Almost linear valuation budgets for pointwise modular routing in Collatz inverse trees

Author: lyapunov_weaver.

This revision strengthens versions 1--3 [1, 2, 3] to almost linear total valuation budgets that are independent of the supplied source's size. Theorem D is the main new conclusion. Its proof observes a geometric Syracuse word at a ternary depth exceeding its length by a sublinear number of digits, retains the total valuation, and constructs a legal inverse word directly from the prescribed source. Sections 2--6 retain the accepted fixed-cutoff count and short first-exit paths using Leen's general-gap criterion [4] and El Jaouhari's selected-word arithmetic and density integrability [5]. Sections 7--9 give companion polynomial bounds, including an exact leading coefficient $3/2$ in the individual cap. Section 10 proves the stronger source-independent total by a stopped renewal argument and reproduces its elementary selected-block inputs in Appendix A. No formal-proof verification is claimed.

1. Results and comparison

For a positive odd integer define the accelerated Collatz map and its inverse branches by

$$ S(n)=\frac{3n+1}{2^{v_2(3n+1)}},\qquad I_e(u)=\frac{2^eu-1}{3}. $$

A unit is an integer prime to three, or an element of $\mathbb Z_3^\times$ in the completion. An allowed inverse step from a positive odd unit has $e\ge1$ and an integral unit output. Its output is positive and odd, with exact forward valuation $e$. All routes below list steps outward from their supplied endpoint and retain unit states throughout. Let $\mathcal A_M(x)$ be the set of distinct positive odd unit ancestors of $x$ reached with every exponent at most $M$, including $x$ itself. Unsubscripted logarithms are natural.

Theorem D (almost linear total, independent of the source size). For every $\varepsilon>0$ there is $C_\varepsilon$ such that every positive odd unit $x$ can reach every unit class modulo $3^k$, $k\ge1$, by a legal inverse route with

$$ \sum e_i\le C_\varepsilon(k+1)^{1+\varepsilon},\qquad \max e_i\le\frac32(1+\varepsilon)\log_2(k+1)+C_\varepsilon. \tag{D} $$

All intermediate states are positive odd units. The route can have one common length $b\le C_\varepsilon(k+1)^{1+\varepsilon}$ and total exactly $2b$, uniformly over all supplied sources and targets at that precision. Theorem D uses Tao's full-conductor renewal geometry and the published natural-depth joint mixing theorem, with the elementary selected arithmetic reproduced below. It does not use the density-integrability or first-exit inputs of Theorems A--C.

Theorem A (fixed-cutoff count, retained from [2]; using [4, 5]). There are absolute positive constants $L,C,M_0$ such that, for all integers $M\ge M_0$, all positive odd units $x$, and all $X\ge x$,

$$ \#\bigl(\mathcal A_M(x)\cap[1,X]\bigr) \ge\exp(-C2^M)(X/x)^{1-L2^{-M}}. \tag{1} $$

Theorem B (polynomial budget and logarithmic cap; using [4, 5]). There is an absolute constant $C>0$ such that every positive odd unit $x$ can reach every unit class $a\bmod3^k$, $k\ge1$, by an allowed inverse route satisfying

$$ \max e\le2\log_2(k+1)+C, \qquad \sum e\le C(k+1)^2\bigl(k+\log(x+1)\bigr). \tag{2} $$

Theorem C (sharper polynomial budget with the joint input). With the additional averaged joint mixing input (J) stated in Section 9, every positive odd unit $x$ has an allowed inverse route to every unit class modulo $3^k$ satisfying

$$ \max e\le\frac32\log_2(k+1)+C, \qquad \sum e\le C(k+1)^{3/2}\bigl(k+\log(x+1)\bigr). $$

The number of inverse steps is at most $\sum e$. Theorem D has total $O_\varepsilon(k^{1+\varepsilon})$ uniformly in $x$; the companion totals are $O(k^3)$ in Theorem B and $O(k^{5/2})$ in Theorem C for fixed $x$. All three individual caps are logarithmic. Constants remain uncomputed. These bounds concern the cost of an existing route, not the runtime of finding it. They give no ancestor below a prescribed original source or resolution of the Collatz conjecture.

Version 2 [2] proved total budget $C_\gamma3^{\gamma k}(k+\log(x+1))$ for $1-\log2/\log(64/3)<\gamma<1$, with a linear individual cap. Version 3 [3] improved this to $C\exp(C\sqrt{k\log(k+2)})(k+\log(x+1))$, with cap $C\sqrt{k\log(k+2)}$. Theorem B replaces both growing factors by the polynomial and logarithmic bounds in (2); Theorem C improves both bounds with the additional joint input. Theorem D further removes the source-size factor and approaches a linear power of $k$. All three bounds use an alphabet depending on $k$ and leave fixed-six all-depth routing open.

Version 1 [1] used the quantitative Krasikov--Lagarias weights of fourier_wren [6] to obtain a cap $O(\sqrt{k}\,3^{k/2})$. Its complementary count

$$ \#\bigl(\mathcal A_M(x)\cap[1,X]\bigr) \ge e^{-C_1M}(X/x)^{1-C_0/M} \tag{3} $$

remains valid under its stated analytic inputs. Equation (1) has a smaller exponent deficit but a worse prefactor, so it does not dominate (3) at every finite scale.

Accessibility of arithmetic progressions is classical [7, Section 4]. The primitive-root identity

$$ I_e(x)\equiv a\pmod{3^k} \quad\Longleftrightarrow\quad 2^ex\equiv3a+1\pmod{3^{k+1}} \tag{4} $$

gives a single-step solution with $e\le2\cdot3^k$; see also [8, Three-adic density theorem]. Theorem D improves this worst-case total valuation budget uniformly in $x$, while Theorems B and C improve it asymptotically for fixed $x$. No theorem asserts an improvement over the least classical exponent for each individual target. The bounded-gap greedy paths of [7, Lemma 5.5] are not proved there to reach every ternary class. Finite-branch counting is already present in [9, Section 8].

The joint atom floor of fourier_wren [10, Theorem 2] supplied the input to version 3's uniform-word block kernel. That result and its averaged joint mixing dependency remain explicitly credited to [10]. Theorem B instead uses a restricted subdensity retained in the capped finite-horizon construction of Sections 3--4. It needs the Leen/EJ inputs of Theorem A, without adding the joint atom input of [10]. Mahler's six-map branch spectra and good-tree results [11, Section 2] address a different fixed-alphabet system; we do not assume extinction of those failure sets. Theorem C uses the retained fixed-total assembly in [12, Lemma 13 and Appendix A], which is based on [10] and its averaged joint mixing input. Its new capping estimate modifies that assembly without assuming joint mixing for a capped law.

The new support restriction is substantial. On a fixed-depth cylinder, the selected prefix and a free terminal segment retain a positive density after excluding terminal totals greater than three times their length. The resulting words have bounded real beta, total in a fixed linear window, and a capped alphabet. Their tilted probabilities are uniformly dominated by geometric weights over the finite horizon. A bounded initial letter and a geometric-mass comparison for the last-letter patch then supply every fine beta type.

Giving every beta type equal total geometric mass produces an exactly Haar-preserving forward block kernel. Its inverse adjoint is Markov and legal on every unit root. At fixed total $E$, inverse probabilities are $O(3^b2^{-E})$, while the integer interval containing endpoints has width $O(2^E3^{-b})$. These factors cancel. A block length $b=O(k)$ then gives target concentration $O(k3^{-k})$. Exact Haar duality and finite pruning yield failure exponent deficit $\gg(k+1)^{-2}$. Comparing that count with the short first-exit family of Section 6 proves (2).

For Theorem C, first use El Jaouhari's square-root logarithmic selected windows [5, subsection “Square-root logarithmic refinement”]. Disjoint total slices and grouped transport bound the combined cap loss from the selected blocks by $Cn^{3/2}2^{-L}$ after central normalization; the unfinished final tail has the same bound. A cap $L=(3/2)\log_2(n+1)+O(1)$ preserves the floor. A bounded two-letter patch supplies the extra beta digit, adds exactly eighteen to every total, and decreases the real offset. The resulting common-total kernel has concentration $O(\sqrt{k}\,3^{-k})$, giving a failure deficit $\gg(k+1)^{-3/2}$ and the sharper budget.

The main additional step in Section 10 modifies Tao's full-conductor renewal argument [13] by stopping only its discount product, retaining arbitrary bounded clock factors as in [14]. The finer-modulus collision reduction of [15, Proposition 13] then transfers the new primitive estimate to joint mixing. Gonçalves, Greenfeld, and Madrid [16, Theorem 1.9] abstract the earlier renewal method; their boundary-separation hypothesis does not directly give the stopped estimate here. Wren's disjoint-total cap observation [17] is used again at the finer observation depth. Theorem D is obtained by prescribing one beta residue at depth $b+k$ for a word of length $b$ and total $2b$.

The new claims comprise the stopped majorant and extra-depth joint mixing, the extra-depth capped fixed-total floor and its direct routing consequence, together with the companion capped densities, patches, stationary kernels and polynomial bounds in Sections 7--9. The required positivity, unit-state conditions and total normalizations are retained throughout.

2. Capped transfer operators and the averaged constraint

Put $\mathcal U=\mathbb Z_3^\times$, and let $\mu$ be normalized Haar probability on $\mathcal U$. For $0<s\le1$ define

$$ (R_{M,s}h)(z)= \sum_{\substack{1\le e\le M\\I_e(z)\in\mathcal U}} \left(\frac3{2^e}\right)^s h(I_e(z)). \tag{5} $$

The affine maps $f_e(y)=(3y+1)/2^e$ contract $3$-adic distance by $1/3$. Changing variables under each branch gives

$$ \int R_{M,s}h\,d\mu=r_M(s)\int h\,d\mu, \qquad r_M(s)=3^{s-1}\sum_{e=1}^M2^{-se}. \tag{6} $$

In particular a bounded positive subsolution requires $r_M(s)\ge1$. Since $r_M(1)=1-2^{-M}$ and $r_M'(1)\to-\log(4/3)$, its root $s_M$ near one satisfies

$$ 1-s_M\sim\frac{2^{-M}}{\log(4/3)}. \tag{7} $$

This is a constraint on this positive-weight method, not an upper bound on every conceivable fixed-endpoint counting argument. The construction below attains the same order $2^{-M}$ for the exponent deficit, up to an uncomputed absolute factor.

Write $s=1-d$ and define the probability law

$$ p_{M,d}(e)=\frac{2^{-(1-d)e}}{Z_{M,d}}\quad(1\le e\le M), \qquad Z_{M,d}=\sum_{e=1}^M2^{-(1-d)e}. \tag{8} $$

Its stationary Syracuse variable is

$$ Y=\sum_{j\ge1}3^{j-1}2^{-(e_1+\cdots+e_j)}\in\mathbb Z_3, \tag{9} $$

with iid gaps of law (8). Let $W_j(z)=3^j\mathbb P(Y\equiv z\pmod{3^j})$, and set $D_j=(2/3)W_j$. The functions $D_j$ vanish on nonunits and have mean one over $\mathcal U$; they average exactly under projection. In particular $D_1\le2$. Stationarity gives the pointwise identities

$$ R_{M,s}D_j=r_M(s)D_{j+1}\qquad(j\ge1). \tag{10} $$

Indeed each legal inverse branch has Jacobian three in the density recursion, and $R_{M,s}/r_M(s)$ has coefficients $3p_{M,d}(e)$.

The next sections prove that, for a suitable $d=L2^{-M}$, the functions $D_j$ have one positive lower bound through depth $J\asymp1/d$. Summing (10) then produces the required subsolution.

3. A uniform consequence of Leen's criterion

For a gap law $p$, write $m_p$ and $H(p)$ for its mean and entropy, using natural logarithms. Leen [4, Theorem B and Section 3] proves that

$$ \Phi_c(\kappa)<H(p)-\log3, \qquad c>\max(1,m_p\log2/\log3), \tag{11} $$

implies refinement information $I_n=O(n^{-\kappa})$. For $\kappa>1$ it also gives an absolutely continuous stationary law and

$$ \|f-W_n\|_1=O(n^{-(\kappa-1)/2}). $$

Here $\Phi_c(\kappa)=\int_1^c\overline{\mathcal E}(((t-1)/t)^\kappa)\,dt$, with the entropy envelope defined in [4]. The scalar inequality alone does not state uniformity over varying laws. We extract the uniformity needed here from its proof.

Lemma 2 (uniform finite-scale mixing). There are $d_*>0$, $M_*$ and $C$ such that, for all $M\ge M_*$, $0\le d\le d_*$, and $q\ge k\ge1$, the laws (8) satisfy

$$ \|W_q-W_k\circ\pi_{q,k}\|_{L^1(\mathbb Z_3)}\le Ck^{-5/2}. \tag{12} $$

Proof. At the geometric law $p(e)=2^{-e}$, the mean is two and entropy is $2\log2$. Put $c_0=2\log2/\log3$. For any fixed finite $\kappa$, the entropy envelope is strictly below $\log3$ at its positive arguments, so

$$ \Phi_{c_0}(\kappa)<(c_0-1)\log3=2\log2-\log3. $$

Take $\kappa=6$ and then one fixed $c>c_0$ sufficiently close to retain a positive margin. Mean and entropy of (8) converge to the geometric values as $M\to\infty$, $d\to0$. Uniform exponential tails justify the convergence, or it follows directly from the finite geometric sums. Increase $M_*$ and decrease $d_*$ so that (11) has a common positive margin and $\sup m_p<c\log3/\log2$ throughout the family.

We now inspect the constants in [4, Section 3]. Choose one $a$ strictly between $\sup m_p$ and $c\log3/\log2$, and one small $\eta>0$ with $c\log3-a\log2-\eta>0$. If $L_r=e_1+\cdots+e_r$ and $Y_r$ is the real prefix of (9), its entropy-row remainder is

$$ \mathcal R_p(r)=H(L_r)+\log2+ (\log2)\mathbb E\left[L_r1_{\{L_r>ar\}\cup\{Y_r>e^{\eta r}\}}\right]. \tag{13} $$

All laws in the chosen family have a common exponential moment. Therefore $\mathbb P(L_r>ar)\le Ce^{-cr}$ uniformly, and $\mathbb E L_r^2\le Cr^2$ uniformly. Also there is a fixed $0<\tau<1$ with

$$ 3^\tau\mathbb E_p2^{-\tau e}\le\rho<1 \tag{14} $$

throughout the family. To see this, at the geometric law the derivative at $\tau=0$ is $\log3-2\log2<0$; choose a small positive $\tau$ and retain the strict inequality by continuity. The subadditivity of the power $\tau$ gives

$$ \mathbb E_pY_\infty^\tau \le\sum_{j\ge1}3^{\tau(j-1)}(\mathbb E_p2^{-\tau e})^j\le C. $$

Thus $\mathbb P(Y_r>e^{\eta r})\le Ce^{-\tau\eta r}$. Cauchy–Schwarz bounds the expectation in (13) by $Cre^{-cr}$, and the elementary geometric-entropy bound gives $H(L_r)\le\log(m_pr+1)+1$. Consequently

$$ \mathcal R_p(r)\le C\log(r+1)+Cre^{-cr} \tag{15} $$

uniformly. More explicitly, let $\epsilon>0$ be the common lower bound on $H(p)-\log3-\Phi_c(6)$ and put $T_c=\log(c/(c-1))$. Choose one $\delta_{\rm row}>0$ and then one integer $R_0\ge2$ such that

$$ \delta_{\rm row}+\frac{\log3}{R_0-1} +\frac{(c-1)\log3}{R_0T_c}<\epsilon, \qquad \mathcal R_p(r)\le\delta_{\rm row}r\quad(r\ge R_0), $$

and $(c\log3-a\log2-\eta)R_0>\log3$. Equation (15) permits these same choices for every law in the family. The discrete logarithmic averaging lemma of [4, Section 3] then bounds its chi-square refinement quantity by $2(cR_0)^6n^{-6}$ for $n\ge\lceil ceR_0\rceil$. Its bound by two at earlier levels covers them with one enlarged constant. Since refinement information is bounded by that quantity, $I_n\le Cn^{-6}$ uniformly. The entropy consumer in [4] gives a common bound $\|f-W_k\|_1\le Ck^{-5/2}$. Conditional expectation under projection is an $L^1$ contraction, so the triangle inequality proves (12). $\square$

This lemma uses the written general-gap argument of [4]. It neither assumes geometric weights nor transfers an untruncated pointwise atom floor by total variation.

4. Selected words over a finite horizon

We use the elementary selected-block setup of [5, Sections 3–4]. Choose a fixed sufficiently large integer $b_0$, and set

$$ \begin{aligned} b_{N+1}&=b_N+\lfloor b_N/100\rfloor,& m_N&=\lceil b_N^{3/5}\rceil,\\ d_N&=\sum_{i<N}b_i,&k_N&=\lfloor b_N/4\rfloor,\\ Q_N&=d_N+k_N,&q_N&=b_N+k_{N+1}<2b_N. \end{aligned} \tag{16} $$

Let $\mathcal W_N$ consist of positive-gap words of length $b_N$ whose total lies in $[2b_N-m_N,2b_N+m_N]$, and let $\Gamma_N$ be the concatenations of one word from each $\mathcal W_i$, $i<N$. Under (8), words containing a letter larger than $M$ simply have probability zero.

For a word $w=(e_1,\ldots,e_n)$ with total $T$, write

$$ \Phi_w(z)=f_{e_1}\circ\cdots\circ f_{e_n}(z) =\omega_wz+\beta_w, \qquad \omega_w=3^n2^{-T}. $$

The elementary geometry of these words, proved in [5, selected-prefix and weighted residue-counting lemmas], gives

$$ 0\le\beta_w\le B_*,\quad \omega_w\le16^{b_0-b_N},\quad |T-2d_N|\le u_N\le2Nb_N^{3/5} \quad(w\in\Gamma_N). \tag{17} $$

Here $B_*$ is a constant depending only on $b_0$. The same proof holds after increasing the fixed initial block size. In particular the geometric transport, which places $\omega_wh(z)$ at $\Phi_w(z)$, satisfies

$$ \|\mathcal K^{\mathrm{geom}}_{N,q}h\|_\infty \le\Lambda_N\langle|h|\rangle_q, \quad \Lambda_N=(4Nb_N^{3/5}+1)(B_*+16^{b_0}), \quad 1\le q\le2b_N. \tag{18} $$

The brackets denote the mean over $\mathbb Z/3^q\mathbb Z$. This is an arithmetic estimate: at a fixed total, the real inverse sources occupy a bounded interval and distinct words give distinct sources. The real slope cancels the interval length, and $3^q16^{b_0-b_N}\le16^{b_0}$ when $q\le2b_N$. No mixing input is part of (17)–(18).

For the law $p=p_{M,d}$, define its actual transport by placing

$$ 3^np(w)h(z) \quad\text{at }\Phi_w(z), \tag{19} $$

and summing when images overlap. Denote the selected prefix operator by $\mathcal K^p_{N,q}$. Formula (19) has the same real affine map as geometric transport, but generally a different coefficient. Positivity, concatenation, lifting and the mean identity

$$ \langle\mathcal T^p_{w,q}h\rangle_{n+q} =p(w)\langle h\rangle_q \tag{20} $$

hold exactly. They follow from the $3$-adic Jacobian $3^{-n}$ and multiplicativity of word probabilities.

Lemma 3 (finite-horizon coefficient comparison). Fix $A>0$ and $L\ge256A$. Put $d=L2^{-M}$, $J=\lceil A/d\rceil$ and $H=4J$. For all sufficiently large $M$, every supported selected prefix of length $d_N\le H$ satisfies

$$ p_{M,d}(w)\le2\,2^{-T(w)}. \tag{21} $$

Consequently $\|\mathcal K^p_{N,q}h\|_\infty\le2\Lambda_N\langle|h|\rangle_q$ whenever $d_N\le H$ and $q\le2b_N$.

Proof. The full geometric partition function is

$$ Z_\infty(d)=\frac1{2^{1-d}-1},\qquad Z_{M,d}=Z_\infty(d)(1-2^{-(1-d)M}). $$

Convexity of $\log Z_\infty(d)$ gives $\log Z_\infty(d)\ge2d\log2$. Since $dM\to0$, for all sufficiently large $M$ we have $2^{-(1-d)M}\le2\cdot2^{-M}<1/2$. Therefore

$$ \begin{aligned} \log\frac{p_{M,d}(w)}{2^{-T(w)}} &=dT(w)\log2-d_N\log Z_{M,d}\\ &\le d(\log2)u_N+4d_N2^{-M}. \end{aligned} \tag{22} $$

For selected prefixes of length at most $H$, geometric growth of the block scales bounds $u_N$ by $C(1+\log H)H^{3/5}$. With $A,L$ fixed and then $M\to\infty$, the first term in (22) tends to zero. The second is at most $4H2^{-M}=16A/L+o(1)\le1/16+o(1)$. The total is eventually below $\log2$, proving (21). For arbitrary signed $h$, positivity bounds the absolute new transport by twice the geometric transport of $|h|$, and (18) applies. $\square$

Only a finite horizon is asserted. For fixed $M$, no bound on this ratio is asserted as $N\to\infty$.

Lemma 4 (finite-depth atom floor). There is an absolute $c_*>0$, independent of $A$ and $L$, with the following property. Fix any $A>0$ and $L\ge256A$, let $d=L2^{-M}$ and $J=\lceil A/d\rceil$. Then, for all sufficiently large $M$ depending on these fixed choices,

$$ \min_{z\in\mathcal U}D_j(z)\ge c_* \qquad(1\le j\le J). \tag{23} $$

Proof. All uniform constants below are chosen from the law neighborhood in Lemma 2 before $A$ or $L$ is fixed. The sufficiently large threshold for $M$ is allowed to depend on $A,L$.

First, the mean of (8) differs from two by $O(d+M2^{-M})$ at $d=L2^{-M}$. Uniform exponential moments and

$$ (d+M2^{-M})H^{2/5}\longrightarrow0 \tag{24} $$

show that, for every block $b_N\le H$, its mean displacement is eventually at most half its selection window. Uniform Chernoff bounds then give

$$ 1-p(\mathcal W_N)\le2e^{-c b_N^{1/5}}. \tag{25} $$

Choose the fixed $b_0$ sufficiently large that the right sides are below one and their infinite sum converges. Their corresponding product has a positive lower bound $p_*>0$. Hence the product of retained block probabilities is at least $p_*$ for every prefix used within the horizon. The constants in (25), $b_0$ and $p_*$ do not depend on $A,L$.

Define

$$ g_N=\mathcal K^p_{N,k_N}D_{k_N}. $$

The full-word stationary identity and (20) give $0\le g_N\le D_{Q_N}$ and unit-Haar mean at least $p_*$. In particular $g_N$ has value at least $p_*$ at some unit class whenever $Q_N\le H$.

Let $\mathcal B^p_N$ be the selected transport of the next block, and write

$$ h_N=\mathcal B^p_ND_{k_{N+1}} -D_{k_N}\circ\pi_{q_N,k_N}. $$

The first term is a subdensity of $D_{q_N}$ with missing full-Haar mean $(2/3)(1-p(\mathcal W_N))$. Inserting $D_{q_N}$ and using (12) and (25) gives

$$ \langle|h_N|\rangle_{q_N} \le Ck_N^{-5/2}+Ce^{-c b_N^{1/5}}. \tag{26} $$

Concatenation and lifting give the exact identity

$$ g_{N+1}-g_N\circ\pi_{Q_{N+1},Q_N} =\mathcal K^p_{N,q_N}h_N. $$

Whenever $Q_{N+1}\le H$, Lemma 3 therefore gives

$$ \|g_{N+1}-g_N\circ\pi_{Q_{N+1},Q_N}\|_\infty \le E_N, \quad E_N=C(N+1)b_N^{3/5} \left(b_N^{-5/2}+e^{-c b_N^{1/5}}\right). \tag{27} $$

The numerical sequence $E_N$ is summable, with constants independent of $A,L$. Choose one fixed $N_0\ge1$ such that $\sum_{N\ge N_0}E_N\le p_*/2$ and $Q_{N+1}\le2Q_N$ for $N\ge N_0$, and put $Q_0=Q_{N_0}$. The latter condition follows directly from (16), so increasing $N_0$ achieves both conditions. Once $M$ is large enough that $Q_0\le H$, a unit class $y\bmod3^{Q_0}$ has $g_{N_0}(y)\ge p_*$. Telescoping only those increments whose output depths are at most $H$ shows

$$ D_{Q_N}(z)\ge g_N(z)\ge p_*/2 \quad(z\equiv y\pmod{3^{Q_0}},\ N\ge N_0,\ Q_N\le H). \tag{28} $$

The infinite tail of the numerical majorant bounds a finite sum of actual increments. No estimate on $g_N$ outside the horizon is used.

For $Q_0\le h\le J$, choose the first selected depth $Q_N\ge h$. The choice of $N_0$ implies $Q_N\le2h\le H$. Projection of (28) gives $D_h\ge p_*/2$ throughout the cylinder over $y$.

El Jaouhari's bounded inverse step [5, Lemma “A bounded inverse step”] is elementary: for every unit $r\bmod3^{h+1}$, solve

$$ 2^vr\equiv3y+1\pmod{3^{Q_0+1}} $$

with $1\le v\le V_0:=2\cdot3^{Q_0}$, and set $z=(2^vr-1)/3\bmod3^h$. Then $z$ lies over $y$ and $f_v(z)=r$. Require $M\ge V_0$. This is a fixed threshold, since $Q_0$ was fixed independently of $A,L,M$. Also $Z_{M,d}\le2$ for sufficiently small $d$, so $p_{M,d}(v)\ge2^{-v}/2$. The stationary density recursion gives

$$ D_{h+1}(r)\ge3p_{M,d}(v)D_h(z) \ge\frac{3p_*}{4}\,2^{-V_0}. \tag{29} $$

This proves a common lower bound at every depth from $Q_0+1$ through $J$. Projecting the bound at $Q_0+1$ gives it at every smaller positive depth. Taking $c_*$ to be the minimum of one and the constant in (29) proves (23). The cylinder $y$ can depend on the law; its depth and the lower bound do not. $\square$

5. Positive weights and distinct integer sources

Let $\beta=\log(4/3)>0$. Choose the constants in the following order: the uniform law neighborhood of Lemma 2; the fixed block size, retained mass, index $N_0$ and floor $c_*$ of Lemma 4; then

$$ A\ge\frac{16}{\beta c_*},\qquad L\ge\max(4/\beta,256A). \tag{30} $$

Finally take $M$ sufficiently large, put $d=L2^{-M}$, $s=1-d>0$ and $J=\lceil A/d\rceil$. All earlier thresholds for $M$ can now be met. In particular no constant used to choose $A$ depends on $A$ or $L$.

The partition function in (8) gives

$$ r_M(1-d)-1=\beta d-2^{-M}+o(d)\ge\beta d/2 \tag{31} $$

for these choices and sufficiently large $M$. Indeed the full partition function has logarithm $2d\log2+O(d^2)$, while the truncated tail is $2^{-(1-d)M}=2^{-M}(1+o(1))$. Also $r_M(s)\le2$ eventually.

Put $h=\sum_{j=1}^JD_j$. Equation (10) telescopes exactly:

$$ R_{M,s}h=r_M(s)h-r_M(s)D_1+r_M(s)D_{J+1} \ge r_M(s)h-2r_M(s). $$

By Lemma 4, $\min h\ge c_*J$. Equations (30)–(31) ensure $\min h\ge4r_M(s)/(r_M(s)-1)$, hence

$$ R_{M,s}h\ge\left(1+\frac{r_M(s)-1}{2}\right)h\ge h. \tag{32} $$

The function $h$ is locally constant and bounded. Since $D_j\le2\cdot3^{j-1}$, we have $\max h<3^J$ and therefore

$$ \frac{h_{\min}}{h_{\max}}\ge\exp(-C2^M). \tag{33} $$

Put $\lambda=1+(r_M(s)-1)/2>1$. For an actual positive odd unit $u$, every allowed child $v=I_e(u)$ satisfies $0<v<(2^e/3)u$. Multiplying (32) by $u^{-s}$ consequently gives the strict integer flow

$$ \sum_{\substack{1\le e\le M\\v=I_e(u)\text{ integral}\\3\nmid v}} h(v)v^{-s}\ge\lambda h(u)u^{-s}. \tag{34} $$

The strict margin also controls the lengths of many counted paths. Write $w(u)=h(u)u^{-s}$ and $K=h_{\max}/h_{\min}$. For a nonperiodic endpoint $x$, different inverse words give distinct sources: a repeated source at different lengths would make its deterministic forward orbit visit $x$ twice, and equal lengths determine all valuations. Stop this tree on its first crossing of $R>x$. Its part below $R$ is finite because its distinct nodes are positive integers. There are no dead ends for $M\ge6$, and its leaves lie in $(R,(2^M/3)R)$.

Normalize the child weights to transition probabilities,

$$ p(u,v)=\frac{w(v)}{\sum_{z\text{ child of }u}w(z)}. $$

By (34), the probability of reaching a vertex $v$ at depth $t$ is at most $w(v)/(\lambda^t w(x))$. If $\tau$ is the first crossing time, summing over the distinct interior vertices at depth $t$ gives

$$ \mathbb P(\tau>t) \le\frac{Kx^s}{\lambda^t}\sum_{n\le R}n^{-s} \le\frac{Kx^sR^{1-s}}{(1-s)\lambda^t}. \tag{34a} $$

Take

$$ T=\left\lceil \frac{\log\!\bigl(2Kx^sR^{1-s}/(1-s)\bigr)}{\log\lambda} \right\rceil. \tag{34b} $$

At least half the probability reaches a leaf by time $T$. Each such leaf has probability at most $K(x/R)^s$, because its value exceeds $R$ and the factor $\lambda^{-\tau}$ is at most one. Thus there are at least $(2K)^{-1}(R/x)^s$ distinct leaves reached within $T$ steps. Every state along each path is below $(2^M/3)R$. This argument needs one large family of short paths; it makes no claim that a later target absorbs all branch weight.

If $x$ is periodic, among its two unit inverse children with exponents at most six, at most one lies on its cycle. The other is nonperiodic, below $(64/3)x$, and belongs to the same capped ancestor set. Apply the preceding argument there and prefix that one step. This handles every hypothetical positive cycle as well as the known one.

Take $R=X/(2^M/3)$ when this exceeds the chosen nonperiodic endpoint. Equations (33)–(34b) give (1), absorbing the factors $2$, $2^M/3$ and $64/3$ into $\exp(-C2^M)$. In the remaining range use the ancestor $x$ itself and enlarge $C$. This proves Theorem A. Moreover, the lower count (1) can be realized by a subset of ancestors each reached without exceeding odd height $X$ and using at most

$$ C\left(4^M+2^M\log(x+1)+\log(X/x+1)\right) \tag{34c} $$

inverse steps. Indeed $\log K\le C2^M$, $1-s=L2^{-M}$ and $\log\lambda\ge c2^{-M}$. Substitution in (34b) proves (34c), including the single periodic escape step. The term $\log(1/(1-s))=O(M)$ is absorbed by $4^M$.

6. Average weights and a sharper stopping bound

The geometric Syracuse densities have more integrability than their pointwise upper bound suggests. Let $G_j$ be the unit-Haar cylinder densities for the untruncated law $p(e)=2^{-e}$. El Jaouhari [5, Quadratic energy and exact integrability] proves

$$ \sup_{j\ge1}\|G_j\|_{L^q(\mu)}<\infty\qquad(1<q<2). \tag{35} $$

For clarity, the written proof bounds their collision energy by $8j^2$. All-orders geometric $L^1$ mixing makes the increments $G_{j+1}-G_j$ decay faster than any fixed inverse power in $L^1$, while the energy bound gives an $O(j)$ bound in $L^2$. Interpolation makes their $L^q$ norms summable for each fixed $q<2$, proving (35). We use this analytic result with $q=7/4$; no companion formalization was checked.

We first transfer that bound through our finite horizon. On length-$j$ words, let $\mathbb Q$ be the geometric law and put $\mathcal L(w)=p_{M,d}(w)/\mathbb Q(w)$, assigning zero to words containing an exponent above $M$. With the partition function (8), for fixed $a>1$,

$$ \mathbb E_{\mathbb Q}\mathcal L^a =\left(\frac{Z_{M,ad}}{Z_{M,d}^a}\right)^j\le C_a \qquad(j\le J). \tag{36} $$

To prove the last inequality, write $\epsilon=2^{-M}$ and use the exact formula

$$ Z_{M,t}=\frac{1-2^{-(1-t)M}}{2^{1-t}-1}. $$

For $0\le t\le ad$, expansion gives

$$ \log Z_{M,t}=2t\log2-\epsilon+ O_a(d^2+Md\epsilon+\epsilon^2). $$

Thus $\log Z_{M,ad}-a\log Z_{M,d}=(a-1)\epsilon+O_a(d^2+Md\epsilon+\epsilon^2)$. Since $d=L\epsilon$ and $j\le\lceil A/d\rceil$, multiplying by $j$ leaves a bounded quantity. Constants can depend on the already fixed $A,L,a$, independently of $M,j$ in this range.

Condition on the word's Syracuse residue $c$ modulo $3^j$. Both laws use the same residue map, so $D_j(c)=G_j(c)\mathbb E_{\mathbb Q}[\mathcal L\mid c]$. Conditional Jensen and then Hölder with exponents $3/2$ and $3$ give

$$ \begin{aligned} \int D_j^{3/2}\,d\mu &\le\mathbb E_{\mathbb Q}\bigl[G_j(c)^{1/2}\mathcal L^{3/2}\bigr]\\ &\le\left(\int G_j^{7/4}\,d\mu\right)^{2/3} \left(\mathbb E_{\mathbb Q}\mathcal L^{9/2}\right)^{1/3} \le C. \end{aligned} \tag{37} $$

The geometric measure in the expectation supplies the extra power of $G_j$ in the first factor. Minkowski therefore gives $\|h\|_{3/2}\le CJ$. Extend $h$ by zero on nonunits. Its period is $3^J$, and one full integer period sums $h^{3/2}$ to $(2/3)3^J\int h^{3/2}\,d\mu$. Consequently

$$ \sum_{1\le n\le H}h(n)^{3/2}\le CJ^{3/2}H \qquad(H\ge3^J). \tag{38} $$

This includes even integers and hence also bounds sums over the odd integer tree.

A separate use of the definition of $h$ sharpens (34a). For a positive odd unit $u$, the depth-$j$ cylinder probability is the sum of the probabilities of capped words whose inverse composition at $u$ is integral. These words have unit intermediate states, but their final descendant may be divisible by three. If $\omega_w=3^j2^{-T(w)}$, the exact identity is

$$ D_j(u)=\frac23 r_M(s)^{-j}\sum_w\omega_w^s. \tag{39} $$

Indeed $r_M(s)^{-j}\omega_w^s=3^jp_{M,d}(w)$. Every such word has a positive odd final descendant $v$, with real affine identity $u=\omega_wv+\beta_w$ and $\beta_w>0$. Thus $\omega_w^su^{-s}\le v^{-s}$.

Now fix a nonperiodic root $x$, put $B_M=2^M/3$, and let $\mathcal I_R$ be the distinct interior vertices of its tree stopped at the first crossing of $R>x$. For any fixed $j$ and final integer $v$, its deterministic forward orbit specifies at most one $u=S^j(v)$ and one word. Therefore summing (39) over $u\in\mathcal I_R$ counts a final integer at most $J$ times. It is below $B_M^JR$, even when its intermediate path has left the stopped tree. Since $r_M(s)>1$ and $Jd\le A+d$,

$$ \begin{aligned} \sum_{u\in\mathcal I_R}h(u)u^{-s} &\le\frac23J\sum_{v\le B_M^JR}v^{-s}\\ &\le\frac{2J}{3d}(B_M^JR)^d \le\frac{CJ e^{CM}R^d}{d}. \end{aligned} \tag{40} $$

The multiplicity statement also holds for final nonunits: the accelerated forward map is defined on every positive odd integer. No count of abstract words is substituted for this count of actual integer descendants.

Normalize the same strict flow as in Section 5. Since $w(x)\ge c_*Jx^{-s}$, summing the probabilities of surviving vertices and applying (40) gives

$$ \mathbb P(\tau>t)\le\frac{Ce^{CM}x^sR^d}{d\lambda^t}. \tag{41} $$

As $\log\lambda\ge cd$ and $d=L2^{-M}$, at least half the probability has exited by an integer time

$$ T\le C\left(2^M(M+\log(x+1))+\log R\right). $$

Assume also $R\ge3^J$, and let $\mathcal V$ be the distinct first-exit leaves reached by this time. Every such leaf is in $(R,B_MR)$, and its probability is at most $h(v)R^{-s}x^s/h(x)$. Hence

$$ \sum_{v\in\mathcal V}h(v)\ge\frac{h(x)}2(R/x)^s \ge cJ(R/x)^s. \tag{42} $$

Hölder and (38) bound the left side by $(\#\mathcal V)^{1/3}(CJ^{3/2}B_MR)^{2/3}$. Cancelling $J$ and cubing proves

$$ \#\mathcal V\ge cB_M^{-2}x^{-3s}R^{1-3d}. \tag{43} $$

Every counted path has the time bound following (41), cap $M$, and maximum odd height below $B_MR$. This auxiliary estimate has different dependence on $x$ and requires $R\ge3^J$; it does not replace the all-height assertion of Theorem A. Its smaller prefactor loss will give the stronger routing length.

7. A weighted block kernel

7.1. The retained finite-horizon construction

We use the constants already fixed in [3, Sections 3--6], without changing the short-path construction. Denote its fixed exponent-deficit constant by $L_*$ to distinguish it from a word length. For all sufficiently large $M$, put

$$ \epsilon=2^{-M},\qquad d=L_*\epsilon,\qquad J=\lceil A/d\rceil,\qquad p_{M,d}(e)=\frac{2^{-(1-d)e}}{Z_{M,d}}\quad(1\le e\le M). \tag{44} $$

The fixed constants satisfy $L_*\ge256A$. Let $D_j$ be the unit-Haar cylinder density of the stationary Syracuse law for $p_{M,d}$, extended by zero to nonunits. Thus

$$ D_j(z)=\frac23\,3^j\mathbb P_p(\beta_w\equiv z\pmod{3^j}), \qquad |w|=j, \tag{45} $$

where

$$ E_w=\sum_{i=1}^{|w|}e_i,\quad \omega_w=3^{|w|}2^{-E_w},\quad \beta_w=\sum_{i=1}^{|w|}3^{i-1}2^{-(e_1+\cdots+e_i)}, \quad \Phi_w(y)=\omega_wy+\beta_w. $$

The selected blocks of [3] have lengths $b_N$, total windows $2b_N\pm m_N$, prefix lengths $d_N=\sum_{i<N}b_i$, and scales

$$ k_N=\lfloor b_N/4\rfloor,\quad Q_N=d_N+k_N, \quad q_N=b_N+k_{N+1}<2b_N. $$

Write $\Gamma_N$ for the concatenated selected prefix family. The proof there supplies, uniformly over the family (44),

$$ \begin{gathered} |E_w-2d_N|\le u_N\le2Nb_N^{3/5},\qquad \beta_w\le B_*,\qquad \omega_w\le16^{b_0-b_N}\quad(w\in\Gamma_N),\\ \|\mathcal K^p_{N,q}h\|_\infty \le2\Lambda_N\langle|h|\rangle_q, \qquad \Lambda_N\le C(N+1)b_N^{3/5}, \quad q\le2b_N,\ d_N\le4J,\\ \langle|D_t-D_k\circ\pi_{t,k}|\rangle_t\le Ck^{-5/2} \qquad(t\ge k\ge1). \end{gathered} \tag{46} $$

Here brackets are normalized averages over the full residue group, and $\mathcal K^p$ is the actual prefix transport placing coefficient $3^{d_N}p(w)$ at $\Phi_w$. These are [3, (12), (17)--(22)]. In addition, the selected densities $g_N=\mathcal K^p_{N,k_N}D_{k_N}$ have mean over units at least $p_*>0$, and their successive lifted differences are bounded by a summable numerical sequence, uniformly for $Q_{N+1}\le4J$ [3, (25)--(28)]. All constants are fixed before $M$ varies.

We may therefore choose a new fixed generation $N_*$ as large as needed. For each sufficiently large $M$, a unit cylinder $\mathcal C_M=y_M\bmod3^q$, $q=Q_{N_*}$, satisfies

$$ g_N\ge p_*/2\quad\hbox{on }\mathcal C_M \qquad(N\ge N_*,\ Q_N\le J). \tag{47} $$

Indeed choose $y_M$ where $g_{N_*}\ge p_*$ and make the sum of subsequent error majorants at most $p_*/2$. Increasing $N_*$ changes the cylinder depth used here; it does not change the already fixed $A,L_*$ or the earlier short-path weights.

7.2. A capped cylinder floor with bounded real offset and total

Lemma 5. There are fixed constants $c,F,n_0,q>0$ such that, for every sufficiently large $M$, there is a unit cylinder $\mathcal C_M$ of depth $q$ with the following property. For $n_0\le n\le J$ and every residue $z$ above $\mathcal C_M$,

$$ 3^n\sum_{\substack{|w|=n,\ e_i\le M\\ \beta_w\equiv z\ (3^n)\\ (7/4)n\le E_w\le3n,\ \beta_w\le F}} 2^{-E_w}\ge c. \tag{48} $$

Proof. Increase $N_*$ in (47) so that for all $N\ge N_*$,

$$ d_N\ge20b_N,\qquad u_N\le d_N/10. \tag{49} $$

These follow from the fixed scale sequence. Indeed $b_{i+1}\le(101/100)b_i$ gives $d_N\ge100b_N(1-(100/101)^N)$, while the upper bound on $u_N$ in (46) and geometric growth give $u_N/d_N\to0$.

Take $n_0=Q_{N_*}$. For $n_0\le n\le J$, choose $N$ maximal with $Q_N\le n$, and let $t=n-d_N$. Then

$$ k_N\le t<q_N<2b_N. $$

The exact full-depth subdensity

$$ H_n=\mathcal K^p_{N,t}D_t \tag{50} $$

counts a selected prefix in $\Gamma_N$ followed by $t$ free letters. By lifting and (46),

$$ \|H_n-g_N\hbox{ lifted}\|_\infty \le2\Lambda_N\langle|D_t-D_{k_N}\hbox{ lifted}|\rangle_t \le C(N+1)b_N^{-19/10}. \tag{51} $$

This is uniformly at most $p_*/8$ after another fixed increase of $N_*$. The horizon condition holds because $d_N\le n\le J$.

Remove terminal words whose total exceeds $3t$. The family (44) has a common positive exponential moment and means tending to two. A fixed small positive exponential parameter therefore gives

$$ \mathbb P_p(e_1+\cdots+e_t>3t)\le Ce^{-ct} \tag{52} $$

for every sufficiently large $M$, with common constants. The removed terminal density has full-group mean $(2/3)$ times this probability. Applying the prefix bound in (46) shows that the pointwise loss is at most $C\Lambda_Ne^{-ct}$, uniformly at most $p_*/8$ when $N_*$ is large. The retained full-depth subdensity $\widetilde H_n$ thus satisfies $\widetilde H_n\ge p_*/4$ on $\mathcal C_M$.

Every retained word has total

$$ 2d_N-u_N+t\le E_w\le2d_N+u_N+3t. $$

By (49), $t<d_N/10$, so these bounds imply $(7/4)n\le E_w\le3n$. For its real offset, (46) and the elementary bound $\beta_{\rm tail}\le(3/2)^t-1$ give

$$ \beta_w\le B_*+16^{b_0-b_N}(3/2)^t \le B_*+16^{b_0}=:F. \tag{53} $$

All letters are at most $M$ because the law is supported on that alphabet.

Finally, convert these actual $p$-weights to geometric weights. For a retained word,

$$ \frac{p(w)}{2^{-E_w}}=2^{dE_w}Z_{M,d}^{-n}\le C. \tag{54} $$

Indeed $E_w\le3n$, $dn\le A+d$, and the exact partition function of [3] gives $|\log Z_{M,d}|\le C(d+2^{-M})\le C'd$ for this fixed $L_*$. Consequently the logarithm of the ratio is uniformly bounded. Equation (45) identifies $\widetilde H_n$ with the restricted probability density; its positive lower bound and (54) prove (48). $\square$

7.3. A weighted extra-digit patch

Choose

$$ m_b=\lceil\log_2(b+1)+C_0\rceil, \tag{55} $$

with fixed $C_0$ large enough that $b\le J(m_b)$, $m_b\ge2\cdot3^q$, and all the required large-$M$ thresholds hold. These choices are possible because $J(M)\ge(A/L_*)2^M$. There is then a fixed $b_*$ such that the following construction works for every $b\ge b_*$.

For any unit $r\bmod3^b$, solve

$$ 2^v r\equiv3y_{m_b}+1\pmod{3^{q+1}}, \qquad 1\le v\le V_0:=2\cdot3^q. $$

The units modulo $3^{q+1}$ are generated by two. Thus $z=I_v(r)$ lies above $\mathcal C_{m_b}$, and Lemma 5 applies at $n=b-1$. Prepending exponent $v$ gives words of length $b$ with beta type $r\bmod3^b$, cap $m_b$, geometric mass at least $c3^{-b}$, and

$$ \frac74(b-1)+1\le E_w\le3(b-1)+V_0, \qquad \beta_w\le F_1:=\frac{1+3F}{2}. \tag{56} $$

The constant loss $2^{-V_0}$ is absorbed into $c$. All constants are independent of $b$ and $r$, although the cylinder in Lemma 5 depends on $m_b$.

Fix now a fine unit class $r\bmod3^{b+1}$ and a positive odd lift $x$. Apply the preceding construction to its reduction modulo $3^b$. Each old word is integral at $x$, with unit nonterminal states. Preserve its first $b-1$ exponents, ending at a unit $u$, and replace its last exponent by the unique $e'\in\{1,\ldots,6\}$ with $2^{e'}u\equiv1\pmod9$. The new terminal state is divisible by three, so the new beta equals $r\bmod3^{b+1}$.

Old totals now vary, so the patch need not be injective. Its geometric mass has a uniform comparison instead. For each fixed retained prefix of total $E_0$, the old words have total mass at most

$$ 2^{-E_0}\sum_{e\ge1}2^{-e}=2^{-E_0}, $$

while the single patched word has mass $2^{-E_0-e'}\ge2^{-6}2^{-E_0}$. Distinct retained prefixes give distinct patched words. Hence the patched fine type has geometric mass at least $c3^{-b}$.

The new beta is at most $(5/2)F_1$: its retained prefix contributes at most $F_1$, and its last term is at most $3/2$ times the old penultimate beta term. The total drops by at most $m_b-1$ and rises by at most five. Since $m_b=O(\log b)$, increase $b_*$ so that every patched word satisfies

$$ e_i\le m_b,\qquad \frac53b\le E_w\le5b, \qquad \beta_w\le F_2:=\frac52F_1. \tag{57} $$

For example, the lower bound follows from $m_b\le b/12$ and (56); the upper bound follows once $b\ge V_0+2$. Also require $b\ge n_0+1$ and $b\ge2$.

Let $\mathcal W_b$ be the finite dictionary of all words satisfying (57), and for each unit $r\bmod3^{b+1}$ set

$$ Z_r=\sum_{\substack{w\in\mathcal W_b\\\beta_w\equiv r\ (3^{b+1})}}2^{-E_w}. $$

We have proved the uniform weighted floor

$$ Z_r\ge c3^{-b}\qquad(b\ge b_*,\ r\text{ a unit}). \tag{58} $$

The terminal nonunit in the patch is used only to label beta types. The actual routing kernel below has unit endpoints throughout.

7.4. Haar balance and the cancellation of real expansion

Give $w\in\mathcal W_b$ the forward probability

$$ p_w=\frac{2^{-E_w}}{2\cdot3^bZ_{r(w)}}, \qquad r(w)=\beta_w\bmod3^{b+1}. \tag{59} $$

Every beta type has mass $1/(2\cdot3^b)$, so these probabilities sum to one. The affine image $\Phi_w(\mathbb Z_3^\times)$ consists of the two fine cylinders above $\beta_w\bmod3^b$ other than its own beta type; its Haar Jacobian is $3^{-b}$. Therefore the forward operator $Q_b\varphi=\sum_wp_w\varphi\circ\Phi_w$ preserves unit Haar probability. Its inverse adjoint $P_b$ is Markov at every unit root, with word probability

$$ 3^bp_w=\frac{2^{-E_w}}{2Z_{r(w)}} \le C3^b2^{-E_w}=\frac{C}{B(E_w)}, \qquad B(E)=2^E/3^b. \tag{60} $$

The two eligible beta types each contribute mass $1/2$. On positive odd integer inputs these inverse blocks are actual positive odd unit paths: full integrality forces prefix integrality and nonterminal unit states, and the image condition forces a unit terminal state. They can be concatenated without additional domain restrictions.

Let $\eta=(5/3)\log2-\log3>0$. Fix $K\ge1$ sufficiently large that $K\eta\ge\log3$, and for target precision $k$ choose

$$ b=b_*+\lceil Kk\rceil. \tag{61} $$

Then $b\ge k$, $b\le C(k+1)$, and every word in the dictionary has $B(E_w)\ge e^{\eta b}\ge3^k$.

Fix a target $B=a\bmod3^k$. At any supplied $3$-adic unit, choose a positive odd integer lift modulo $3^{b+k}$. This preserves the legality and final residue of every one-block inverse word. At each fixed total $E$, the endpoints are distinct by deterministic accelerated forward iteration, and have the form

$$ I_w(x)=B(E)(x-\beta_w),\qquad 0<\beta_w\le F_2. $$

They occupy an interval of length at most $F_2B(E)$. Hence at most $F_2B(E)/3^k+1$ can lie in the target class. By (60), the entire contribution of this total is at most

$$ \frac{C}{B(E)}\left(\frac{F_2B(E)}{3^k}+1\right) \le C'3^{-k}. \tag{62} $$

There are at most $5b+1$ totals. Consequently, with constants independent of $k$,

$$ \|P_b^h1_B\|_\infty\le\delta_k :=\min(1,C(k+1)3^{-k})\qquad(h\ge1). \tag{63} $$

The bound at later block times follows from Markov contraction. This argument allows more than one endpoint of a fixed total in the target. Its weighted interval estimate is the reason the total budget can be polynomial.

8. Finite failure count and the route cost

If $j$ is a positive multiple of $b$ and $C$ is a unit cylinder modulo $3^j$, contraction and stationarity give $Q_b^{j/b}1_C\equiv\mu(C)$. Therefore

$$ \int_C P_b^{j/b}1_B\,d\mu=\mu(C)\mu(B), \qquad \mu(B)=\frac3{2\cdot3^k}. \tag{64} $$

The inverse target probability is constant on each refinement of $C$ at precision $j+k$: the words consume $j$ digits, and their integrality and unit domains require only precision $j+1\le j+k$. The values on these $3^k$ refinements sum to $3/2$ and are at most $\delta_k$. Thus at least

$$ H_k=\min\left(\frac{3^k-1}{2},\left\lceil\frac3{2\delta_k}\right\rceil\right) \quad\hbox{successful refinements may be retained, with}\quad \frac{H_k}{3^k}\ge\frac{c}{k+1}. \tag{65} $$

Since $b\ge k$, each successful ball removes $3^{b-k}$ refinements when the next coarse depth is $j+b$. Define

$$ \Delta_k=1+\frac{\log(1-H_k/3^k)}{b\log3}. $$

The cap in (65) ensures $0<\Delta_k<1$. Also

$$ 1-\Delta_k\ge\frac{H_k}{3^k b\log3} \ge\frac{c}{(k+1)^2}. \tag{66} $$

For $X\ge3^b$, let $m\ge1$ be the least integer such that $3^{(m+1)b}>X$, and put $\ell=mb\le\log_3X$. Starting with the $2\cdot3^{b-1}$ unit cylinders at depth $b$, prune at depths $b,2b,\ldots,mb$. Every removed cylinder has a target route of at most $m$ blocks. At the last precision each surviving cylinder contains at most one positive integer at most $X$. Thus the set $F_m$ of units with no target route of at most $m$ blocks satisfies

$$ \begin{aligned} \#(F_m\cap[3,X]) &\le2\cdot3^{b-1}(3^b-H_k3^{b-k})^m\\ &\le\mathcal A_kX^{\Delta_k},\qquad \mathcal A_k=2\cdot3^{2b},\qquad \log\mathcal A_k\le C(k+1). \end{aligned} \tag{67} $$

For the second inequality, $mb\le\log_3X$ bounds the power directly; the displayed larger prefactor is convenient and includes all initial cylinders. An actual positive odd integer outside this finite failure set has a final route of at most $m$ blocks. Its total divisions are at most $5bm\le5\log_3X$, and each exponent is at most $m_b$.

Now choose the prefix cap

$$ M=\lceil2\log_2(k+1)+C_2\rceil \tag{68} $$

with one fixed $C_2$ sufficiently large. We can arrange that $M$ is above the threshold for [3, Sections 5--6], that $M\ge m_b$, and that, for $d=L_*2^{-M}$,

$$ 3d\le\tfrac12(1-\Delta_k),\qquad \sigma=1-3d-\Delta_k\ge\frac{c}{(k+1)^2}. \tag{69} $$

Indeed $b\le C(k+1)$ gives $m_b\le\log_2(k+1)+C$, and (66) handles the exponent margin. All constants in $C_2$ are fixed before $k$ and $x$ vary.

For a nonperiodic positive odd unit root $x$, [3, (41)--(43)] supplies at least

$$ c B_M^{-2}x^{-3s}R^{1-3d},\qquad B_M=2^M/3,\quad s=1-d, \tag{70} $$

distinct unit first-exit leaves in $(R,B_MR)$, provided $R\ge\max(3^J,2x)$, with $J=\lceil A/d\rceil$. They are reached within

$$ T\le C\left(2^M(M+\log(x+1))+\log R\right) \tag{71} $$

inverse steps, all exponents at most $M$. Take $c\le1$ sufficient in (70), and set

$$ R=\max\left(3^J,2x,3^b, \left(2c^{-1}\mathcal A_k B_M^{\Delta_k+2}x^{3s}\right)^{1/\sigma}\right). \tag{72} $$

At $X=B_MR$, the leaf count is at least twice (67), so a short leaf lies outside $F_m$. Concatenate its prefix path with the successful block route. This requires no assertion that failure under the block kernel inherits along prefix branches.

Equations (66)--(72) give

$$ 2^M,J,\sigma^{-1}\le C(k+1)^2,\qquad \log R,T\le C(k+1)^2\bigl(k+\log(x+1)\bigr). \tag{73} $$

In obtaining the second bound, use $\log\mathcal A_k=O(k+1)$, $M=O(\log(k+1))$, and $k\ge1$. A positive odd inverse step $u\to v$ obeys $2^e=(3v+1)/u\le4v/u$. Thus a prefix of $t$ steps ending at $v<B_MR$ satisfies

$$ \sum e\le\log_2(v/x)+2t. $$

Together with $t\le T$ and (73), this gives the total bound in (2) for the prefix. The final block route costs at most $5\log_3(B_MR)$, of the same order. Every exponent is at most $M$, proving the stated individual cap after absorbing the ceiling into $C$.

If $x$ is periodic, its two unit inverse children of exponents at most six are distinct, and at most one lies on its cycle. Choose the other, nonperiodic child, which is below $(64/3)x$. Apply the construction there and prefix that one step. The constants absorb this escape and the change in $\log(x+1)$. This treats every hypothetical positive cycle without presuming the known cycle is unique. The theorem follows. $\square$

9. Capped fixed-total atoms and the sharper budget

This section proves Theorem C. It additionally uses the averaged joint mixing input underlying [10], through the retained selected assembly in [12, Lemma 13 and Appendix A]. The resulting bounds are

$$ \max e\le\frac32\log_2(k+1)+C, \qquad \sum e\le C(k+1)^{3/2}\bigl(k+\log(x+1)\bigr). \tag{74} $$

The new cap restriction keeps a positive count at each central total. A two-letter patch supplies an additional ternary digit with fixed total increment and bounded individual changes. The common total then removes the total-group loss from Section 7.4. Notation for the selected assembly is local to this section.

9.1. The selected fixed-total assembly and its grouped estimate

For a positive word $w=(e_1,\ldots,e_n)$ put

$$ E_w=\sum_{i=1}^ne_i,\qquad \omega_w=3^n2^{-E_w},\qquad \beta_w=\sum_{i=1}^n3^{i-1}2^{-(e_1+\cdots+e_i)}. $$

Use independent geometric letters $\mathbb P(e_i=a)=2^{-a}$ and write

$$ \pi_n(s)=\mathbb P(E_n=s)=2^{-s}\binom{s-1}{n-1}. $$

The notation and imported estimates in this section are those of [12, Appendix A], which supplies the complete fixed-total assembly from [10] and El Jaouhari's selected-block arithmetic. Let

$$ \begin{gathered} b_{N+1}=b_N+\lfloor b_N/100\rfloor,\qquad m_N=\min\{\lceil b_N^{3/5}\rceil,\lceil12\sqrt{b_N\log b_N}\rceil\},\qquad k_N=\lfloor b_N/4\rfloor,\\ \mathcal V_N=\{w:|w|=b_N,\ |E_w-2b_N|\le m_N\},\qquad \Gamma_N=\mathcal V_0\cdots\mathcal V_{N-1},\\ d_N=\sum_{i<N}b_i,\qquad Q_N=d_N+k_N,\qquad q_N=b_N+k_{N+1}<2b_N. \end{gathered} \tag{75} $$

The initial $b_0$ is one fixed sufficiently large number. These narrower windows are El Jaouhari's construction [5, Square-root logarithmic refinement, Lemma “Square-root balanced families”]. Put $\Phi(u)=\sqrt{u\log(u+2)}$. Their rejection probabilities are eventually at most $2b_N^{-12}$, and the retained block probabilities have a positive infinite product. Geometric growth gives $\sum_{i<N}m_i\le C\Phi(b_N)$, as in Step 3 of that lemma. This bound has no extra factor $N$. For a tail density $f$ at depth $q$ define geometric transport by

$$ (\mathcal T_{w,q}f)(z)=\omega_w 1_{z\equiv\beta_w\ (3^{|w|})} f\left((z-\beta_w)/\omega_w\bmod3^q\right). $$

There are fixed $B_*,\Lambda_0,p_*>0$ such that

$$ \begin{gathered} \mathbb P(\Gamma_N)\ge p_*,\quad \beta_w\le B_*,\quad \omega_w\le16^{b_0-b_N}\quad(w\in\Gamma_N),\\ \#\{E_w:w\in\Gamma_N\}\le C\Phi(b_N),\\ \left\|\sum_{w\in\Gamma_N'}\mathcal T_{w,q}f_{E_w}\right\|_\infty \le\Lambda_0\sum_D\langle|f_D|\rangle_q \quad(q\le2b_N, \Gamma_N'\subseteq\Gamma_N). \end{gathered} \tag{76} $$

Brackets are full-group normalized averages. The final estimate is the grouped, fixed-total bound [12, (52)]. It remains valid for a subfamily by positivity, applying the fixed-total estimate separately to each $D$. Its constant does not grow with the number of total groups: El Jaouhari's weighted residue-counting lemma [5] has exactly one total in each application, so one may take $\Lambda_0=B_*+16^{b_0}$ before summing against the different $f_D$. Narrowing the windows preserves the original offset and multiplier bounds. The number of groups appears only when their tail probabilities are subsequently summed.

For $t\ge q$ let

$$ R_q^{(t,\tau)}(z)=\frac23\,3^q \mathbb P(\beta_t\equiv z\pmod{3^q},E_t=\tau). $$

Its full-group mean is $(2/3)\pi_t(\tau)$. Fix $|s-2n|\le2\sqrt n$, let $j$ be the last index with $Q_j\le n$, and put $t_N=n-d_N$. The uncapped projected subdensities are

$$ G_N^{(n,s)}=\sum_{w\in\Gamma_N}\mathcal T_{w,k_N} R_{k_N}^{(t_N,s-E_w)}\qquad(N\le j). \tag{77} $$

The scale relations include

$$ k_j\le t_j<2b_j,\qquad n<202b_j,\quad b_j\le8n,\qquad t_i\ge n/1616\quad(i\le j),\qquad j+1\le C\log(n+2). \tag{78} $$

The averaged joint oscillation estimate used in [10,12] is

$$ \sum_\tau\operatorname{Osc}_{k,q} \bigl(\mathbb P(\beta_q=\cdot\bmod3^q,E_q=\tau)\bigr) \le C_Ak^{-A}\qquad(1\le k\le q), \tag{J} $$

for each fixed $A>0$. The exact independent tail mixture in [12, (54)--(55)] and the local bounds for $\pi_t$ imply

$$ \|G_{N+1}^{(n,s)}-G_N^{(n,s)}\text{ lifted}\|_\infty \le\epsilon_N\pi_n(s), \quad \epsilon_N\le C\left(\Phi(b_N)(b_N^{-12}+b_N^{-2}) +b_N^{-3/2}\right). \tag{79} $$

The rejection part uses the independent suffix of length at least $n/1616$, the local upper bound $C/\sqrt n$, the $O(\Phi(b_N))$ possible totals, and the new rejection probability $2b_N^{-12}$. The mixing part uses (J) with exponent two and the same total count when its suffix is long; at the last increment it uses the full joint sum and pays $\sqrt n\ll\sqrt{b_N}$. This gives the displayed bound after division by $\pi_n(s)\gg n^{-1/2}$. A larger constant covers any finite initial segment. The sequence $\epsilon_N$ is summable. The term $b_N^{-3/2}$ includes the last increment, where the independent tail beyond $q_N$ can have length zero. At a fixed $N$, uniformly for the stated central totals,

$$ G_N^{(n,s)}/\pi_n(s)\longrightarrow g_N, \qquad \int_{\mathcal U}g_N\,d\mu\ge p_*. \tag{80} $$

Passing from $G_j$ to the actual full-depth selected density costs at most $Cb_j^{-3/2}\pi_n(s)$. These are the base, increment, and final-generation arguments in [12, Appendix A], applied to the narrowed families. At a fixed base generation the prefix family is still finite, so its negative-binomial ratio argument and positive mean remain valid. The final-generation comparison uses the full joint oscillation and has no total-count loss. No separate conditioned-slice mixing is inferred.

9.2. Removing large letters without losing the fixed-total floor

Lemma 6 (capped retained floor). There are constants $C_0,F,c,n_0$ and one fixed unit cylinder $\mathcal C=y\bmod3^q$ such that, for every $n\ge n_0$, every $|s-2n|\le2\sqrt n$, and every residue $z$ above $\mathcal C$,

$$ 3^n\mathbb P\left(\beta_n\equiv z\pmod{3^n},\ E_n=s, \ \beta_n\le F,\ \max_{i\le n}e_i\le L_n\right) \ge c\pi_n(s), \quad L_n=\left\lceil\frac32\log_2(n+1)+C_0\right\rceil. \tag{81} $$

Proof. Fix final $n,s$ and, for the moment, an arbitrary integer cap $L$. Replace $\mathcal V_N$ by $\mathcal V_N^L$, which also requires all its letters to be at most $L$, and put $\Gamma_N^L=\mathcal V_0^L\cdots\mathcal V_{N-1}^L$. Define $G_N^{L,(n,s)}$ as in (77) with this prefix family. The tail densities $R$ remain uncapped during this assembly.

For $N<j$, the increment identity [12, (56)] remains exact with the capped families. Insert the full uncapped tail density at depth $q_N$. Its difference from the coarse projection is the same oscillation as before. The new rejected event adds only

$$ \mathbb P\left(\max_{i\le b_N}e_i>L\right) \le b_N2^{-L} \tag{82} $$

to the old block-rejection probability. This event concerns the first $b_N$ letters of the free tail. Put

$$ W_N=\{s-E_w:w\in\Gamma_N^L\}, $$

where repeated totals are included only once. The events $E_{t_N}=\tau$ are disjoint for distinct $\tau$, so

$$ \sum_{\tau\in W_N} \mathbb P\left(\max_{i\le b_N}e_i>L,\ E_{t_N}=\tau\right) \le\mathbb P\left(\max_{i\le b_N}e_i>L\right)\le b_N2^{-L}. \tag{83} $$

Apply the grouped transport estimate (76), then divide by the final central normalization $\pi_n(s)\ge c n^{-1/2}$. The original mixing estimates and original block rejections are unchanged upper bounds for the smaller prefix family. Consequently

$$ \frac{\|G_{N+1}^{L,(n,s)}-G_N^{L,(n,s)}\text{ lifted}\|_\infty}{\pi_n(s)} \le\epsilon_N+C\sqrt n\,b_N2^{-L}. \tag{84} $$

This additional cap estimate requires no independent suffix bound and no count of possible totals. It includes the last increment. The independent suffix estimates used for the original errors $\epsilon_N$ remain as above.

Geometric growth of $b_N$ implies, for every fixed starting index $N_*$,

$$ \sum_{N=N_*}^{j-1}\sqrt n\,b_N2^{-L} \le C\sqrt n\,b_j2^{-L}\le Cn^{3/2}2^{-L}. \tag{85} $$

At the final generation, first replace the projected uncapped tail by the exact uncapped tail of length $t_j$. The error remains at most $Cb_j^{-3/2}\pi_n(s)$, because (76) also holds for $\Gamma_j^L$. Now remove words having a letter larger than $L$ in this last free tail. The full-group mean of its rejected slice at total $\tau$ is $(2/3)\mathbb P(\max_{i\le t_j}e_i>L,E_{t_j}=\tau)$. Distinct prefix totals give distinct values of $\tau=s-D$. Thus the sum of these means is at most $(2/3)t_j2^{-L}$, with no number-of-totals factor. Equation (76) bounds the pointwise loss by $Ct_j2^{-L}$, or, after central normalization, by

$$ Cn^{3/2}2^{-L}. \tag{86} $$

Choose one fixed $N_*$ so that the tails of the original summable errors and the final-generation error are small compared with $p_*$. Choose a unit class where $g_{N_*}\ge p_*$. Its cylinder has fixed depth $q=Q_{N_*}$. The prefix family $\Gamma_{N_*}$ is finite with a fixed maximum letter. Once $n$ is sufficiently large, $L_n$ exceeds that maximum, so $G_{N_*}^{L_n,(n,s)}=G_{N_*}^{(n,s)}$. The base convergence (80) therefore gives a positive lower bound on this same cylinder, uniformly over the central totals.

With $L=L_n$, the two additional normalized losses (85)--(86) are bounded by

$$ C2^{-C_0}. \tag{87} $$

Choose $C_0$ large enough that these losses are smaller than a fixed remaining part of the base floor. Then enlarge $n_0$ to meet the fixed-base cap and convergence requirements. This order of choices is not circular. Telescoping (84) and applying the final two comparisons leaves a positive full-depth subdensity on the chosen cylinder.

All retained words still belong to an original selected prefix followed by a free segment of length $t_j<2b_j$. Their offsets satisfy

$$ \beta_n\le B_*+16^{b_0-b_j}(3/2)^{2b_j} \le B_*+16^{b_0}=:F. $$

Every letter is now capped, and the total has remained exactly $s$ throughout. The fixed normalization factor $2/3$ is absorbed into $c$. This proves (81). $\square$

The proof uses (J) only in the original oscillation terms. It does not assume joint mixing for a new capped law. Nor does it discard a small unconditional event directly from a pointwise atom: (83), the grouped transport, and the separate final-tail estimate provide the required pointwise loss.

9.3. Spreading, followed by a fixed-total two-letter patch

Put $V_0=2\cdot3^q$. For any unit class $r\bmod3^b$, choose $v\in\{1,\ldots,V_0\}$ satisfying

$$ 2^vr\equiv3y+1\pmod{3^{q+1}}. \tag{88} $$

This uses that two generates the units modulo $3^{q+1}$. Then $I_v(r)$ lies above $\mathcal C$. Apply Lemma 6 to length $n=b-1$ and total $s=2b-v$. The central condition is $|2-v|\le2\sqrt{b-1}$, which holds for all $b$ above one fixed onset. After prepending $v$, the resulting words have length $b$, total exactly $2b$, beta type $r\bmod3^b$, bounded offset

$$ \beta_w\le F_1:=(1+3F)/2, $$

and cap at most $\frac32\log_2(b+1)+C$. Since all words in the tail slice have probability $2^{-(2b-v)}$, Lemma 6 gives a count at least

$$ c3^{-b}\binom{2b-v-1}{b-2} \ge c'\frac{(4/3)^b}{\sqrt b}. \tag{89} $$

The last comparison is uniform over the finitely many choices of $v$ and follows from the central binomial estimate with a fixed shift. All constants and the required onset are fixed before $b$ varies.

Fix now a unit class $r\bmod3^{b+1}$ and a positive odd lift $x$. Use the old words from (89) for its coarse reduction. Preserve the first $b-2$ exponents of such a word, ending at a positive odd unit $u$, and write its last pair as $(a,e)$ with $S=a+e$. The old two-letter inverse output is

$$ \frac{2^Su-2^e-3}{9}. $$

Thus $e=c+6h$, where $1\le c\le6$ is determined by

$$ 2^c\equiv2^Su-3\pmod9, \qquad h\ge0. $$

Choose $1\le d\le18$ with

$$ 2^d\equiv2^{S+18}u-3\pmod{27}. $$

The right side is a unit, so the choice is unique. Write $h=3j+t$, $0\le t\le2$, and set

$$ e'=d+18j,\qquad a'=S+18-e'. \tag{90} $$

The exact changes are

$$ a'-a=c+6t+18-d\in[1,35],\qquad e'-e=d-c-6t\in[-17,17],\qquad a'+e'=S+18. \tag{91} $$

In particular both new letters are positive. The new two-letter output is integral and divisible by three. Reducing its numerator modulo three first proves the first step integral; the second step's integrality makes its intermediate state a unit. All states are positive and odd. The full patched beta is therefore congruent to $x$ modulo $3^{b+1}$.

The real two-letter offset decreases, since

$$ 2^{-a'}+3\,2^{-(S+18)}<2^{-a}+3\,2^{-S}. \tag{92} $$

Its contribution to the full beta is multiplied by a positive prefix multiplier, so the full offset is still at most $F_1$. The maximum letter increases by at most thirty-five.

For a fixed fine class and retained prefix, the old total $2b$ determines $S$. Hence $u,c,d$ are fixed. The new last letter determines $j$, and the three possible values of $t$ give at most three old last pairs. Different retained prefixes give different patched words. The patch therefore has multiplicity at most three.

Let $\mathcal D_b$ be all words of length $b$ satisfying

$$ E_w=2b+18,\qquad \beta_w\le F_1,\qquad \max e_i\le M_b:=\left\lceil\frac32\log_2(b+1)+C_1\right\rceil, \tag{93} $$

where $C_1$ covers the fixed spreading exponent and the patch increment. For some fixed $b_*$, every $b\ge b_*$ and every fine unit beta type $r\bmod3^{b+1}$ have

$$ N_r:=\#\{w\in\mathcal D_b:\beta_w\equiv r\pmod{3^{b+1}}\} \ge c\frac{(4/3)^b}{\sqrt b}. \tag{94} $$

The nonunit terminal output in the patch labels the fine beta type only. The actual kernel in the next section uses unit endpoints and has no nonunit intermediate states.

9.4. A common-total kernel and its concentration bound

Give each $w\in\mathcal D_b$ forward probability

$$ p_w=\frac1{2\cdot3^bN_{r(w)}},\qquad r(w)=\beta_w\bmod3^{b+1}. \tag{95} $$

Each fine beta type has total probability $1/(2\cdot3^b)$. The affine image $\Phi_w(\mathcal U)$ consists of the two other fine cylinders above its coarse beta type, with Haar Jacobian $3^{-b}$. Thus the forward block operator preserves unit Haar probability. Its inverse adjoint $P_b$ is Markov at every unit root; the two eligible beta types contribute $1/2$ each. A particular eligible word has probability

$$ \frac1{2N_{r(w)}}\le C\frac{\sqrt b}{B_b}, \qquad B_b=2^{18}(4/3)^b. \tag{96} $$

This is the same exact Haar balance used in [3] and Section 7.4. On positive odd inputs, integrality at the end gives the complete legal inverse word, and its image condition gives unit terminal state. Blocks can be concatenated.

Increase $b_*$ to be at least two and choose

$$ b=b_*+\left\lceil\frac{k\log3}{\log(4/3)}\right\rceil. \tag{97} $$

Then $b\ge k$, $b\le C(k+1)$, and $B_b\ge3^k$. For a fixed unit root and a target $A=a\bmod3^k$, choose a positive odd lift of the root modulo $3^{b+k}$. This preserves every block's legality and final residue. All block endpoints are distinct by deterministic forward iteration and lie in an interval of length at most $F_1B_b$, since they are $B_b(x-\beta_w)$. At most $F_1B_b/3^k+1$ lie in the target. Multiplying by (96) gives

$$ \|P_b1_A\|_\infty\le\delta_k :=\min(1,C\sqrt{k+1}\,3^{-k}). \tag{98} $$

The same bound holds after any positive number of blocks by Markov contraction. The interval may exceed the target modulus: its expansion cancels the reciprocal expansion in (96). There is one total group.

9.5. Pruning and comparison with short ancestors

Write $Q_b$ for the forward block operator and $\mu$ for unit Haar probability. If $\ell$ is a positive multiple of $b$ and $C$ is a unit cylinder of depth $\ell$, contraction and stationarity give $Q_b^{\ell/b}1_C\equiv\mu(C)$. Hence

$$ \int_C P_b^{\ell/b}1_A\,d\mu=\mu(C)\mu(A), \qquad\mu(A)=\frac3{2\cdot3^k}. $$

The inverse probability is constant on the $3^k$ refinements of $C$ at depth $\ell+k$. Their values sum to $3/2$ and are at most $\delta_k$. Therefore at least

$$ H_k=\min\left(\frac{3^k-1}{2},\left\lceil\frac3{2\delta_k}\right\rceil\right) \quad\text{refinements succeed},\qquad \frac{H_k}{3^k}\ge\frac c{\sqrt{k+1}}. \tag{99} $$

Prune these holes at depths $b,2b,\ldots$. Because $b\ge k$, each hole deletes $3^{b-k}$ refinements at the next coarse depth. Put

$$ \Delta_k=1+\frac{\log(1-H_k/3^k)}{b\log3}. $$

Then $0<\Delta_k<1$ and

$$ 1-\Delta_k\ge\frac c{(k+1)^{3/2}}. \tag{100} $$

For $X\ge3^b$, let $m\ge1$ be the least integer with $3^{(m+1)b}>X$. The surviving cylinders contain at most one integer at most $X$, so the set $F_m$ with no route to $A$ in at most $m$ blocks obeys

$$ \#(F_m\cap[1,X])\le\mathcal A_kX^{\Delta_k}, \qquad\mathcal A_k=2\cdot3^{2b},\quad \log\mathcal A_k\le C(k+1). \tag{101} $$

A positive odd integer outside $F_m$ has a final route with total valuation at most $(2b+18)m\le20\log_3X$ and cap $M_b$.

Now use the retained short first-exit theorem of [3, Sections 5--6]. Choose

$$ M=\left\lceil\frac32\log_2(k+1)+C_2\right\rceil, \qquad d=L_*2^{-M},\quad s=1-d,\quad J=\lceil A_*/d\rceil,\quad B_M=2^M/3. \tag{102} $$

Here $L_*,A_*$ are the already fixed constants of that theorem; $A_*$ is not the target cylinder. Choose $C_2$ so that $M$ meets its threshold and

$$ \sigma:=1-3d-\Delta_k\ge\tfrac12(1-\Delta_k) \ge c(k+1)^{-3/2}. \tag{103} $$

For a nonperiodic positive odd unit $x$, that theorem gives at least

$$ c B_M^{-2}x^{-3s}R^{1-3d} \tag{104} $$

distinct unit first-exit leaves in $(R,B_MR)$, reached with cap $M$ in at most

$$ T\le C\left(2^M(M+\log(x+1))+\log R\right) $$

steps, provided $R\ge\max(3^J,2x)$. Take $c\le1$ valid in (104) and choose

$$ R=\max\left(3^J,2x,3^b, \left(2c^{-1}\mathcal A_kB_M^{\Delta_k+2}x^{3s}\right)^{1/\sigma}\right). \tag{105} $$

At $X=B_MR$, the first-exit count is at least twice the bound in (101). One such leaf therefore has a final block route to the target. This comparison does not require failure to inherit along prefix branches or require $M\ge M_b$.

All constants being fixed, (97) and (100)--(105) give

$$ 2^M,J,\sigma^{-1}\le C(k+1)^{3/2},\qquad \log R,T\le C(k+1)^{3/2}\bigl(k+\log(x+1)\bigr). \tag{106} $$

For a positive odd inverse step $u\to v$, $2^e=(3v+1)/u\le4v/u$. The total exponent along a prefix of length $t$ is therefore at most $\log_2(v/x)+2t$. Apply this with $t\le T$ and $v<B_MR$, then add the final cost $20\log_3(B_MR)$. Equation (106) proves the total bound in (74). The prefix cap is at most $(3/2)\log_2(k+1)+C$, and (93),(97) bound the final cap by $\frac32\log_2(k+1)+C$.

If $x$ is periodic, it has two distinct unit inverse children with exponents at most six, and at most one is on its cycle. Choose the other, nonperiodic child, smaller than $(64/3)x$, and prefix this one step to the route just proved. This treats every hypothetical positive cycle and completes the theorem. $\square$

10. Sublinear extra precision and a direct routing word

Let $e_i$ be independent with $\mathbb P(e_i=a)=2^{-a}$ for $a\ge1$, and put

$$ E_i=e_1+\cdots+e_i,\qquad \beta_N=\sum_{i=1}^N3^{i-1}2^{-E_i}. \tag{107} $$

We distinguish conductor depth $q$ from word length $N$. All modular evaluations of dyadic rationals use the inverse of two in the indicated ternary ring. Let $\chi_{q,\xi}(x)=\exp(-2\pi i\xi x/3^q)$, where $\xi$ is a unit modulo $3^q$.

10.1. The full geometry and a stopped product

Use the full conductor-$q$ black and white sets $B,W$ of [13, Section 7], on $1\le j\le K:=\lfloor q/2\rfloor$. Fix the small parameter $\epsilon$ as in that argument and write $\lambda=\epsilon^3$. The black triangles are not intersected with a shorter strip. Let $V=(J,L)$ be Tao's marked holding time: it records the first pair whose sum is three. Then $J$ is geometric with success probability $1/4$, $\mathbb EV=(4,16)$, and $V$ has exponential tails.

For $0\le H\le K$ and $R=K-H$, define

$$ Q_H(j,l)=\mathbb E\exp\left[-\lambda\sum_{t\ge0} 1_W((j,l)+V_1+\cdots+V_t) 1_{j+J_1+\cdots+J_t\le H}\right]. \tag{108} $$

The vector sum at $t=0$ is zero; its white-point indicator is included. Thus $Q_H(j,l)=1$ for $j>H$. For $j\le H$ it obeys the usual recursion

$$ Q_H(j,l)=e^{-\lambda1_W(j,l)}\mathbb E Q_H((j,l)+V). \tag{109} $$

We use the following precise ingredients of Tao's proof. They concern the full triangles and are uniform in $q,\xi$.

1. If $(j,l)$ belongs to a triangle with top height $l_\Delta$, put $s=l_\Delta-l$ and $m=K-j$. Its right vertex lies before $K$, so $0\le s\le(\log9/\log2)m$.
2. The first marked passage $\tau$ with $L_1+\cdots+L_\tau>s$ has horizontal location centered at $s/4$, with the Gaussian and exponential tails in [13, Lemma “Distribution of first passage location”]. Its endpoint is white with probability bounded below by an absolute positive constant, by the separation of the full triangles.
3. If $s>m/\log^2m$, then for every $\eta>0$ there are fixed $P,M_\eta$ such that for $m\ge M_\eta$,

$$ \mathbb E\exp\left[-\lambda\sum_{p=0}^{P-1} 1_W((j,l)+V_1+\cdots+V_{\tau+p})\right]\le\eta. \tag{110} $$

This is the conclusion of Tao's Case 3, obtained from his two lemmas “Many triangles usually implies many white points” and “Large triangles are rarely encountered shortly after a lengthy crossing.” It is the estimate labelled cheap-0 in the arXiv v7 source. Taking its exponent parameter sufficiently large gives any prescribed $\eta$. For this fixed $P$, the same first-passage and holding-time bounds give

$$ \mathbb P(J_1+\cdots+J_{\tau+P}>0.9m)\le C_Pe^{-cm}. \tag{111} $$

Indeed $s/4\le(\log9/(4\log2))m<0.8m$, and $P$ further increments have exponentially small probability of using another $0.1m$.

Lemma 7 (stopped majorant). For every $A>0$ there is $C_A$ such that, uniformly in $q,\xi,H$,

$$ \mathbb E Q_H(V)\le C_A\left(\frac{R+1}{q}\right)^A. \tag{112} $$

Here an upper bound greater than one is harmless. All constants may be enlarged to cover the finitely many small conductor depths.

Proof. We give the modification of Tao's maximal-expression induction. For integer $m\ge1$ put

$$ \mathcal M_m=\sup_{j\ge\max(1,K-m),\ l\in\mathbb Z} \max(K-j,1)^A Q_H(j,l). \tag{113} $$

It is finite and nondecreasing, with $1\le\mathcal M_m\le\max(m,1)^A$. Let

$$ M_0=\left\lceil\max\{C_{A,\epsilon},20(R+1)\}\right\rceil. \tag{114} $$

The constant $C_{A,\epsilon}$ will absorb all fixed thresholds below. We claim $\mathcal M_m\le\mathcal M_{m-1}$ for $m>M_0$ and $m<K$. It suffices to prove

$$ Q_H(j,l)\le m^{-A}\mathcal M_{m-1},\qquad j=K-m. \tag{115} $$

These starting points lie before $H$, since $m>R$. Every later marked state has horizontal coordinate at least $j+1$, so its $Q_H$ is bounded using $\mathcal M_{m-1}$, even if it has passed $H$ or $K$.

If $(j,l)$ is white, (109) and the exponential moment of $J$ give

$$ Q_H(j,l)\le e^{-\lambda}\mathcal M_{m-1} \mathbb E\max(m-J,1)^{-A} \le e^{-\lambda/2}m^{-A}\mathcal M_{m-1}. \tag{116} $$

This is Tao's Case 1. It remains valid for any white starting point before $H$ with $m$ above a fixed constant, irrespective of whether the next mark crosses the stopping line.

Suppose next that the starting point is black and $0\le s\le m/\log^2m$. Stop at its first vertical passage $\tau$. Put $U=J_1+\cdots+J_\tau$ and $a_m=m/\log^2m$. The first-passage estimate gives $\mathbb P(U>a_m)\le C e^{-ca_m}$, uniformly for $s\le a_m$. On $U\le a_m$ the endpoint remains before $H$, because $m-R\ge0.95m$. Its new full remaining distance is large enough to apply the white estimate (116), with remaining distance $m-U$ and maximal expression $\mathcal M_{m-U-1}\le\mathcal M_{m-1}$, if it is white. This white estimate needs only $m-U$ above a fixed constant; it does not require the induction hypothesis at that distance. Dropping all pre-passage discounts, the strong Markov property gives $Q_H(j,l)\le\mathbb E Q_H(j+U,l+L_1+\cdots+L_\tau)$. On the bad event use the bound by $\mathcal M_{m-1}$ for this future value. As in Tao's Case 2, using

$$ \max(m-u,1)^{-A}\le m^{-A} \exp\left(O\left(Au\log m/m\right)\right) \tag{117} $$

and the first-passage moment bound gives an upper bound for $m^AQ_H(j,l)/\mathcal M_{m-1}$ of

$$ 1+O(A/\log m)-c_\epsilon +O_A(m^Ae^{-cm/\log^2m}). \tag{118} $$

Indeed on $U\le a_m$ the remaining-distance ratio is at most $\exp(O(A/\log m))$, whereas it is at least one when paying the white discount. The probability of a white endpoint in this event is at least the full white-exit probability minus $Ce^{-ca_m}$. Therefore one may take any fixed $c_\epsilon$ smaller than the product of that positive white-exit lower bound and $1-e^{-\lambda/2}$, once $m$ is large. The bad event, including any passage beyond $H$, is paid by the final term. Its polynomial weight is at most $m^A$, since the subsequent remaining-distance factor is at most one. Enlarging $C_{A,\epsilon}$ makes (118) less than one.

Finally suppose $s>m/\log^2m$. Choose $\eta=10^{-A-2}$ in (110) and the corresponding fixed $P$. Stop at the passage plus $P$ marked steps. On the event in which the total horizontal advance is at most $0.9m$, all those counted marks lie before $H$: the available distance is $m-R\ge0.95m$. Hence the stopped product includes every discount in (110), and the remaining-distance weight is at most $10^Am^{-A}$. On the complementary event use (111), the bound by one for a product, and the bound $\mathcal M_{m-1}$ for the future $Q_H$. Thus

$$ \frac{m^AQ_H(j,l)}{\mathcal M_{m-1}} \le10^A\eta+C_Pm^Ae^{-cm}<1 \tag{119} $$

for a sufficiently large fixed $C_{A,\epsilon}$. This proves (115). No geometric statement about triangles cut at $H$ has been used.

The trivial base bound and induction give, for every $j\ge1$,

$$ Q_H(j,l)\le\min\left(1, \frac{M_0^A}{\max(K-j,1)^A}\right). \tag{120} $$

If $M_0\ge K-1$, the same statement follows directly from $Q_H\le1$. Average (120) at the first holding time. On $J\le K/2$ the denominator is at least $K/2$, and $\mathbb P(J>K/2)$ is exponentially small. Since $M_0\le C_{A,\epsilon}+21(R+1)$, this proves (112). $\square$

Corollary 8 (shortened word, retained clock). For every $A>0$, $0<\theta<1$ and fixed $B<\infty$, there is $C_{A,\theta,B}$ such that whenever $1\le N\le q$ and $q-N\le BN^\theta$,

$$ \sup_{|\gamma|\le1} \left|\mathbb E\left[\chi_{q,\xi}(\beta_N)\gamma(E_N)\right]\right| \le C_{A,\theta,B}q^{-A}. \tag{121} $$

Proof. Condition on all pair totals and, for odd $N$, the last singleton. The total $E_N$ and the factor $\gamma(E_N)$ are measurable there. Internal pair splits remain independent and uniform. Tao's pair bound gives exactly the majorant with $H=\lfloor N/2\rfloor$ and the full conductor-$q$ white set. The singleton phase has absolute value one. Equation (112), with an exponent larger than $A/(1-\theta)$, proves (121). The floor difference is at most $(q-N+1)/2$ and causes no loss. $\square$

For $N\ge q$, the same conclusion with no length-deficit condition follows from the usual weighted primitive estimate: reducing $\beta_N$ modulo $3^q$ removes its terms after $q$, and conditioning the remaining sum turns $\gamma(E_N)$ into a bounded function of $E_q$.

10.2. Joint oscillation with extra precision

For a subprobability $h$ on $\mathbb Z/3^q\mathbb Z$, let $\operatorname{Osc}_{m,q}(h)$ be its $L^1$ distance from the uniform lift of its projection modulo $3^m$. Normalized averaging is applied to the mass function, exactly as in Tao's definition. Write

$$ \mathcal J_{m,q}^{(N)}= \sum_s\operatorname{Osc}_{m,q} \left(\mathbb P(\beta_N=\cdot\bmod3^q,E_N=s)\right). \tag{122} $$

Theorem 9. Fix $0<\theta<1$ and $B<\infty$. Uniformly for $N\le q\le N+BN^\theta$ and $1\le m\le q$,

$$ \mathcal J_{m,q}^{(N)}\le C_{A,\theta,B}m^{-A} \tag{123} $$

for every fixed $A>0$.

Proof. First let $m\ge0.9q$ and set $r=\lfloor17q/20\rfloor$. For sufficiently large $q$, $r<N$. Split

$$ \beta_N=\beta_r+3^r2^{-E_r}\beta'_{N-r}, \tag{124} $$

where the prime denotes an independent suffix. Put $d_0=\lceil q\log_2 3\rceil$, $u=1+1/(4q)$ and

$$ f_d(z)=\mathbb P(E_r=d,\beta_r=z\bmod3^q),\qquad d\ge d_0. $$

The global fixed-total collision bound of [15] gives

$$ \left(\sum_{d\ge d_0}u^{-d}\right) 3^q\sum_{d\ge d_0,z}u^d f_d(z)^2\le12q^2. \tag{125} $$

For clarity, this estimate does not require $N=q$. It only concerns the prefix length $r\le q$ and modulus $3^q$. Distinct rational numerators at a fixed total are distinct integers. In one residue class they are spaced by $3^q$, giving the elementary squared-count bound used in [15]. Its geometric-word expectation yields a bound linear in $r$, and $\sum_{d\ge d_0}u^{-d}\le4q$ supplies the displayed polynomial factor.

For a high frequency $\xi\notin3^{q-m}\mathbb Z/3^q\mathbb Z$, put $j=v_3(\xi)<q-m$. Its suffix conductor is $\ell=q-r-j\ge0.05q$ for large $q$. The suffix length is $N-r$, and any positive deficit satisfies

$$ \ell-(N-r)=q-N-j\le BN^\theta\le B'\ell^\theta. \tag{126} $$

Also $N-r\ge q/8$ eventually, because $q-N=o(q)$. Thus both the suffix length and its conductor are comparable to $q$, and the right side of (126) is at most a fixed multiple of $(N-r)^\theta$ as required by Corollary 8. If the suffix is shorter than its conductor, Corollary 8 applies; if it is at least as long, use the usual weighted primitive estimate. Therefore the suffix Fourier transform with an arbitrary unit-modulus clock twist is bounded by $\delta=C_Dq^{-D}$ for every fixed $D$, uniformly in the prefix total $d$ and the high frequency.

Let $g_t$ be the suffix subprobability at exact total $t$ on $\mathbb Z/3^{q-r}\mathbb Z$. Its Fourier transform below is taken on that group, while $\widehat f_d$ is taken at depth $q$; this convention absorbs the factor $3^r$ in (124). Fourier expansion gives gives the retained joint coefficient

$$ \widehat\mu_s(\xi)=\sum_{d\ge d_0} \widehat f_d(\xi)\widehat g_{s-d}(2^{-d}\xi). \tag{127} $$

Parseval in the clock twist yields $\sum_t|\widehat g_t(2^{-d}\xi)|^2\le\delta^2$. Weighted Cauchy--Schwarz with weights $u^{-d}$, followed by (125) and Plancherel in $z$, consequently gives

$$ \sum_s\sum_{\xi\notin3^{q-m}\mathbb Z/3^q\mathbb Z} |\widehat\mu_s(\xi)|^2\le12q^2\delta^2. \tag{128} $$

Retain $s\le4q$ and apply Cauchy--Schwarz to these at most $4q$ slices. The normalized spatial $L^1$ norm is bounded by its $L^2$ norm. The discarded mass contributes at most twice its probability to oscillation. Thus

$$ \mathcal J_{m,q}^{(N)}\le\sqrt{48}\,q^{3/2}\delta +2\mathbb P(E_r<d_0)+2\mathbb P(E_N>4q). \tag{129} $$

Both probabilities are exponentially small: $2r/q\to1.7>\log_2 3$, and $N\le q$. Taking $D$ sufficiently large proves the high-scale case.

Eventually $N\ge0.9q$. If $m\le N$, insert precision $N$ between $m$ and $q$. The triangle inequality gives the original natural-depth joint estimate at $(m,N)$ plus the just-proved estimate at $(N,q)$, bounded by $Cm^{-A}+Cq^{-A}$. If $m>N$, the difference from its $m$-projection is at most twice the difference from its $N$-projection, by contraction of conditional expectation in $L^1$. The high-scale estimate applies again. Small parameters are absorbed in the constant. $\square$

10.3. A capped fixed-total floor at the extra depth

We now combine Theorem 9 with the selected fixed-total assembly of [12, Appendix A], using its original windows. The selected-block geometry is due to El Jaouhari [5]; the fixed-total assembly uses Wren's unnormalized tail-mixture argument [10]. The cap loss follows the disjoint-total observation credited in [17]. Appendix A below reproduces the elementary geometric and assembly estimates used in this section. In particular, the direct consumer does not use a density integrability or first-exit theorem.

Choose a sufficiently large fixed $b_0$ and set

$$ \begin{gathered} b_{i+1}=b_i+\lfloor b_i/100\rfloor,\quad k_i=\lfloor b_i/4\rfloor, \quad d_i=\sum_{h<i}b_h,\quad Q_i=d_i+k_i,\\ m_i=\lceil b_i^{3/5}\rceil,\quad \mathcal V_i=\{w:|w|=b_i,\ |E_w-2b_i|\le m_i\},\quad \Gamma_i=\mathcal V_0\cdots\mathcal V_{i-1}. \tag{130} \end{gathered} $$

For a word $w$ put $\omega_w=3^{|w|}2^{-E_w}$ and use the transport

$$ (\mathcal T_{w,h}f)(z)=\omega_w 1_{z\equiv\beta_w\ (3^{|w|})} f((z-\beta_w)/\omega_w\bmod3^h). \tag{131} $$

The retained selected estimates provide fixed constants $B_*,\Lambda_0,p_*>0$ such that $\mathbb P(\Gamma_i)\ge p_*$, $\beta_w\le B_*$, $\omega_w\le16^{b_0-b_i}$ on $\Gamma_i$, and

$$ \left\|\sum_{w\in\Gamma'_i}\mathcal T_{w,h}f_{E_w}\right\|_\infty \le\Lambda_0\sum_D\langle|f_D|\rangle_h, \qquad h\le2b_i,\quad\Gamma'_i\subseteq\Gamma_i. \tag{132} $$

Brackets are normalized full-group means. This applies one fixed-total arithmetic residue-counting bound at a time and then sums over distinct $D$; its constant does not grow with the number of groups. One may take $\Lambda_0=B_*+16^{b_0}$. Positivity permits the displayed subfamilies. These geometric estimates are imported from [12, (51)--(52)] and [5].

Put $\pi_n(s)=2^{-s}\binom{s-1}{n-1}$, with zero mass for $s<n$. For any positive $t,h$ define

$$ R_h^{(t,\tau)}(z)=\frac23\,3^h \mathbb P(\beta_t\equiv z\pmod{3^h},E_t=\tau). \tag{133} $$

This definition also allows $h>t$. Its mean is always $(2/3)\pi_t(\tau)$. Fix $|s-2n|\le2\sqrt n$, let $j$ be the last index with $Q_j\le n$, and put $t_i=n-d_i$. On $i\le j$, let

$$ G_i^{(n,s)}=\sum_{w\in\Gamma_i}\mathcal T_{w,k_i} R_{k_i}^{(t_i,s-E_w)}. \tag{134} $$

The fixed-total assembly [12, Appendix A], reproduced below, gives summable errors $\epsilon_i$, independent of $n,s$, with

$$ \|G_{i+1}^{(n,s)}-G_i^{(n,s)}\text{ lifted}\|_\infty \le\epsilon_i\pi_n(s),\qquad G_i^{(n,s)}/\pi_n(s)\longrightarrow g_i \quad(n\longrightarrow\infty) \tag{135} $$

for each fixed $i$, uniformly over the central totals. Here $g_i$ is a nonnegative function on the quotient of depth $Q_i$ with unit-group mean at least $p_*$. Its errors satisfy $\epsilon_i\le C((i+1)b_i^{3/5}e^{-cb_i^{1/5}} +(i+1)b_i^{-7/5}+b_i^{-3/2})$. The independent-tail bound is used in the long-tail increments and the full joint sum is used in the last increment, as detailed in [12]. No individual conditional-slice bound is assumed.

Lemma 10 (extra-depth retained floor). Fix $0<\theta<1$ and $B<\infty$. There are $C_0,F,c,n_0$ and one fixed unit cylinder $\mathcal C=y\bmod3^{h_0}$ such that for $n\ge n_0$, $0\le K\le Bn^\theta$, $|s-2n|\le2\sqrt n$, and every residue $z$ above $\mathcal C$ at depth $n+K$,

$$ 3^{n+K}\mathbb P\left(\beta_n\equiv z\pmod{3^{n+K}},E_n=s, \beta_n\le F,\max_{a\le n}e_a\le L_n\right)\ge c\pi_n(s), \quad L_n=\left\lceil\frac32\log_2(n+1)+C_0\right\rceil. \tag{136} $$

Proof. For a temporary cap $L$, replace each $\mathcal V_i$ by its subfamily having all letters at most $L$, and denote the resulting prefix family by $\Gamma_i^L$. Define $G_i^L$ using that family in (134). The tails in this intermediate construction remain uncapped.

The exact concatenation identity underlying (135) is unchanged. Its new rejected event concerns the first $b_i$ letters of a free tail. For the set $W_i=\{s-E_w:w\in\Gamma_i^L\}$ of distinct totals,

$$ \sum_{\tau\in W_i} \mathbb P(\max_{a\le b_i}e_a>L,E_{t_i}=\tau) \le b_i2^{-L}. \tag{137} $$

Apply (132), whose group mean of a rejected slice is $(2/3)$ times its probability, and divide by $\pi_n(s)\ge c/\sqrt n$. Original errors remain upper bounds by positivity. Thus

$$ \frac{\|G_{i+1}^L-G_i^L\text{ lifted}\|_\infty}{\pi_n(s)} \le\epsilon_i+C\sqrt n\,b_i2^{-L}. \tag{138} $$

Geometric growth and $b_j\le8n$ bound the sum of the additional terms by $Cn^{3/2}2^{-L}$.

Only the last tail comparison now uses the extra depth. The exact scale inequalities are

$$ k_j\le t_j<b_j+k_{j+1}\le1.2525b_j,\quad n<202b_j,\quad t_j\ge n/1616. \tag{139} $$

The strict upper bound follows from $n<Q_{j+1}$, and $b_{j+1}\le1.01b_j$. Since $K\le Bn^\theta=o(b_j)$ uniformly, $h=t_j+K\le2b_j$ for all sufficiently large $n$. Also $K\le B(1616t_j)^\theta$. Therefore (132) applies at depth $h$, and Theorem 9 applies to the actual tail of length $t_j$ at that depth. Compare the lifted $G_j^L$ with

$$ H_{n,s,K}^L= \sum_{w\in\Gamma_j^L}\mathcal T_{w,t_j+K} R_{t_j+K}^{(t_j,s-E_w)}. \tag{140} $$

For each total $\tau$, the mean difference of its tail densities is $(2/3)\operatorname{Osc}_{k_j,t_j+K} (\mathbb P(\beta_{t_j}=\cdot,E_{t_j}=\tau))$. Distinct prefix totals give distinct $\tau$. Summing is bounded by the full joint oscillation from Theorem 9. Consequently, choosing its exponent two, the new final comparison costs

$$ \|H_{n,s,K}^L-G_j^L\text{ lifted}\|_\infty \le C_{\theta,B}k_j^{-2},\qquad \frac{\|H_{n,s,K}^L-G_j^L\text{ lifted}\|_\infty}{\pi_n(s)} \le C'_{\theta,B}b_j^{-3/2}. \tag{141} $$

Now cap every letter in this last free tail. Its rejected slice has mean $(2/3)\mathbb P(\max_{a\le t_j}e_a>L,E_{t_j}=\tau)$ even at the finer depth $t_j+K$. Disjoint totals and (132) give pointwise loss at most $Ct_j2^{-L}$, hence normalized loss at most $Cn^{3/2}2^{-L}$. There is no extra factor $3^K$ because the transport estimate acts on the normalized density at its actual observation depth.

Choose a fixed base $i_*$ for which the tail of $\sum\epsilon_i$ and the eventual bound (141) are small relative to $p_*$. Choose a unit value of $g_{i_*}$ at least $p_*$ and its cylinder of depth $h_0=Q_{i_*}$. The prefix family $\Gamma_{i_*}$ is finite. As $n$ grows, $L_n$ exceeds its largest letter, and base convergence in (135) gives a positive floor on that same cylinder, uniformly in the central total. Choose $C_0$ large enough that both extra cap losses $Cn^{3/2}2^{-L_n}\le C2^{-C_0}$ fit within the remaining margin. Finally enlarge $n_0$ for the base convergence, base cap, and depth inequalities. This order of choices is independent of $K,n,s$ within the stated ranges. Telescoping (138) and applying (141) and the tail cap therefore leaves a fixed positive normalized floor at depth $n+K$.

These are actual words of length $n$, not longer words projected down. Their real offsets obey

$$ \beta_n\le B_*+16^{b_0-b_j}(3/2)^{t_j} \le B_*+16^{b_0}=:F, \tag{142} $$

using $t_j<2b_j$. Every letter has been capped and the total has remained $s$. Absorb the fixed factor $2/3$ into the floor constant. This proves (136). $\square$

10.4. A direct route with a valuation budget independent of the source

Lemma 11 (spreading at fixed total). Fix $0<\theta<1$ and $B<\infty$. There are $b_*,F_1,C,c>0$ such that for every $b\ge b_*$, $0\le K\le B(b-1)^\theta$, and unit residue $r\bmod3^{b+K}$, there are at least

$$ c\,\frac{4^b}{3^{b+K}\sqrt b} \tag{143} $$

words of length $b$ with $E_w=2b$, $\beta_w\equiv r\pmod{3^{b+K}}$, $\beta_w\le F_1$, and $\max e_i\le\frac32\log_2(b+1)+C$.

Proof. Use the cylinder from Lemma 10. Since two generates the unit group modulo $3^{h_0+1}$, there is $v\in\{1,\ldots,2\cdot3^{h_0}\}$ with $2^vr\equiv1+3y\pmod{3^{h_0+1}}$. Thus $z=(2^vr-1)/3\bmod3^{b-1+K}$ lies above that cylinder. Apply Lemma 10 with $n=b-1$ and $s=2b-v$; the central condition follows from $|2-v|\le2\sqrt{b-1}$ once $b$ exceeds a fixed onset. Prepend $v$ to every tail word. Its offset is $(1+3\beta_{b-1})/2^v$, so its residue is $r$ at the required extra depth and its offset is at most $F_1=(1+3F)/2$. Its total is exactly $2b$; the bounded first letter is absorbed in the cap constant.

All tail words in that slice have probability $2^{-(2b-v)}$. Their number is at least $c3^{-(b-1+K)}\binom{2b-v-1}{b-2}$, which implies (143) by the central binomial estimate uniformly over the finitely many choices of $v$. Prepending the same letter is injective. $\square$

Theorem D (source-independent almost-linear total). For every $\varepsilon>0$ there is $C_\varepsilon$ such that, for every positive odd integer $x$ prime to three, every $k\ge1$, and every unit target $a\bmod3^k$, there is a legal inverse route from the supplied $x$ to that target with

$$ \sum e_i\le C_\varepsilon(k+1)^{1+\varepsilon},\qquad \max e_i\le\frac32(1+\varepsilon)\log_2(k+1)+C_\varepsilon. \tag{144} $$

All intermediate states are positive odd units. The constants do not depend on the size of $x$. More precisely, a sufficiently long word of length $b\le C_\varepsilon(k+1)^{1+\varepsilon}$ with total exactly $2b$ suffices, without a preliminary first-exit tree or a terminal patch.

Proof. Set $\theta=(1+\varepsilon)^{-1}$ and use Lemma 11 with $B=1$. Choose $b$ above its fixed onset, with $k\le(b-1)^\theta$ and $b\le C_\varepsilon(k+1)^{1+\varepsilon}$. Prescribe the unit residue

$$ r=x-3^b2^{-2b}a\pmod{3^{b+k}}. \tag{145} $$

Lemma 11 supplies a word $w=(e_1,\ldots,e_b)$ with this beta type and total $2b$. The affine composite satisfies

$$ I_w(x)=\frac{2^{2b}x-2^{2b}\beta_w}{3^b} =\frac{2^{2b}(x-\beta_w)}{3^b}\equiv a\pmod{3^k}, \tag{146} $$

and is integral. Here $2^{2b}\beta_w$ is an integer. To check every intermediate step, reduce the numerator modulo three: its first term factors as $2^{2b-e_1}(2^{e_1}x-1)$, and all other terms are divisible by three. Thus $I_{e_1}(x)$ is integral. Substituting that integer into the remaining composite and repeating proves integrality at every prefix. For a positive odd integer $u$, an integral $(2^eu-1)/3$ is again positive and odd. If a nonterminal state were divisible by three, the next numerator would be $-1\pmod3$ and could not be integral. The terminal state is a unit by (146), since $k\ge1$. Every step also has exact forward valuation $e$, because $3I_e(u)+1=2^eu$ with $u$ odd. Hence this is a legal route from the actual supplied integer, not only from a compatible residue lift.

Its total is $2b$ and Lemma 11 bounds its largest exponent. The chosen bound on $b$ gives (144). $\square$

10.5. Scope and relation to prior work

Theorem D remains a modular routing result. Its endpoint is less than $(4/3)^b x$ and every intermediate value is at most $4^b x$. The result gives no descent below the original source and does not prove Collatz convergence. In particular, an appended low-valuation corridor still has to pay for the source's actual growth. Nor does the proof establish a linear total in $k$: the stopped bound (112) alone gives no arbitrarily small polynomial error when $q-N$ is a positive proportion of $N$.

The new analytic step is the stop-before-boundary estimate in Lemma 7, with the complete conductor geometry retained. Tao [13] supplies all its renewal and triangle estimates. Gonçalves, Greenfeld, and Madrid [16, Theorem 1.9] abstract that renewal method to general positive increments. Their stated geometric hypotheses include separation from the observation boundary, so applying that theorem to triangles cut at $H$ would not justify Lemma 7. We instead preserve the original triangles and pay for the omitted terminal strip through the induction base.

The joint-coordinate formulation and collision reduction are retained from [14,15]. Lemma 10 extends the final observation in the existing capped fixed-total assembly [17,12]; its disjoint-total cap-loss argument is due to Wren's observation credited in [17]. El Jaouhari's selected-block geometry and the prior natural-depth joint theorem remain analytic inputs. Neither a finite cover census nor an unproved capped-law mixing claim is used.

The author checked the three stopped cases, clock and spatial Fourier normalizations, actual final-tail slack, cap-loss means at the finer depth, and the literal inverse reconstruction. The elementary selected geometry and the fixed-total increment and base estimates are reproduced in Appendix A, and were also checked directly against [12,5]. Fourier Wren independently desk-checked Lemma 7 against Tao, Corollary 8, and Lemma 10, including the final-tail slack and disjoint-total cap losses, and reported no gap [18]. He subsequently checked Appendix A's geometric, grouped transport and increment estimates, including the restriction-by-positivity step; he read but did not verify the fixed-base limit (A8). The high-scale proof of Theorem 9, the collision proposition, GGM, and the Section 10.4 reconstruction remain outside his stated check. The complete chain has not received formal review. No numerical check establishes this analytic chain or resolves the main problem.

11. Reproduction and limits

Lemma 5 retains support information in the finite-horizon tilted construction, removes a terminal exponential tail, and transfers its weights. Lemma 6 instead caps the selected blocks in the existing geometric fixed-total assembly. Its combined selected-block cap loss and its final-tail cap loss are each at most $Cn^{3/2}2^{-L}$ after final central normalization. It leaves the tail law uncapped during the increments, so its joint mixing input remains exactly (J). Neither result assumes a pointwise floor for one fixed capped law at all depths.

The maximum ordinary Collatz height along the Theorem B route is at most $\exp(C(k+1)^3)(x+1)^{C(k+1)^2}$. For Theorem C the corresponding bound is $\exp(C(k+1)^{5/2})(x+1)^{C(k+1)^{3/2}}$. Theorem D bounds the endpoint by $(4/3)^b x$ and the route peak by $4^b x$, where $b=O_\varepsilon(k^{1+\varepsilon})$. None of these height bounds forces descent below a prescribed original source. Fixed-six all-depth routing and a total cost linear in $k$ remain open. No practical value is supplied for the constants inherited from the analytic constructions. These are existence and route-cost bounds; exhaustive search can examine far more candidates.

The new weighted normalization and variable-total patch identities were checked once with five CPU workers on the fixed pairs $(b,M)=(2,6),(2,7),(3,6),(3,7),(4,6)$. The dictionaries contain $1{,}940$ words altogether. All $5{,}820$ patch instances, $11{,}640$ old and patched inverse/forward replays, and $3{,}880$ unit-kernel path replays pass. The exact rational patch-mass inequality, beta bound, Haar pushforward and inverse Markov normalization pass. Normalizing raw geometric probabilities without equalizing beta-type masses fails the Haar check in every case, as expected.

The two-letter fixed-total patch was separately checked once with five workers on all old words of lengths two through six and total $2b$, at their three fine lifts. All $1{,}908$ patches and $3{,}816$ old/new path replays pass: fixed increment eighteen, decreasing beta, the additional ternary digit, positivity, exact valuations, and multiplicity at most three. The largest observed multiplicity is two; the proof permits three. A zero-increment control has invalid outputs in every case. Both audit outputs are retained without enlarging or rerunning them for the subsequent capped fixed-total deduction.

These finite audits check the weighted and fixed-total patch algebra. They cannot verify any uniform analytic floor, the stopped majorant, the asymptotic constants, or (J). Section 10 is an analytic proof and does not rely on a new cover census or simulation. The earlier finite-horizon, descendant, and block audits remain available with [3] and were not rerun. Assessment of the analytic claims concerns their written proofs; no formal-proof development was rebuilt.

The supplement retains the five code/result pairs from version 3 and adds the weighted and two-letter patch checkers and their retained outputs, with a README and checksums. Run python3 src/weighted_block_patch_audit_20261026.py and python3 src/fixed_total_two_letter_patch_audit_20261026.py from the extracted root. Each uses five workers, exact integer/fraction arithmetic and no randomness, and completes in under a minute. Code is MIT licensed and text is CC BY 4.0. The earlier reproduction instructions and unchanged results are also included.

Fourier Wren suggested applying El Jaouhari's narrower windows [5] to Lemma 6 and the disjoint-slice estimate (83), which removes the extra log-log term from the cap. He reproduced the two-letter patch audit and desk-checked the capped construction, spreading, patch, Haar eligibility, pruning and route-cost comparison in Section 9. He subsequently desk-checked the retained Sections 2--6, including the first-exit theorem. His check leaves imported El Jaouhari density integrability and selected-block geometry estimates unverified. The author subsequently checked and reproduced the elementary selected geometry in Appendix A. His subsequent scoped check of the new stopped argument is described in Section 10.5. The window construction and accumulated-window bound are due to El Jaouhari; the new cap-loss estimate does not depend on the window width.

Appendix A. The retained geometry and fixed-total assembly

This appendix records the inputs to (132) and (135), including their normalizations. These arguments are due to El Jaouhari [5] and to Wren's fixed-total adaptation [12,10]. We include them to identify exactly which parts of those constructions the extra-depth consumer needs.

Block probabilities and real bounds. Take $b_0\ge10^6$. For $b\ge b_0$, $b/200\le\lfloor b/100\rfloor\le b/100$ and $\lceil b^{3/5}\rceil\le b/10$. Thus $b_i$ grows geometrically and $k_i\ge b_i/8$. For a geometric letter, $\psi(t)=\log\mathbb E e^{t(e-2)}=-t-\log(2-e^t)$ satisfies $\psi(t)\le3t^2$ for $|t|\le1/4$, because $\psi(0)=\psi'(0)=0$ and $\psi''(t)\le6$ there. Chernoff with $t=m_i/(6b_i)$ gives

$$ \mathbb P(|E_{b_i}-2b_i|>m_i) \le2\exp(-m_i^2/(12b_i)) \le2\exp(-b_i^{1/5}/12). \tag{A1} $$

The all-two word has positive probability in every block family. Summability of the rejection probabilities gives a positive infinite product $p_*$. Unique fixed-length block cuts and independence then give $\mathbb P(\Gamma_i)\ge p_*$. For each selected block $w$ at scale $b_i$, positivity of letters yields $\beta_w\le(3/2)^{b_i}-1\le2^{b_i}$, and

$$ \omega_w\le(3/4)^{b_i}2^{m_i} \le(7/8)^{b_i} \le\tfrac12\,16^{-\lfloor b_i/100\rfloor}. \tag{A2} $$

For the last inequality use $(7/8)^6<1/2$ and $b_i\ge100\lfloor b_i/100\rfloor$. Multiplying (A2) gives $\omega_w\le16^{b_0-b_i}$ for $w\in\Gamma_i$. Backward induction on the remaining blocks bounds the total offset by $B_*=2^{b_0+1}$: if the remaining offset is at most $2^{b_{h+1}+1}$, then the preceding block gives at most $2^{b_h}+2^{b_h}2^{-3\lfloor b_h/100\rfloor}\le2^{b_h+1}$. The terminal empty offset is zero. Also all prefix totals lie within $2d_i\pm2i b_i^{3/5}$, so the number of distinct totals is at most $4i b_i^{3/5}+1$.

Grouped arithmetic transport. At fixed length $d$ and total $D$, the integer numerator $C_w=2^D\beta_w$ determines the word. Indeed $C_w-3^{d-1}=2^{e_d}C_{w^-}$, where $C_{w^-}$ is odd. This recovers $e_d$ by its binary valuation, then the shorter numerator, and finally $e_1$ from the fixed total. For $d=1$ the total already determines the word. The empty word is unique.

Fix an integer output representative $M$ in (131) and a fixed-total group. Its nonempty inverse fibers correspond to distinct integers

$$ R_w=(M-\beta_w)/\omega_w \in[(M-B_*)/\omega_w,M/\omega_w]. \tag{A3} $$

The multiplier $\omega_w$ is constant in the group. At most $B_*/(\omega_w3^h)+1$ of these integers occupy any particular residue modulo $3^h$. Multiplication by the common transport weight bounds its coefficient by $B_*/3^h+\omega_w$. Thus this group contributes at most $(B_*+3^h16^{b_0-b_i})\langle|f_D|\rangle_h$ to the absolute output. For $h\le2b_i$ the coefficient is at most $B_*+16^{b_0}$. Summing over distinct totals proves (132), also for subfamilies. No assertion about equidistribution of offsets is used.

The increment and its clock. For $k\le h\le t$, terms after $h$ in $\beta_t$ vanish modulo $3^h$, and the later exponents are independent. With $\pi_0$ the point mass at zero,

$$ \mathbb P(\beta_t=z\bmod3^h,E_t=\tau) =\sum_\sigma \mathbb P(\beta_h=z\bmod3^h,E_h=\sigma) \pi_{t-h}(\tau-\sigma). \tag{A4} $$

If $W$ is any finite set of distinct totals, the triangle inequality for oscillation and the two bounds on the sum of $\pi_{t-h}$ give

$$ \begin{split} &\sum_{\tau\in W}\operatorname{Osc}_{k,h} (\mathbb P(\beta_t=\cdot\bmod3^h,E_t=\tau))\\ &\quad\le\min(1,|W|\sup_u\pi_{t-h}(u))\, \mathcal J_{k,h}^{(h)}. \tag{A5} \end{split} $$

Only the original natural-depth joint theorem is used for $\mathcal J_{k,h}^{(h)}\le C_A k^{-A}$. For $i<j$, set $h_i=b_i+k_{i+1}$ and

$$ u_i^\tau= \sum_{v\in\mathcal V_i}\mathcal T_{v,k_{i+1}} R_{k_{i+1}}^{(t_i-b_i,\tau-E_v)} -R_{k_i}^{(t_i,\tau)}\text{ lifted to depth }h_i. \tag{A6} $$

Concatenation gives $G_{i+1}-G_i\text{ lifted}=\sum_{w\in\Gamma_i} \mathcal T_{w,h_i}u_i^{s-E_w}$. Inserting the full density $R_{h_i}^{(t_i,\tau)}$ into (A6), its omitted part has group mean $(2/3)\mathbb P(\operatorname{rej}_i,E_{t_i}=\tau)$; the other difference has mean absolute value equal to $(2/3)$ times its joint oscillation. This identity requires no division by a tail slice probability.

Since $d_j\le200b_j$, the last-index inequalities give $n<202b_j$ and $t_i\ge t_j\ge k_j\ge n/1616$. Rejection depends only on the first $b_i$ letters of the free tail, whose independent suffix has length $t_{i+1}\ge n/1616$. The local upper bound $\sup\pi_t\le C/\sqrt t$ bounds its total rejection over the relevant $W$ by $C|W|n^{-1/2}\mathbb P(\operatorname{rej}_i)$. For mixing in (A5), if $i+1<j$, then $t_i-h_i=b_{i+1}-k_{i+1}+t_{i+2}\ge n/1616$, and the same factor $C|W|n^{-1/2}$ applies. If $i+1=j$, this independent tail can be empty; use the factor one and $\sqrt n\le C\sqrt{b_i}$ instead. Apply (132), $|W|\le4ib_i^{3/5}+1$, (A1), and natural-depth joint mixing with exponent two. Dividing by the final central lower bound $\pi_n(s)\ge c/\sqrt n$ proves (135) with

$$ \epsilon_i\le C\left((i+1)b_i^{3/5}e^{-cb_i^{1/5}} +(i+1)b_i^{-7/5}+b_i^{-3/2}\right). \tag{A7} $$

These errors are summable by geometric growth. Positivity gives exactly the same original error upper bounds if the prefix family is restricted by an additional cap; the new cap rejection is handled separately in (137)--(138).

Fixed base. For fixed $i$, the family $\Gamma_i$ is finite. Use (A4) in (134). For each fixed prefix total $D$ and fixed $\sigma$, the local negative-binomial ratio gives $\pi_{n-Q_i}(s-D-\sigma)/\pi_n(s)\to1$, uniformly for $|s-2n|\le2\sqrt n$. The ratios are uniformly bounded for all $\sigma$ by the local upper bound and central lower bound. Truncate the fixed law of $E_{k_i}$, pass to the limit on its finite part, and then let the truncation grow. This proves the base convergence in (135), with

$$ g_i=\sum_{w\in\Gamma_i}\mathcal T_{w,k_i}\mathcal D_{k_i}, \qquad \mathcal D_{k_i}(z)=\frac23\,3^{k_i} \mathbb P(\beta_{k_i}=z\bmod3^{k_i}). \tag{A8} $$

The full-group mean of $\mathcal T_{w,k_i}f$ is $2^{-E_w}\langle f\rangle_{k_i}$, by summing over its injective image. Hence $g_i$ has full-group mean $(2/3)\mathbb P(\Gamma_i)$ and unit-group mean $\mathbb P(\Gamma_i)\ge p_*$. This supplies the stated base floor. All estimates used in Section 10.3 have now been identified or reproduced.

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