+D-1$ with equal counts, and let $T_D$ be their first simultaneous counted-fusion time. All logarithms below are natural. **Theorem 1.** There is an absolute constant $C_0$ such that, for every $D\ge2$ and integer $t\ge0$, $$ \mathbb P(T_D>t)\le \min\left\{1,\frac{C_0(1+\log D)^4}{\sqrt{t+1}}\right\}. \tag{1} $$ Consequently, for $0t)$ satisfies $\kappa_D\le C_0(1+\log D)^4$. The exact pair chain and binary residue realization are due to Said Duran [1]. Leen [2] proves the uniform pair tail, sharp coefficients, and a remaining-time estimate from every integer canonical state. Leen [3] proves the existence of the fixed-family coefficients. These are imported results. The growing-family lower bounds in [4] give $\kappa_D\ge c\log D$ and a $(\log D)^2$ lower time scale. We do not match those lower powers. Leen [5, Section 4] obtains simultaneous agreement of every member of blocks of length $o(\sqrt{\log n})$ by summing adjacent-pair failures. His fixed-family theorem [3] supplies no error uniform in family size. Theorem 1 replaces the linear dependence on the number of adjacent edges by a polylogarithmic dependence on the block length. Its new ingredient is a counting argument for stopped parity prefixes: except for a small collection of minimum-numerator representatives, each prefix certifies a nearby positive displacement that counted-fuses with the original source. This certificate controls the probability of a long interval belonging to one class next to an unfused adjacent pair. Earlier constructions address different quantifiers. Valbuena [6] gives an independent proof of Said Duran's existence theorem for arbitrarily large translates of every fixed finite pattern with equal first hitting times of one. Novgorodtsev [7] gives exact clocks and peak envelopes for a specified positive-density family of consecutive pairs. Neither supplies a probability bound uniform over the size of every-member blocks. The parity-word affine formula and the need to check actual cylinder realization are classical; the new contribution is the stopped representative count and its family-tail consumer. The bound is uniform for the whole interval, so it also covers any subset of its offsets, even if that subset is selected after seeing the input. We give two actual-source survival comparisons: unrestricted selection from a controlled interval, and selection from a much wider range using only a short source prefix. The latter is uniform under arbitrary weights on that prefix, including rare prefix conditions. It uses the published scalar chain directly. Neither comparison establishes relative mixing inside an arbitrary invariant basin or convergence of every positive integer. ## Scalar inputs and a prefix estimate A canonical pair has the form $(z,3^kz+c)$, with count gap $k\ge0$, integer offset $c$, and normalized offset $U=|c|/3^k$. The fused state has $k=U=0$. Fix $0<\alpha<1$. We use two results from [2, Section 2]: $$ \mathbb P_{k,c}(\sigma>s) \le \min\left\{1,\frac{A_\alpha(k+1+U^\alpha)}{\sqrt{s+1}}\right\}, \tag{3} $$ and the one-step bound $$ U'\le A_bU+\frac12,\qquad A_0=\frac12,\quad A_1=\frac32,\qquad \chi_\alpha=\frac{2^{-\alpha}+(3/2)^\alpha}{2}<1. \tag{4} $$ Here the next anchor bit $b$ is fair conditional on the actual finite prefix. Equation (3) is [2]'s “Remaining absorption from a scalar state” estimate; replacing its $\sqrt s$ denominator by $\sqrt{s+1}$ changes an absolute constant. The count gap changes by at most one per shortcut step. **Lemma 2.** Uniformly over integer canonical states, $$ \mathbb P_{k,c}(\sigma>s) \le \min\left\{1, A\sqrt{\frac{1+k^2+\log(2+U)}{s+1}}\right\}. \tag{5} $$ For the adjacent pair started from $(0,1)$, every actual prefix of length $M$ therefore satisfies, for $t\ge2M$, $$ \mathbb P(\sigma>t\mid\mathcal F_M) \le\frac{A'(M+1)}{\sqrt{t+1}}. \tag{6} $$ **Proof.** Concavity and (4) give $\mathbb E U_B^\alpha\le\chi_\alpha^B U^\alpha+ 2^{-\alpha}/(1-\chi_\alpha)$. Choose a deterministic $B=O(1+\log(2+U))$ making this bounded by a constant. The canonical six-row kernel in [2] also gives $\mathbb E k_B^2\le k^2+B$, including absorption: the conditional squared-level increment is at most one. For $s\ge2B$, condition at $B$, apply (3), and use $\mathbb E k_B\le\sqrt{k^2+B}$. For $s<2B$, the right side of (5) is at least one after enlarging $A$, so the trivial bound suffices. This is [2]'s uniform-pair contraction argument with a nonzero initial level retained; no new scalar mechanism is claimed. From $(0,1)$, a deterministic $M$-step prefix has $k_M\le M$ and $U_M\le2(3/2)^M$ by (4). Apply (5) for the remaining $t-M$ steps. The estimate is valid on every prefix separately; it remains available after retaining an arbitrary $\mathcal F_M$-measurable event. $\square$ ## Actual sources with the same counted future For a binary parity word $w=(w_0,\ldots,w_{a-1})$, let $q_s$ count its ones before position $s$, and put $q=q_a$. Its affine numerator is $$ N(w)=\sum_{s:w_s=1}3^{q-q_{s+1}}2^s,\qquad T^a(x)=\frac{3^qx+N(w)}{2^a}. \tag{7} $$ The word is realized by exactly the source residue $x\equiv-N(w)3^{-q}\pmod{2^a}$. This is the usual parity-word bijection, including the fresh Haar suffix after conditioning on a prefix. **Lemma 3.** Suppose $w,v$ have the same length $a$, the same number $q$ of ones, and $N(w)\equiv N(v)\pmod{3^q}$. Then, for $$ d=\frac{N(w)-N(v)}{3^q}, \tag{8} $$ every source $x$ realizing $w$ counted-fuses with $x+d$ by time $a$. **Proof.** The displacement is integral and $3^q(x+d)+N(v)=3^qx+N(w)$. The parity-word residue formula consequently places $x+d$ in the cylinder for $v$. Equation (7) gives equal endpoints and equal counts. Distinct words with the same $(a,q)$ have distinct numerators: equal numerators would give the same source residue and hence the same parity word. $\square$ The conclusion concerns actual Haar sources and every integer lift of their finite cylinders. For $d>0$, positive integer sources have positive siblings. ## Stop the prefix at a prescribed inverse slope Define $$ B_s=\frac{2^s}{3^{q_s}},\qquad \tau_L=\inf\{a\ge1:B_a\ge L\},\qquad L>1. $$ An even bit doubles $B_s$; an odd bit multiplies it by $2/3$. At the first crossing, $$ L\le B_{\tau_L}<2L. \tag{9} $$ All earlier $B_sM$, or that the stopped word is one of the chosen representatives. This event is measurable from the first $M$ source parity bits, equivalently its binary residue modulo $2^M$, and $$ \mathbb P(F_{L,M})\le\mathbb P(\tau_L>M)+\frac ML. \tag{11} $$ Outside this event, the asserted positive sibling exists. The competing word is selected within the same stopping-length and count class; words of different lengths are never equated. For $D\ge4096$, choose $$ M=\lceil256\log D\rceil,\qquad L=\frac{D-1}{M}>1. $$ Then $d<(D-1)/3M$, then $M\log2-q_M\log3<\log L\le M/256$. The inequality $$ \log2-\frac58\log3 =\frac18\log\frac{256}{243} \ge\frac{13}{2048}>\frac1{256} $$ forces $q_M>5M/8$. The parity bijection gives $q_M\sim\operatorname{Bin}(M,1/2)$, so Hoeffding yields $\mathbb P(\tau_L>M)\le e^{-M/32}\le D^{-8}$. Writing $F_D=F_{L,M}$, we have proved $$ \mathbb P(F_D)\le D^{-8}+\frac{M^2}{D-1}. \tag{12} $$ ## The family tail Write $\theta_D(t)=\mathbb P(T_D>t)$, and let $\theta_1(t)=0$. Persistence, transitivity, and Haar translation invariance give $$ \theta_D(t)-\theta_{D-1}(t) =\mathbb P\bigl(T_{\{0,1\}}>t,\ T_{[1,D)}\le t\bigr). \tag{13} $$ For $t\ge M$, the event on the right is contained in $F_D$. Indeed, outside $F_D$, the source at offset zero has a counted sibling at some integer offset $1\le da+s)\ge p_s(a),\qquad s\ge0. $$ More explicitly, write $x=r+2^az$, $0\le r<2^a$. The offsets $(-r)\bmod2^a$ and $(-1-r)\bmod2^a$ select the two words. At time $a$ their canonical pair state is $(a,-1)$ when $r>0$, and $(a,3^a-1)$ when $r=0$. Apply [4]'s uniform surviving-prefix clock to this actual selected pair, killed on first reaching gap one from height $a-1$. For each fixed $u>0$, reflection and the clock conversion give $$ \liminf_{D\to\infty}\mathbb P(T_D>u(\log D)^2) \ge\operatorname{erf}\!\left(\frac1{\sqrt u\log2}\right). $$ Indeed, for any fixed $\delta>0$, survival through $N=\lceil u(\log D)^2/(2-\delta)\rceil$ mixed steps forces a physical duration exceeding the deadline except for a vanishing relative error. The reflection limit is $2\Phi(\sqrt{2-\delta}/(\sqrt u\log2))-1$; let $\delta\downarrow0$ and note $a=o((\log D)^2)$. Fatou and the Brownian hitting-time moment integral also give the lower moment coefficient $\Gamma(1/2-p)/(\sqrt\pi(\log2)^{2p})$, for $00. \tag{15} $$ **Proof.** The range is a nonnegative submartingale: its conditional drift is half the sum of the mixed-parity indicators of the minimum and maximum count groups, as in [3]. At any fixed time $b$, select one member of each extreme group before revealing further bits. They cannot counted-fuse before their gap $K_b$ returns to zero. Its mixed-step walk is symmetric, while physical time dominates mixed time. If $p_s(v)$ is simple-walk survival through $s$ steps from $v$, then $\mathbb P(T_D>b+s)\ge\mathbb E p_s(K_b)$. Since $K_b\le b$ and $\sqrt s\,p_s(v)\to\sqrt{2/\pi}\,v$, (1) implies $\mathbb E K_b\le A\ell_D$. Doob's inequality and an increasing union over $b$ give (15). $\square$ **Theorem 5.** Let $h\ge1$, $D\ge2$, and $c_0\in\mathbb Z$. On one Haar source $x$, start the lower trajectory $x$ with formal count zero and every higher trajectory $3^{h+1}x+c$, $c_0\le ct\}$ and $B_{c'}=\{\tau_{c'(x)}>t\}$. Set $r=h/\ell_D$. For $r$ sufficiently large in terms of $a,A$, both selected events have positive probability and $$ \left\|\mathcal L(x\mid B_c)-\mathcal L(x\mid B_{c'})\right\|_{\rm TV} \le A_{a,A}\frac{\sqrt{\log(2+r)}}r. \tag{16} $$ More strongly, outside a coupling failure of this size, the sources agree and the chosen higher trajectories have counted-fused by the common time $$ b=\max\left\{1,\min\left(\lfloor t/2\rfloor, \left\lfloor\frac{h^2}{64\log(2+r)}\right\rfloor\right)\right\}. $$ Both labeled pair futures after $b$ and their first-exit records agree. An explicit exit record is $(\tau,T^\tau(x),T^\tau(3^{h+1}x+c),p_\tau,q_\tau)$ for finite $\tau$, and $(\infty,\dagger)$ otherwise, where $p,q$ are the absolute formal counts. The selectors may depend on the entire source; no adaptedness assumption is imposed. The same assertion holds exactly on a block $I$ of $2^R$ positive integer sources, with $R\ge t$ and all higher starts positive, for arbitrary selectors on $I$. **Proof.** Use $c_0$ as baseline. Its killed height starts at $h$ and has bounded fair martingale increments. The maximal Azuma bound gives $\mathbb P(\min_{s\le b}H_0(s)\le h/2)\le e^{-h^2/(8b)}$. The higher-family root is Haar because multiplication by $3^{h+1}$ is invertible on $\mathbb Z_2$. Combine (1), (15), and this bound. There is a prefix event $G_b$, on which all higher trajectories have counted-fused by $b$ and every lower-higher pair remains alive through $b$, with $$ \mathbb P(G_b^c)\le \frac{A'\ell_D}{\sqrt{b+1}}+\frac{A'\ell_D}{h} +e^{-h^2/(8b)}. \tag{17} $$ Explicitly, intersect higher-family fusion by $b$, baseline height above $h/2$ through $b$, and higher-count range below $h/2$ through $b$. The baseline height minus that range bounds every other signed height below. On $G_b$, all pair futures and first-exit records agree. For large $r$, the chosen $b$ makes (17) at most $\epsilon=A_a\sqrt{\log(2+r)}/r$. The baseline physical exit time dominates a simple-walk hitting time from $h$. Since $t\le Ah^2$, its survival event $B_0$ has probability at least a positive $c_A$, uniformly in $c_0,h,t$. On $G_b$, every exit time is identical and exceeds $b$, so $B_c\mathbin\triangle B_0\subseteq G_b^c$ for every selector, even one depending on the full future. For $\epsilont\}}X \le \frac{C_0(1+\log D)^4}{\sqrt{t+1}}+\frac{2^t}{X}. \tag{18} $$ There is no factor $D$ in the residue-counting error. **Corollary 6.** Let $D(n)\ge1$ be any prescribed integer function satisfying $$ \log(D(n)+1)=o\bigl((\log n)^{1/8}\bigr). \tag{19} $$ For every fixed $0<\delta<1$, a natural-density-one set of $n$ has the following property: writing $2^j\le n<2^{j+1}$, all $D(n)$ starts $n,\ldots,n+D(n)-1$ agree at ordinary time $2\lfloor\delta j\rfloor$, and at every later common time. **Proof.** In each shell use the maximum prescribed width $D_j$, which satisfies the same little-oh bound without a monotonicity assumption. Apply (18) with $X=2^j$ and $t=\lfloor\delta j\rfloor$. Its exceptional proportion tends to zero. Summing dyadic shell counts proves natural density one. Counted fusion by shortcut time $t$ yields a common ordinary time $t+q\le2t$, since the odd counts agree. Determinism then gives agreement at the fixed time $2t$. $\square$ The descent transfers in [5, Section 4] now apply with the larger width (19): * For every $\log_4 3n^c$, so the singleton descent witness must occur afterward. * For every $f(n)\to\infty$, logarithmic-density-almost every $n$ has some common ordinary time and value $yt) -\mathcal L_{\mu_W}(x\mid\tau_{c'}>t)\right\|_{\rm TV} \le C_{a,A}\frac{\sqrt{\log(2+r)}}r. \tag{20} $$ A coupling with this failure bound preserves the actual source, both labeled pair futures after $b=m+s$, and the full first-exit record, where $$ s=\left\lfloor\frac{h^2}{128\log(2+r)}\right\rfloor. $$ The same statement holds exactly on a complete positive-integer block of length $2^R$, $R\ge t$, when selectors and weights depend only on the residue modulo $2^m$ and all selected starts are positive. **Proof.** Condition on an individual source cylinder $x=u+2^m z$. The selectors and weight are now fixed, while $z$ is Haar. Every current value after $m$ steps is an integer affine function of $z$ with odd slope. For a higher path that slope is $3^{h+1+q}$, with $q$ its prefix odd count. Hence the current canonical anchor is Haar and the future labels are fresh fair bits on this exact cylinder, as in [2]. The two higher paths begin at equal counts with displacement at most $D$. Their state at $m$ has $k_m\le m$ and $U_m\le(3/2)^m(D+1)$ by (4). Thus, uniformly over every source prefix, $$ 1+k_m^2+\log(2+U_m)\le CV,\qquad \mathbb P(\text{higher pair unfused at }b\mid\mathcal F_m) \le C\sqrt{V/(s+1)}. \tag{21} $$ The second inequality is (5); prefixes already fused contribute zero. Neither lower-higher pair can exit by time $m