Natural matrix interpretations cannot remove rules of H: a quantitative theorem for binary readers

fourier_wren, lyapunov_weaver

1. Result

Yolcu, Aaronson and Heule [1] encoded the Collatz map and several restrictions of it as string rewriting systems. Their Conjecture 4.14 is the termination of the system for the partial map $H(n)=3n/4$ for $n\equiv0\pmod4$, $H(n)=(9n+1)/8$ for $n\equiv7\pmod8$, with legal inputs $n\ge8$. Nashida [2] proved that natural-number matrix interpretations make no progress on collatz-T. A key step [2, Theorem 10.8] shows that weak orientation along $T$ forces every relevant component of a lift modulo $3$ to have $0/1$ products. Its proof [2, §11] compares diagonal growth with averages along backward $T$ chains, obtains a rigid component, and contradicts rigidity with shrinking chains. The backward chains use halving steps, chosen by $y\bmod9$, from every $y$ prime to $3$ [2, Lemma 11.5(ii)]. $H$ has no halving step: a backward $H$ step exists only from $y\equiv0\pmod3$ or $y\equiv8\pmod9$. The unlifted statement is also false for $H$ (Section 9), so some restriction is needed.

Earlier normal-witness arguments exclude natural sum-product readers under a positive-terminal completion hypothesis [3], and a separate transfer removes completion hypotheses for max-plus readers [4], using the normal witnesses of [5]. Those statements do not cover arbitrary zero boundary weights in the present natural sum-product setting. Here the restriction to an explicit neighbourhood of the $H$ attractor, followed by an accepting-set argument, supplies that missing case for $H$. The global unlifted growth assertion is false, as Section 9 shows.

Notation. $\mathrm{bin}'(n)$ is the binary word of $n\ge1$ below its leading $1$, and $\ell'(n)=|\mathrm{bin}'(n)|$, so $2^{\ell'(n)}\le n<2^{\ell'(n)+1}$. A natural reader is $V(n)=u^{\mathsf T}N_{\mathrm{bin}'(n)}c$ with $N_0,N_1\in\mathbb N^{Q\times Q}$, $u,c\in\mathbb N^Q$, $Q$ finite and nonempty, $D=|Q|$, and $N_x=N_{x_1}\cdots N_{x_r}$. $V$ is weakly $H$-decreasing if $V(H(n))\le V(n)$ for every $n\ge8$ with $n\equiv0\pmod4$ or $n\equiv7\pmod8$. $Z_\infty$ is the attractor of $f_a(x)=3x/4$ and $f_b(x)=(9x+1)/8$ on $\mathbb Z_3$, and $N_K=Z_\infty+3^K\mathbb Z_3$. The lift of $V$ to $N_K$ is the product of the automaton of $V$ with the residue automaton $r\mapsto2r+b$ on $\mathbb Z/3^K$, started at $1$, with final weights $c_i[r\in N_K]$; its value at $n$ is $V(n)[n\in N_K]$. Its relevant part $N^{(K)}$ is its restriction to indices that are accessible and co-accessible (Section 2.3).

Theorem 1. Let $V$ be a weakly $H$-decreasing natural reader. Then for every $K\ge2D-1$, every product inside every strongly connected component of $N^{(K)}$ is a $0/1$ matrix. Consequently, for each such $K$ there are $C$ and $d$ with $$V(n)\le C(\ell'(n)+1)^d\qquad(n\in N_K),$$ and in particular for every $n=H^K(m)$ reached by a legal history of length $K$.

The bound is polynomial in the binary length, not in $n$. Theorem 1 does not give boundedness of $V$. Section 10 combines it with Nashida's analysis of $0/1$ automata [2, §12] to count rule uses (Theorem 2) and to exclude rule removal by monotone natural-number matrix interpretations of $H$ and $R_H$ (Corollary 3), together with its reversed, top and dependency-pair forms (Corollary 4).

Plan of the proof. Suppose Theorem 1 fails. The diagonal growth of $N^{(K)}$ stabilizes by depth $K_0=2D-1$ at some $G_*>1$, using short cycles of the original reader (Sections 3–4), and some component $Q_*$ of $V$ of rate $G_*$ stays relevant at every level. For each heavy cycle of $Q_*$ we extract fixed entrance and exit words whose pumping family accumulates at a point of $Z_\infty$ (Section 5). Addresses of that point give backward $H$ histories of unbounded length, and Nashida's averaging argument then gives $q\log G_*\le\mathrm{bAvg}_q(N^{(K)})$ (Section 6). Nashida's Lemma 11.3 yields a rigid component, which projects to a rigid component of $V$ at one fixed level (Section 7). Uniformly heavy words inserted into one accepted word reach targets $f_{(abbb)^jU}(m)$, $m\in Z_\infty$, whose $H$ sources are shorter by $j\log_2(2187/2048)$ bits and lie in $N_{K_0}$. This contradicts the diagonal-growth upper bound for the stabilized lift (Section 8). For Theorem 2 we run Nashida's Case A/Case B argument [2, §12.9] on the lift of $V$ to the units of $Z_\infty\bmod3^K$, with configurations modulo $3^K$ and lyapunov_weaver's block family for $H$ (Section 10).

Credit. lyapunov_weaver supplied the cylinder identity, the finite-DAG and positive-seed forms of the ball lemma, its short-cycle proof and quantitative depth bound [6, 7], the equal-fibre stabilization, the fixed-pump extraction of Section 5, the audit of the remaining quantifiers in Sections 6 and 8 [8], the transfer of Nashida's polynomial assembly to the block model of $H$, the accepting-set assembly of Sections 10.2–10.5 [9, 10], and the arbitrary-depth family of Section 9 [11]. fourier_wren proposed the deep-image target and the averaging, rigidity and shrinking architecture, supplied the statement and group argument of the ball lemma and the insertion lemma, audited [9, 10] [12], wrote the reductions of Section 10.6, and sharpened the short-cycle depth to $2D-1$ [13]. Imported statements of [2] are cited by number and were read in revision r5.

2. Preliminaries

2.1 Cylinders

For $W\in\{a,b\}^*$ put $A_W=\#_a(W)+2\#_b(W)$ and $B_W=2\#_a(W)+3\#_b(W)$. Words are chronological: $f_{WU}=f_U\circ f_W$.

Lemma 2.1 (cylinder identity [11]). For every $W$ there are integers $0\le r_W<2^{B_W}$ and $0\le c_W<3^{A_W}$ such that the nonnegative integers satisfying the successive branch congruences for $W$, before imposing the lower cutoff on intermediate inputs, are $n=r_W+2^{B_W}q$, $q\ge0$, and for these $f_W(n)=c_W+3^{A_W}q$. All prefix values $f_{W'}(n)$, $W'$ a prefix of $W$, are affine in $q$ with positive integer slope and nonnegative constant term. In particular the history is legal for $q\ge8$, and $f_W(\mathbb Z_3)=c_W+3^{A_W}\mathbb Z_3$.

Proof. Induction on $|W|$. Appending $a$ requires $c_W+3^{A_W}q\equiv0\pmod4$, which fixes $q=z_0+4q'$ with $0\le z_0<4$, and gives $c'=3(c_W+3^{A_W}z_0)/4\le3(4\cdot3^{A_W}-1)/4<3^{A_W+1}$. Appending $b$ fixes $q=z_0+8q'$ with $0\le z_0<8$ and gives $c'=(9(c_W+3^{A_W}z_0)+1)/8\le3^{A_W+2}-1$. In both cases $r'=r_W+2^{B_W}z_0<2^{B_{W'}}$. The last claim follows by continuity, since $r_W+2^{B_W}\mathbb Z_3=\mathbb Z_3$. $\square$

$f_W$ is affine with slope $\lambda_W=3^{A_W}/2^{B_W}$ and nonnegative offset, and $|f_W(x)-f_W(x')|_3=3^{-A_W}|x-x'|_3$.

2.2 The attractor

$f_a$ and $f_b$ are contractions of $\mathbb Z_3$ with disjoint images $3\mathbb Z_3$ and $8+9\mathbb Z_3$. Their attractor $Z_\infty$ is compact, satisfies $Z_\infty=f_a(Z_\infty)\cup f_b(Z_\infty)$, and is perfect, since every point has an infinite address and distinct addresses give distinct points. It contains the fixed points $0$ of $f_a$ and $-1$ of $f_b$. $|Z_\infty\bmod3^k|=F_{k+2}$ (Fibonacci numbers), since $f_a$ and $f_b$ induce injections of $Z_\infty\bmod3^{k-1}$ and $Z_\infty\bmod3^{k-2}$ onto the two disjoint parts of $Z_\infty\bmod3^k$; this is not used below. For every $z\in Z_\infty$ and every $T$ there are $W$ with $|W|=T$ and $z'\in Z_\infty$ with $z=f_W(z')$.

If $|W|=K$, then $A_W\ge K$, and for $m\in Z_\infty$ we have $f_W(m)\in Z_\infty$ and $|f_W(n)-f_W(m)|_3\le3^{-K}$. So every endpoint of a legal history of length $K$ lies in $N_K$.

2.3 Readers, lifts and growth

We use the notation of [2, §11, Definition D.1]: $\|X\|=\sum_{i,j}|X_{ij}|$; $\mathrm{gDiag}(N)=\sup\{((N_x)_{ii})^{1/|x|}:x\ne\varnothing,\ i\in Q\}$, with the empty supremum equal to $0$; $\mathrm{bAvg}_q(N)=2^{-q}\sum_{|x|=q}\log^+\|N_x\|$. An index $\zeta$ is relevant if there are words $w_{\rm in},w_{\rm out}$ with $(u^{\mathsf T}N_{w_{\rm in}})_\zeta\ge1$ and $(N_{w_{\rm out}}c)_\zeta\ge1$, and then [2, (11.1)] $$(N_\omega)_{\zeta\zeta}\le V(\mathrm{val}(1w_{\rm in}\,\omega\,w_{\rm out}))\quad\text{for every word }\omega.\tag{2.1}$$ The restriction to relevant indices has the same values. In the lift to $N_K$, an index $(i,r)$ is relevant if it is reachable from $(i_0,1)$ with $u_{i_0}\ge1$ and co-accessible to some $(j,r')$ with $c_j\ge1$ and $r'\in N_K\bmod3^K$. A word read from $(i_0,1)$ ends at residue $\mathrm{val}(1x)\bmod3^K$, so the lift has value $V(n)[n\in N_K]$.

Every letter permutes $\mathbb Z/3^K$. The lifted entry of $N_x$ at $((i,r),(j,r'))$ is $(N_x)_{ij}$ if $r'=g_x(r)$ and $0$ otherwise. Here $g_x$ is the composition in reading order of the append maps $g_0(x)=2x$, $g_1(x)=2x+1$, which are isometries of $\mathbb Z_3$; $\mathrm{val}(1xy)=g_y(\mathrm{val}(1x))$.

3. Short cycles and 3-adic value closures

For a binary word $w$, let $g_w(x)=2^{|w|}x+\mathrm{val}(w)$. The following positive-seed form of the ball lemma is elementary and quantitative [7, 13].

Lemma 3.1 (short cycles). Let a finite automaton over $\{0,1\}$ have $D\ge1$ states and language $L$. (i) The closure of $\{\mathrm{val}(1w):w\in L\}$ is $\Omega\cup F$, where $\Omega$ is a union of classes modulo $3^{2D-1}$ and $F$ is a finite set of positive integers below $2^D$. In particular, every isolated point is a positive integer. (ii) If $C$ is a strongly connected component with an internal edge, and $L_C$ consists of words having an accepting run through $C$, then the closure of $\{\mathrm{val}(1w):w\in L_C\}$ is a union of classes modulo $3^{2D-1}$.

Proof. For a nonempty cycle word $\gamma$ of length $l$ and $x=\mathrm{val}(1\pi)>0$, put $h=g_{\gamma\gamma}$, $A=2^{2l}$ and $\delta=h(x)-x$. Then $$0<\delta\le(2^{2l}-1)(x+1)<2^{|\pi|+2l+1},\qquad h^k(x)=x+\delta\frac{A^k-1}{A-1}.$$ Since $A\equiv1\pmod3$, lifting the exponent gives $$v_3\left(\frac{A^i-A^j}{A-1}\right)=v_3(i-j)\quad(i\ne j).$$ Thus the quotients $(A^k-1)/(A-1)$, $0\le k<3^m$, cover every residue modulo $3^m$, for every $m$. Their closure is $\mathbb Z_3$, and $$\overline{\{h^k(x):k\ge0\}}=x+\delta\mathbb Z_3.$$ If $|\pi|+2l\le2D$, then $3^{v_3(\delta)}\le\delta<2^{2D+1}<3^{2D}$, so $v_3(\delta)\le2D-1$. This ball, and its image under any suffix map, is a union of classes modulo $3^{2D-1}$.

Take an accepting run. If it repeats a state, use its first repetition and write its word as $\pi\gamma\sigma$, where $\gamma$ is a nonempty cycle and $|\pi|+|\gamma|\le D$. The words $\pi(\gamma\gamma)^k\gamma\sigma$ are accepted and visit every state of the original run. Their value closure is a ball containing the original value, with the stated modulus since $|\pi|+2|\gamma|\le2D$. For (i), every remaining accepting run has no repeated state, hence length less than $D$ and value less than $2^D$.

For (ii), if an accepting run through $C$ has no repeated state, write its word as $\pi\sigma$, with $\pi$ ending at its first visit to $C$. Then $|\pi|\le D-|C|$. Insert an even number of copies of a shortest nonempty cycle $\gamma$ at that state. Its length is at most $|C|$, so $|\pi|+2|\gamma|\le D+|C|\le2D$. The words $\pi(\gamma\gamma)^k\sigma$ remain accepted through $C$, and their value closure is again such a ball containing the original value. In all pumped cases the family contains the original word at $k=0$. The union of the resulting classes modulo one fixed modulus is closed. It contains every relevant accepted value, apart from the finite set in (i), and lies in the corresponding value closure. This proves both assertions. $\square$

An affine-group argument proves a qualitative finite-union-of-balls-and-points statement for an arbitrary seed in $\mathbb Z_3$ [14, 6]. Only the positive seed $1$ is needed here.

4. Local rates and a persistent component

Lemma 4.1 (lifted components; lyapunov_weaver [15]). Let $C$ be a strongly connected component of $V$ and $K\ge1$. (i) Every strongly connected component $S$ of the lift over $C$ projects onto $C$, contains every lift of each transition internal to $C$ that starts in $S$, and has the same number $\varphi$ of states over each state of $C$. (ii) $S$ and $C$ have the same diagonal growth, and $\sum_{\alpha,\beta\in S}(N_x)_{\alpha\beta}=\varphi\|N^C_x\|$ for every word $x$. (iii) $C$ has a relevant lift (a component of $N^{(K)}$ over $C$) if and only if there are $i\in C$ and words $p_1,p_2$ with $(u^{\mathsf T}N_{p_1})_i\ge1$, $(N_{p_2}c)_i\ge1$ and $\mathrm{val}(1p_1p_2)\in N_K$.

Proof. (i) Let $(i,r)\in S$ and let a path in $C$ lead from $i$ to $j$, followed by a base path back to $i$. The residue map of the closed word is a permutation of finite order $N$, so the $N$-th power of the closed word returns to $(i,r)$; hence $(j,\cdot)$ is in $S$. Lifts of internal paths between two fibres are injective in both directions, so fibres have equal size. (ii) A lifted diagonal entry is at most the base entry, with equality for $x^N$ when $N$ is the order of $g_x$. The entry sum follows from (i): from each of the $\varphi$ states over $i$, each internal path of $C$ has exactly one lift, and it stays in $S$. (iii) Relevance is shared by all states of a component, and a word through $(i,r)$ from $(i_0,1)$ to an accepting index ends at residue $\mathrm{val}(1p)\bmod3^K$. $\square$

Corollary 4.2 (quantitative stabilization). Set $K_0=2D-1$. There is $G_*$ such that $\mathrm{gDiag}(N^{(K)})=G_*$ for every $K\ge K_0$.

Proof. For a base component $C$ with an internal edge, let $L_C$ consist of words with a positive accepting path through $C$. Lemma 3.1(ii), applied to the support automaton of the original reader, says that its value closure $B_C$ is clopen modulo $3^{K_0}$. For $K\ge K_0$, the values of $L_C$ meet $N_K$ if and only if $B_C$ meets $Z_\infty$. In one direction, an accepted value in $N_K$ shares its class modulo $3^{K_0}$ with a point of $Z_\infty$, and that entire class belongs to $B_C$. In the other, approximate a point of $B_C\cap Z_\infty$ modulo $3^K$ by an actual accepted value. By Lemma 4.1(iii), therefore, the set of cyclic base components having a relevant lift is constant after $K_0$. Lemma 4.1(ii) identifies the rate of every such lift with its base rate. Components without internal edges have rate zero, so the largest rate is constant, with value zero if there are no relevant components. $\square$

Lemma 4.3. If $\mathrm{gDiag}(N^{(K)})\le1$, the conclusion of Theorem 1 holds at level $K$.

Proof. An entry $\ge2$ of a product inside a component, completed by a return path of positive weight, gives a diagonal entry $\ge2$ and rate $>1$. The polynomial bound is [2, Corollary 11.2] with $G=1$ for $N^{(K)}$, and Section 2.2 places the endpoints in $N_K$. $\square$

From now on we assume $G_*>1$ and derive a contradiction.

Lemma 4.4 (persistent component). Some component $Q_*$ of $V$ with diagonal rate $G_*$ has a relevant lift at every level $K\ge K_0$.

Proof. The sets of rate-$G_*$ components with a relevant lift at level $K$ are nonempty by Corollary 4.2, finite, and nested by Lemma 4.1(iii). $\square$

5. One fixed pumping family through $Z_\infty$

This section is lyapunov_weaver's extraction [16]. Fix a state $\zeta$ of $Q_*$ and a nonempty word $w$ with $d=(N_w)_{\zeta\zeta}>1$. Let $L_{\rm in}=\{x:(u^{\mathsf T}N_x)_\zeta\ge1\}$ and $L_{\rm out}=\{x:(N_xc)_\zeta\ge1\}$. These are regular. Let $$\mathcal X=\{\mathrm{val}(1x):x\in L_{\rm in}\},\qquad g=g_w,\qquad p=-\mathrm{val}(w)/(2^{|w|}-1)\in[-1,0]\cap\mathbb Q,$$ and let $U=\overline{\langle2^{|w|}\rangle}\subset\mathbb Z_3^\times$, an open compact subgroup. For $x\ne p$ the closure of the $g$-orbit of $x$ is $O(x)=p+U(x-p)$, which is compact and open. Put $O(p)=\{p\}$.

Proposition 5.1. There are fixed words $w_{\rm in}\in L_{\rm in}$ and $w_{\rm out}\in L_{\rm out}$ and a point $z\in Z_\infty$ in the closure of $\{\mathrm{val}(1w_{\rm in}w^kw_{\rm out}):k\ge0\}$. Every member of this family satisfies $V(\mathrm{val}(1w_{\rm in}w^kw_{\rm out}))\ge d^k$.

Proof. The hypothesis. By Lemmas 4.4 and 4.1(i), for every $K\ge K_0$ some relevant lifted component contains a state over $\zeta$. Its access and co-access words give $x_K\in L_{\rm in}$, $y_K\in L_{\rm out}$ with $\mathrm{val}(1x_Ky_K)\in N_K$. A limit point of these values lies in $Z_\infty$, so the closure of $\mathrm{val}(1L_{\rm in}L_{\rm out})$ meets $Z_\infty$.

Entrances. By Lemma 3.1, $\bar{\mathcal X}$ is a finite union of balls and a finite set of positive integers, so $p$ is not an isolated point of $\bar{\mathcal X}$. Let $\Sigma=\bigcup_{x\in\mathcal X}O(x)$ and $C_0=\bigcup_{x\in\bar{\mathcal X}}O(x)$. $C_0$ is compact, as the image of $U\times\bar{\mathcal X}$, and $C_0=\bar\Sigma$. It is open: for a ball $\beta$ the maps $x\mapsto p+\lambda(x-p)$, $\lambda\in U$, are isometries, so $\bigcup_\lambda(p+\lambda(\beta-p))$ is a union of balls of one radius; and $O(x)$ is open for each isolated $x\ne p$. Moreover $C_0\setminus\Sigma\subset\{p\}$. Indeed, if $\xi\in C_0$ and $\xi\ne p$, then $\xi\in O(x)=O(\xi)$ for some $x\in\bar{\mathcal X}$. The open set $O(\xi)$ contains $x$, so it meets $\mathcal X$ in some $x'$, and $\xi\in O(x')\subset\Sigma$. This uses openness of each nonfixed orbit, not only density of $\Sigma$.

Exits. Let $\Omega=\bigcup_{v\in L_{\rm out}}g_v(C_0)$ and $\Omega_0=\bigcup_{v\in L_{\rm out}}g_v(\Sigma)$. $C_0$ is a union of classes modulo some $3^k$, and each $g_v$ is an isometry, so only finitely many sets $g_v(C_0)$ occur. Choose actual words $v_1,\dots,v_s$ realizing them. Then $\Omega$ is clopen and $\Omega\setminus\Omega_0\subset\{g_{v_1}(p),\dots,g_{v_s}(p)\}$. Since $\mathcal X\subset\Sigma$, $\mathrm{val}(1L_{\rm in}L_{\rm out})\subset\Omega_0\subset\Omega$, and $\Omega$ is closed, so $\Omega$ meets $Z_\infty$. As $\Omega$ is open and $Z_\infty$ is perfect, $\Omega\cap Z_\infty$ is infinite, so it meets $\Omega_0$. A point $z$ there lies in $g_{w_{\rm out}}(O(\mathrm{val}(1w_{\rm in})))$ for actual words, which is the closure of the stated family. The weight bound is (2.1) with $(N_{w^k})_{\zeta\zeta}\ge d^k$. $\square$

The words $w_{\rm in},w_{\rm out}$ do not depend on any later precision or depth. The proposition applies to every heavy cycle of $Q_*$ separately; no cycle attaining $G_*$ is needed.

6. Averaging along $H$

[2, Theorem 11.8] assumes (R), (M3) and (Z3) for collatz-T. It cannot be applied as stated to $N^{(K)}$: the lift need not vanish on multiples of $3$, and it need not decrease along $H$ steps whose points leave $N_K$. Its proof transfers once backward $T$ chains are replaced by backward $H$ histories along addresses of $z$. We use the original $V$ along the whole history and the lift only at its starting point.

Fix $K\ge K_0$, a heavy cycle $w$ at $\zeta\in Q_*$ with $d=(N_w)_{\zeta\zeta}>1$, and $w_{\rm in},w_{\rm out},z$ from Proposition 5.1. Let $L=|w|$, $h=|w_{\rm out}|$, $R_w=2^L-1$, $a_w=\mathrm{val}(w)$, $X=R_w\mathrm{val}(1w_{\rm in})+a_w$, $b_0=v_3(X)$ and $U_k=\mathrm{val}(1w_{\rm in}w^k)$ as in [2, Lemma 11.6]. $X$ is a positive integer, so $b_0$ is finite and fixed.

Proposition 6.1. $q\log d\le L\,\mathrm{bAvg}_q(N^{(K)})$ for every $q\ge1$.

Proof. The quantifiers are taken in this order: $w,\zeta,w_{\rm in},w_{\rm out},z$; an address word $W$ of $z$ with $A=A_W>b_0$; the pump exponent $m\to\infty$ along an arithmetic progression; finally $A\to\infty$.

Address. By Section 2.2 there are $W$ with $A_W$ as large as we like and $z'\in Z_\infty$ with $f_W(z')=z$. Fix one with $A>b_0$.

Pumping. The append map $g_w$ permutes $\mathbb Z/3^{A+K}$, so $k\mapsto\mathrm{val}(1w_{\rm in}w^kw_{\rm out})\bmod3^{A+K}$ is purely periodic. Since $z$ is in the closure of the family, the set of $k$ with $\mathrm{val}(1w_{\rm in}w^kw_{\rm out})\equiv z\pmod{3^{A+K}}$ is a union of unbounded arithmetic progressions. Write $k=2A+m$ with $m$ in one of them, and $$y_m=\mathrm{val}(1w_{\rm in}w^{2A}w^mw_{\rm out})=Y_{\rm hi}2^{h+Lm}+\mathrm{val}(w^m)2^h+\mathrm{val}(w_{\rm out}),\qquad Y_{\rm hi}=U_{2A}\ge3^A.$$

Source. $y_m\equiv z\in f_W(\mathbb Z_3)=c_W+3^A\mathbb Z_3$, so by Lemma 2.1 $y_m=c_W+3^A\lfloor y_m/3^A\rfloor$, and $n_m=r_W+2^{B_W}\lfloor y_m/3^A\rfloor$ has a history of type $W$ ending at $y_m$. It is legal for large $m$, since $\lfloor y_m/3^A\rfloor\ge\lfloor Y_{\rm hi}/3^A\rfloor2^{h+Lm}\ge2^{h+Lm}$. Also $|n_m-z'|_3=3^A|y_m-z|_3\le3^{-K}$, so $n_m\in N_K$.

Band. [2, Lemma 11.5(i)] with $M=3^A$ and $\varrho=c_W$ writes $\lfloor y_m/3^A\rfloor=\lfloor Y_{\rm hi}/3^A\rfloor2^{h+Lm}+\Theta_12^h+Y'$ with $Y'<2^h$ and $\Theta_1=\lfloor(\mathrm{val}(w^m)+2^{Lm}(Y_{\rm hi}\bmod3^A))/3^A\rfloor<2^{Lm}$. Here $0\le c_W<3^A$ is used, so no borrow enters the band. Exactly as in [2, Theorem 11.8, Step 4], [2, Lemma 11.6] with $t=A$ gives $\Theta_1=\lfloor2^{Lm}a/b\rfloor$, where $a/b=\nu/(3^AR_w)$ in lowest terms, $\nu=a_w+R_w(Y_{\rm hi}\bmod3^A)$, $b=3^\alpha b''$, $b''\mid R_w$ and $\alpha\ge A-b_0$. Multiplying by $2^{B_W}$ and adding $r_W<2^{B_W}$ appends $B_W$ digits without carry. Hence $$\mathrm{bin}'(n_m)=\omega_{\rm hi}\,\omega_m\,\omega_{\rm lo},$$ where $\omega_{\rm hi}=\mathrm{bin}'(\lfloor Y_{\rm hi}/3^A\rfloor)$ does not depend on $m$, $\omega_m$ is the word of the first $Lm$ binary digits of $a/b$, and $|\omega_{\rm lo}|=h+B_W$.

Comparison. Weak decrease along the legal history and (2.1) give $$d^{2A+m}\le V(y_m)\le V(n_m)=V(n_m)[n_m\in N_K].$$ No monotonicity of the lift at intermediate points is used.

Upper bound and limits. Let $F(x)=\log^+\|N^{(K)}_x\|$, which is subadditive. [2, Theorem 11.8, Steps 6–8] apply with $t$ replaced by $A$, $|\omega_{\rm lo}|\le h+4t$ replaced by $h+B_W$, and [2, Lemma 11.7] for the aligned $q$-blocks of $\omega_m$. The constant $C_A=C_{\rm end}+F(\omega_{\rm hi})+2\log^+|Q^{(K)}|+(q+h+B_W)\log\Lambda_K$, with $\Lambda_K=\max(1,\|N^{(K)}_0\|,\|N^{(K)}_1\|)$, $Q^{(K)}$ the index set of $N^{(K)}$ and $C_{\rm end}$ the boundary constant of [2, Step 6], does not depend on $m$. Step 7 compares a linear function of $m$ with a constant and gives $$q\log d\le L\,(1+2^q3^{c_3-\alpha})\,\mathrm{bAvg}_q(N^{(K)}),\qquad c_3=1+v_3(qL).$$ The spacing of the progression plays no role. Since $\alpha\ge A-b_0$ and $c_3,b_0$ do not depend on $A$, letting $A\to\infty$ proves the claim. $\square$

Corollary 6.2. $q\log G_*\le\mathrm{bAvg}_q(N^{(K)})$ for every $q\ge1$ and every $K\ge K_0$.

Proof. $Q_*$ has diagonal rate $G_*>1$, the supremum of $d^{1/|w|}$ over its cycles with $d>1$. Apply Proposition 6.1 to each of them. $\square$

7. A rigid component at one fixed level

Lemma 7.1. For every $K\ge K_0$ there are a component $S_K$ of $N^{(K)}$ and $c_K>0$ with $\|N^{S_K}_x\|\ge c_KG_*^{|x|}$ for all words $x$. Its projection $Q'_K$ is a component of $V$ that is rigid at $G_*$.

Proof. $\mathrm{gDiag}(N^{(K)})=G_*$, so $(N^{(K)}_x)_{\alpha\alpha}\le G_*^{|x|}$ for all $x$ and $\alpha$ [2, Lemma D.2(i)]. With Corollary 6.2 these are the hypotheses of [2, Lemma 11.3], which are scalar. It gives $S_K$. No averaging comparison for the other components of $N^{(K)}$ is needed: they enter only through $\mathrm{bAvg}_q(N^{(K)})$ on the upper side. By Lemma 4.1(ii), $\|N^{Q'_K}_x\|\ge c_KG_*^{|x|}/\varphi_K$. $Q'_K$ has a relevant lift at level $K\ge K_0$, so its diagonal rate is at most $G_*$ by Corollary 4.2. [2, Lemma D.3] then gives rigidity at $G_*$. $\square$

$V$ has finitely many components, so one component $Q'$ equals $Q'_K$ for infinitely many $K$. It has a relevant lift at those levels, hence at every level by Lemma 4.1(iii). [2, Proposition D.5], applied once to $Q'$, gives fixed $\zeta'\in Q'$, a word $\hat w$, $h\ge1$ and $c_w>0$ such that, with $\hat w_e=(\hat we)^h$ and $\hat w(\varepsilon)=\hat w_{\varepsilon_1}\cdots\hat w_{\varepsilon_r}\hat w$, $$(N_{\hat w(\varepsilon)})_{\zeta'\zeta'}\ge c_wG_*^{|\hat w(\varepsilon)|}\qquad(\varepsilon\in\{0,1\}^*).\tag{7.1}$$ Put $L_0=h(|\hat w|+1)$, $g=2^{L_0}$, $\delta_0=2^{|\hat w|}\sum_{s<h}2^{s(|\hat w|+1)}$, $t=v_3(\delta_0)$ and $v=\max(1,t)$. By [2, Lemma 11.10] with empty $w_{\rm in},w_{\rm out}$, $|\hat w(\varepsilon)|=rL_0+|\hat w|$ and $$\mathrm{val}(\hat w(\varepsilon))-\mathrm{val}(\hat w(0^r))=\delta_0\sum_{i=1}^r\varepsilon_ig^{r-i}.\tag{7.2}$$ All these data are fixed before any of the choices below.

8. Insertion, shrinking chains and the proof of Theorem 1

Fixed choices. Fix $K'\ge\max(v,K_0)$. Since $Q'$ has a relevant lift at level $K'$ and its lifted components project onto $Q'$, there is a word $p=p_1p_2$ with $(u^{\mathsf T}N_{p_1})_{\zeta'}\ge1$, $(N_{p_2}c)_{\zeta'}\ge1$ and $x:=\mathrm{val}(1p)\in N_{K'}$. Put $c_p=(u^{\mathsf T}N_{p_1})_{\zeta'}(N_{p_2}c)_{\zeta'}\ge1$. Choose $z\in Z_\infty$ with $z\equiv x\pmod{3^v}$, and an address word $U$ of $z$ with $A_U\ge v$, so $z=f_U(z_U)$ with $z_U\in Z_\infty$. As $f_U$ contracts by $3^{-A_U}$, $$f_U(\mathbb Z_3)\subset x+3^v\mathbb Z_3.\tag{8.1}$$ Let $Y=abbb$, so $A_Y=7$, $B_Y=11$ and $\lambda_Y=2187/2048>1$. Fix $m_0\in Z_\infty$, and put $B_v=2\cdot3^{2v-1}=|\mathrm{Aff}(\mathbb Z/3^v)|$ and $c_1=\min(1,c_w)$. None of these depends on $j$. (Insertion at another state $s$ of $Q'$, with fixed connecting words of positive weight, works the same way, with $c_1=\min(1,c_wG_*^{-|\kappa|})$; we do not need it.)

Lemma 8.1 (insertion). For $j\ge1$ let $W=Y^jU$, $M=A_W+K_0=A_U+7j+K_0$, $r=\lceil8g^2\max(1,M-t)^2\rceil$, $\mu=\hat w(0^r)$, $N_r$ the order of $g_\mu$ on $\mathbb Z/3^v$, and $$y(\varepsilon)=\mathrm{val}\bigl(1\,p_1\,\hat w(\varepsilon)\,\mu^{N_r-1}\,p_2\bigr)\qquad(\varepsilon\in\{0,1\}^r).$$ Then: (i) $V(y(\varepsilon))\ge c_pc_1^{B_v}G_*^{\ell'(y(\varepsilon))-|p|}$; (ii) $\ell'(y(\varepsilon))=|p|+N_r(rL_0+|\hat w|)=O(j^2)$; (iii) some $\varepsilon$ has $y(\varepsilon)\equiv f_W(m_0)\pmod{3^M}$.

Proof. (i) $g_\mu$ is affine with unit multiplier, so $1\le N_r\le B_v$. The diagonal entry of a product is at least the product of the diagonal entries, and each of the $N_r$ factors satisfies (7.1). So $(N_{\hat w(\varepsilon)\mu^{N_r-1}})_{\zeta'\zeta'}\ge c_1^{B_v}G_*^{N_r|\mu|}$, and (2.1) with the boundary factor $c_p$ gives the bound. Every inserted copy is heavy along its whole length, so the constant does not depend on $j$. (ii) is the count; $r=O(j^2)$.

(iii) With $\varepsilon=0^r$ the inserted map is $g_\mu^{N_r}$, the identity modulo $3^v$, so $y(0^r)\equiv x\pmod{3^v}$. Only the first block depends on $\varepsilon$, and by (7.2) $$y(\varepsilon)-y(0^r)=2^{d_r}\delta_0\sum_{i=1}^r\varepsilon_ig^{r-i},\qquad d_r=(N_r-1)|\mu|+|p_2|.$$ $2^{d_r}$ is a $3$-adic unit, so the valuation of $2^{d_r}\delta_0$ is exactly $t$ for every $j$. [2, Lemma D.6] at modulus $3^{M-t}$, with $r\ge8g^2(M-t)^2$, shows that the $y(\varepsilon)$ cover the whole class $y(0^r)+3^t\mathbb Z$ modulo $3^M$. By (8.1), $f_W(m_0)=f_U(f_Y^j(m_0))\equiv x\equiv y(0^r)\pmod{3^v}$, and $v\ge t$. $\square$

Proof of Theorem 1. Suppose $G_*>1$. For large $j$ take $\varepsilon$ from Lemma 8.1(iii) and put $y=y(\varepsilon)$. Since $y\equiv f_W(m_0)\in c_W+3^{A_W}\mathbb Z_3$, Lemma 2.1 gives the integer $n=r_W+2^{B_W}\lfloor y/3^{A_W}\rfloor$ with a history of type $W$ ending at $y$. As $\ell'(y)\ge rL_0$ grows quadratically in $j$ and $A_W$ linearly, $\lfloor y/3^{A_W}\rfloor\ge8$ for large $j$, so every input of the history is at least $8$ and the history is legal. Moreover $|n-m_0|_3=3^{A_W}|y-f_W(m_0)|_3\le3^{-K_0}$, so $n\in N_{K_0}$. $f_W$ has slope $\lambda_U\lambda_Y^j$ and nonnegative offset, so $n\le y/(\lambda_U\lambda_Y^j)$ and $$\ell'(n)\le\ell'(y)-j\log_2\tfrac{2187}{2048}+O(1).$$ Weak decrease, Lemma 8.1(i), $V(n)=V(n)[n\in N_{K_0}]$ and [2, Corollary 11.2] for $N^{(K_0)}$, whose diagonal growth is $G_*$, give, with $D_{\rm lift}=|Q^{(K_0)}|$, $$c_pc_1^{B_v}G_*^{\ell'(y)-|p|}\le V(y)\le V(n)\le C(\ell'(n)+1)^{D_{\rm lift}}G_*^{\ell'(n)}.$$ With Lemma 8.1(ii) this gives $G_*^{\,j\log_2(2187/2048)}\le C'(j+1)^{2D_{\rm lift}}$ for all large $j$, which is false since $G_*>1$. Hence $G_*\le1$, and Lemma 4.3 proves Theorem 1. $\square$

$\log_2(2187/2048)\approx0.0947$. The chain $Y=abbb$ is one convenient choice; any $W$ with $3^{A_W}>2^{B_W}$ works.

9. Sharpness

Some restriction is needed. The reader $V(n)=2^{\ell'(n)}[n\equiv1\pmod3]$ is a natural reader of dimension $3$, weakly orients all eleven rules of the rewriting system for $H$, and so is weakly $H$-decreasing [17]. It grows like $n$, and its support $1+3\mathbb Z_3$ misses $N_1$. So Theorem 1 fails without the lift.

No uniform depth works. Fix $v\ge1$, put $M=3^v$ and $Z_v=Z_\infty\bmod M$, and consider the graph on $\mathbb Z/M$ with edges $r\to a(r)=3r\cdot4^{-1}$ and $r\to b(r)=(9r+1)\cdot8^{-1}$. A path of $v$ edges follows some $W$ with $A_W\ge v$, so by Lemma 2.1 it ends in $Z_v$. Also $Z_v$ is forward invariant. Hence every path ending outside $Z_v$ lies outside $Z_v$ and has fewer than $v$ edges. For $r\notin Z_v$ let $\beta(r)$ be the largest number of $b$ edges on a path ending at $r$, the empty path included, let $B=\max\beta$, and put $$c(r)=2^{B-\beta(r)}\ (r\notin Z_v),\qquad c(r)=0\ (r\in Z_v).$$ An edge that ends outside $Z_v$ also starts outside $Z_v$, so $$c(a(r))\le c(r),\qquad 2c(b(r))\le c(r)\qquad(r\in\mathbb Z/M).\tag{9.1}$$

Proposition 9.1. Interpret $\mathtt f,\mathtt t$ by $2P_0,2P_1$, where $P_b$ is the matrix of $r\mapsto2r+b$ on $\mathbb Z/M$, and the ternary digits $d$ by $4Q_d$, where $Q_d$ is the matrix of $r\mapsto3r+d$. Interpret $/$ by the linear map whose first row is $e_1^{\mathsf T}$ and whose other rows are zero, and $.$ by the constant map $c$; here $e_1$ indexes residue $1$. This interpretation of dimension $3^v$ weakly orients all eleven rules of $H$. Its reader $V_v(n)=2^{\ell'(n)}c(n\bmod M)$ is weakly $H$-decreasing. It satisfies $V_v(n)\ge2^{\ell'(n)}>n/2$ on the class $3^{v-1}\bmod3^v$, which lies in $N_{v-1}$, and $V_v=0$ on $N_v$. So in Theorem 1 this reader needs exactly $K\ge v$. In particular, any depth bound valid for all readers of dimension $D$ must be at least $\log_3D$ along this family; Theorem 1 gives the general upper bound $2D-1$.

Proof. Both sides of a carry rule act by $r\mapsto6r+3b+d$ with weight $8$. A left-end rule compares $4e_{3+d}^{\mathsf T}$ with $2^{\ell'(3+d)}e_{3+d}^{\mathsf T}$, and $\ell'(3+d)\le2$. The rule $\mathtt{ff.}\to\mathtt{0.}$ compares $4c(4r)$ with $4c(3r)$, and $\mathtt{ttt.}\to\mathtt{22.}$ compares $8c(8r+7)$ with $16c(9r+8)$. With $s=4r$ and $s=8r+7$ these are the two inequalities (9.1) at $s$, since $a(4r)=3r$ and $b(8r+7)=9r+8$. The class $3^{v-1}\bmod3^v$ lies in $N_{v-1}$ because $0\in Z_\infty$. A point of $Z_\infty$ in that class would have $v-1$ forced inverse $a$ steps, reaching $4^{v-1}\equiv1\pmod3$, and $Z_\infty\bmod3=\{0,2\}$. So the class misses $Z_v$ and has $c\ge1$. $\square$

For $v=1$ this is the reader $2^{\ell'(n)}[n\equiv1\pmod3]$ above. The readers on $161\bmod243$ and $242\bmod729$ in [17] are of the same kind, and the supplement checks their rules directly. Conversely, for interpretations of this shape the dynamic rules force (9.1), and (9.1) forces $c=0$ on $Z_v$: $b(-1)=-1$ gives $c(-1)=0$, and the forward orbit of $-1$ contains $a^v(-1)=0$ and hence all of $Z_v$. So the support of $c$ in the construction is the largest possible. The construction is fourier_wren's [17]; the proof for every $v$ is lyapunov_weaver's [18].

History threshold. lyapunov_weaver's finite-history theorem [11] shows that every nonnegative eventually quasipolynomial $H$-decreasing function of period $2^u3^vq$, with $\gcd(q,6)=1$, is bounded on endpoints of histories with $A_W\ge v$. The readers $V_v$ above are not quasipolynomial, and the threshold $v$ is also sharp within that class. Let $C_j=\{n:n\equiv3^j\pmod{3^{j+1}}\}$ for $0\le j<v$, let $w=3^{v-1-j}4^j$ on $C_j$ and $w=0$ elsewhere, and put $$U_v(n)=1+n\,w(n).\tag{9.2}$$ This is a natural reader of dimension $2M+1$. It is the direct sum of a one-state constant reader with the reader with initial row $(1,1)\otimes e_1^{\mathsf T}$, matrices $\begin{pmatrix}2&0\\b&1\end{pmatrix}\otimes P_b$ and terminal column $(1,0)^{\mathsf T}\otimes w$. Every $b$-image is $2\bmod3$, where $w=0$. An $a$-image never lies in $C_0$. If it lies in $C_j$ with $j\ge1$, then its source lies in $C_{j-1}$, and $3w(C_j)=4w(C_{j-1})$. So $U_v$ is weakly $H$-decreasing. It grows linearly on $C_{v-1}\subset N_{v-1}$, and it equals $1$ on $N_v$ by the forced inverse $a$ steps.

10. Rule removal

This section combines Theorem 1 with Nashida's automaton analysis for collatz-T [2, §12] and with lyapunov_weaver's transfer of that analysis to the block model of $H$ [9, 10]. Nashida's top-window length $K=L_\tau+9$ [2, Definition 12.20(b)] is written $K_\tau$ here, so that $K$ keeps its meaning from Theorem 1.

10.1 Statements

An interpretation assigns each letter $s$ an affine map $[s](y)=M_sy+v_s$ over $\mathbb N^d$ and composes maps in written word order. A rule $\ell\to r$ is weakly oriented when $M_\ell\ge M_r$ and $v_\ell\ge v_r$ entrywise, and strictly oriented when in addition $(v_\ell)_1>(v_r)_1$. The interpretation is monotone in the sense used here when $(M_s)_{11}\ge1$ for every letter $s$ [2, Definition 9.1]. Comparison in the first component means that only the first matrix rows and first vector entries are compared. These definitions also apply to marked letters in §10.6. The homogeneous matrix of $s$ is $\hat M_s=\begin{pmatrix}M_s&v_s\\0&1\end{pmatrix}$.

The canonical system for $H$ has eleven rules: the dynamic rules $\mathtt{ff.}\to\mathtt{0.}$ and $\mathtt{ttt.}\to\mathtt{22.}$, the left-end rules $/d\to/\mathrm{bin}'(3+d)$ for $d\le2$, and six carry rules $bd\to d'b'$ with $3b+d=2d'+b'$ [2, §2]. For $n$ legal, $\mathrm{cd}_H(n)$ is the list of rules of the canonical derivation $\mathrm{can}(n)\to^+\mathrm{can}(H(n))$, $\mathrm{can}(n)=/\mathrm{bin}'(n).$ [2, Lemma 2.3(ii)]. The legal inputs are exactly Nashida's $\mathrm{Dom}(H)$, which excludes $4$ and $7$. Let $\#_\rho(n,p)$ be the number of uses of $\rho$ in $\mathrm{cd}_H(n),\dots,\mathrm{cd}_H(H^{p-1}(n))$. The system $R_H$ has the same dynamic and carry rules and the four left-end rules $L0\to L$, $L1\to L\mathtt t$, $L2\to L\mathtt{tf}$ and $L\mathtt f\to L$; its canonical derivations from $\mathrm{can}_{R_H}(n)=L\,t\,\mathrm{bin}'(n).$ use only ten of them, the set $R_H^u=R_H\setminus\{L0\to L,L\mathtt f\to L\}$ [2, Lemma 2.3(iii)]. Define $\#^{R_H}_{\rho'}(n,p)$ in the same way.

Theorem 2. Let $V$ be a weakly $H$-decreasing natural reader and $K_1\in\mathbb N$. There are $n\ge K_1$ with $3\nmid n$ and $p\ge1$ such that $H^i(n)$ is legal for $i<p$ and $$V(n)<\#_\rho(n,p)\ \text{ for each of the eleven rules }\rho\text{ of }H,\qquad V(n)<\#^{R_H}_{\rho'}(n,p)\ \text{ for each }\rho'\in R_H^u.$$

Corollary 3. Let $J$ be a monotone natural-number matrix interpretation of any dimension [2, Definition 9.1]. (i) If $J$ weakly orients all eleven rules of $H$, it strictly orients none of them. (ii) If $J$ weakly orients $R_H^u$, it strictly orients none of its rules. If it weakly orients some $R'$ with $R_H^u\subseteq R'\subseteq R_H$, every rule of $R'$ that it strictly orients is $L0\to L$ or $Lf\to L$. (iii) Hence rule removal by such interpretations, over any number of stages, removes no rule of $H$, and from $R_H$ at most $L0\to L$ and $Lf\to L$.

Proof of Corollary 3 from Theorem 2. Put $\Phi(w)=([w](0))_1$. In homogeneous coordinates $V(n)=\Phi(\mathrm{can}(n))$ is a natural reader: $N_0,N_1$ are the homogeneous matrices of $\mathtt f,\mathtt t$, $u$ is the first row of that of $/$, and $c=(v_.,1)$. For $R_H$, $u$ is the first row of the homogeneous matrix of $L\,t$ [2, proof of Theorem 3.1(ii)]. All matrices are nonnegative, so weak orientation of a rule survives every context. Thus along canonical derivations $V(H(n))\le V(n)$ for every legal $n$, and $V$ is weakly $H$-decreasing. Now let $J$ be monotone and strictly orient a rule $\rho=(\ell\to r)$ used by the derivations. One application of $\rho$ inside $x\ell y$ lowers $\Phi$ by $$e_1^{\mathsf T}M_x\big([\ell](z)-[r](z)\big)\ge(M_x)_{11}\ge1,\qquad z=[y](0),$$ because the bracket is entrywise nonnegative with first entry at least $1$, and $(M_x)_{11}\ge\prod(M_s)_{11}\ge1$. Summing along a segment gives $V(n)\ge V(H^p(n))+\#_\rho(n,p)\ge\#_\rho(n,p)$, which contradicts Theorem 2. The second claim of (ii) follows, since a weakly oriented $R'$ contains a weakly oriented $R_H^u$. Part (iii) follows by induction on the stages. $\square$

Monotonicity is needed only to turn strict orientation into a decrease in every context. Without it the statement is false for $R_H$: Nashida gives a non-monotone one-dimensional interpretation that weakly orients all rules of $R_H$ and strictly orients $\mathtt{ff.}\to\mathtt{0.}$, while $\Phi\equiv0$ on canonical strings [19, §6]. Theorem 2 holds for it trivially.

Corollary 3 answers the rule-removal question in Nashida's Question 5(a) [2, §15, Q5(a)]. For natural-number interpretations, Corollary 3(ii) also gives the second part of [19, Conjecture 6.2], which [2, Q5(a)] notes is contained in that question for $R_H$. We do not treat the typical-degree part of that conjecture, or interpretations with real entries. Corollary 4 supplies the reversed and top forms, and the dependency-pair forms for $H$ and $H^{\rm rev}$. Dependency pairs of $R_H$ remain outside our result, so we do not claim to settle all of Q5(a).

10.2 The unit lift at a deep level

Fix $V$ and the level $K$ of Theorem 1, and put $M=3^K$ and $S_K=(Z_\infty\bmod M)\cap(\mathbb Z/M)^\times$. $S_K$ contains $-1$, the fixed point of $f_b$. Let $L$ be the lift of $V$ with accepting residues $S_K$, restricted to its relevant indices, so that $L(n)=V(n)[n\bmod M\in S_K]$.

Lemma 10.1. $L$ satisfies (Z01): every product inside every strongly connected component of $L$ is a $0/1$ matrix.

Proof. In the product of $V$ with the residue automaton, accessibility and co-accessibility are constant on strongly connected components. So for every accepting set the relevant indices form a union of whole components. Replacing $Z_\infty\bmod M$ by $S_K$ only shrinks the terminal support. Hence each component of $L$ is a component of $N^{(K)}$ with the same matrices, and Theorem 1 applies. $\square$

Lemma 10.2. Let $B\in\{X,Y\}^*$, with $X=abb$ and $Y=abbb$, have at least $\lceil K/5\rceil$ blocks. Then every endpoint of a legal history following $B$ is in $S_K$ modulo $M$.

Proof. Each block has $A\ge5$, so the history has $A_B\ge K$. By §2.2, $x_1\in f_B(\mathbb Z_3)$ is congruent modulo $M$ to $f_B(z)\in Z_\infty$ for any $z\in Z_\infty$. The last step is $b$, so $x_1\equiv(9y+1)/8\equiv2\pmod3$. $\square$

The length condition is needed. For example $c_X=80$ is in $Z_\infty\bmod3^6$ but not in $Z_\infty\bmod3^7$: a point of $Z_\infty$ congruent to $80\bmod3^7$ would have the backward address $80\leftarrow71\leftarrow63\leftarrow84\leftarrow112$ (steps $b,b,a,a$, which use up $2+2+1+1=6$ of the $7$ ternary digits), and $112\equiv1\pmod3$ is not in $Z_\infty\bmod3=\{0,2\}$ [10]. So no fixed residue class can serve as the source class at every level. This corrects an earlier single-class formulation [12].

10.3 Configurations modulo $3^K$

Nashida's §§12.1–12.8 use only (Z01) of the automaton they are applied to [2, end of §12.9]. We apply them to $L$, by Lemma 10.1. Choose a setting for $L$ as in [2, Definition 12.20], with $R=|Q_L|+1$. The prefix $b_0$ is an $H$ block prefix $B_0$ whose residue word is $[r_{B_0}]_{s'}=y''(u^\sharp)^Rz$. It exists by Win(r) for $H$, as in the proof of [2, Lemma 12.21(i)] [9, §6]. We change parts (e) and (f) as follows.

(e′) The configuration automaton has states $(s,r)\in M_s\times\mathbb Z/M$ and transitions $(s,r)\mapsto(s\bar N^s_b,2r+b)$.

(f′) For $c=(s,r)$, $\mathrm{end}(c)=2^{|z|}r+\mathrm{val}(z)\bmod M$, and the configuration set is $$\mathcal K_S=\{c\text{ reachable from }c_{\rm top}:\ s(c)e=s(c),\ \mathrm{end}(c)\in S_K\}.$$

Lemma 10.3 ([2, Lemma 12.21] modulo $M$). (ii′) $\mathrm{conf}(w_{\rm mid})\in\mathcal K_S$ if and only if $\mathrm{val}(1\omega(w_{\rm mid}))\bmod M\in S_K$, and $\mathcal K_S\neq\emptyset$. (iii) If $s\in K_s$, then $(s,r)$ is recurrent for every $r\in\mathbb Z/M$. (iv) For every $(s,r)$ with $se=s$ and every word $w$, there is a state $d$ reachable from $(s,r)$ from which $w(u^\sharp)^R$ leads back to $(s,r)$. (v) There are $L_c$ and a period $P_{\rm cyc}$ such that, for every $c\in\mathcal K_S$ and every word $w$, there is a state $d$ from which $ww_{\rm top}$ leads to $c$. Moreover, for every $F\ge L_c$ such that $d$ lies in the cyclic class $F$ of $c_{\rm free}$, at least $2^{F-L_c}$ values $\upsilon<2^F$ have $[\upsilon]_F$ leading from $c_{\rm free}$ to $d$.

Proof. This is Nashida's proof with the exponent $6=|\mathrm{Aff}(\mathbb Z/3)|$ replaced by $N=|\mathrm{Aff}(\mathbb Z/M)|=2\cdot3^{2K-1}$. A word $v$ acts on residues by $r\mapsto2^{|v|}r+\mathrm{val}(v)$, an element of $\mathrm{Aff}(\mathbb Z/M)$, and its $N$-th power is the identity by Lagrange's theorem.

• (ii′) The first component of a configuration ends with $\bar N^s_{u^\sharp}=e$, so $se=s$. Reachability from $c_{\rm top}$ holds by construction. The register starts at $1$ and $z$ is the remaining suffix, so $\mathrm{end}(\mathrm{conf}(w_{\rm mid}))=\mathrm{val}(1\omega)\bmod M$. For nonemptiness, middle words of length $F\ge K\log_23$ enter $\mathrm{val}(1\omega)$ through $\mathrm{val}(w_{\rm mid})$ with multiplier $2^{|y''(u^\sharp)^Rz|}$, a unit, so every residue modulo $M$ occurs.
• (iii) If $v$ leads from $(s,r)$ to $(s',r')$, [2, Lemma 12.1(iii)] gives $v'$ with $s\bar N^s_{vv'}=s$. Then $v'(vv')^{N-1}$ leads back to $(s,r)$.
• (iv) $e\bar N^s_we$ lies in the group $G_e$, so $(e\bar N^s_we)^j=e$ for some $j\ge1$. Then $(w(u^\sharp)^R)^{jN}$ returns both coordinates, and $d$ is the state after $(w(u^\sharp)^R)^{jN-1}$.
• (v) By (iii), $c_{\rm top}$ is recurrent. $c_{\rm free}$ lies in its class, and that class is closed. The counting fact (A3) of [2, proof of Theorem 6.8] holds for every recurrent state of a finite automaton. $\square$

$L_c$ and $P_{\rm cyc}$ now depend on $K$ as well. $K$ is fixed by $V$ before the setting is chosen, so no later choice depends on it.

10.4 The block family of $H$

The following is lyapunov_weaver's transfer [9], which fourier_wren checked against the written proofs of [2] [12]. Use the blocks $X=abb$ and $Y=abbb$ of [2, Proposition 7.2(ii)]. They have $(m,r,c,A)=(8,84,80,5)$ and $(11,1364,1457,7)$ and take three and four $H$ steps. Draw blocks independently with probabilities $3/10$ and $7/10$ after the fixed prefix $B_0$. For $B=B_0B'$ with $m=m_B$, $A=A_B$, $p$ steps and $r_B=r_{B_0}+2^{s'}r'$, and for $\tau=\mathrm{val}(1w_{\rm top}0^8)$, $n'\ge K_\tau+s'$, $F=n'-K_\tau-s'$ and $0\le\upsilon<2^F$, put $$t=\tau2^{n'-K_\tau}+2^{s'}\upsilon+\varrho,\qquad x_0=r_B+2^mt,\qquad x_1=c_B+3^At=H^p(x_0),$$ where $0\le\varrho<2^{s'}$ solves $c_B+3^A\varrho\equiv r_{B_0}\pmod{2^{s'}}$. Then $$\mathrm{bin}'(x_0)=\mathrm{bin}'(\tau)[\upsilon]_F[\varrho]_{s'}[r']_{m-s'}[r_{B_0}]_{s'}$$ and $x_0\equiv x_1\equiv r_{B_0}\pmod{2^{s'}}$. Every point of the history is legal and at least $8$ [2, Proposition 7.2(i),(ii)]. This is [2, Lemma 6.1(iv) and (12.3)], with Proposition 7.2(i) in place of the $T$ iterate. The inputs of [2, Proposition 12.31] are replaced as follows.

• Key(r) and Win(r) for $H$ follow from the swap [2, Proposition 7.2(iv)] by (G1), (G2) and [2, Theorems B.7, B.8], uniformly over the earlier block prefix.
• (S1), the top window, is [2, Proposition 12.19] with the same $(m,A)$ counts. It has probability at least some $c_\tau>0$.
• (S2), window sums on the residue part, is [2, Lemma 12.16(i)] via Win(r) and [2, Theorem 12.13].
• (S3), expansion: the mean excess per block is $6.4\log_23-10.1>0$, so [2, Lemma 12.29(ii)] holds with $c_e=1/256$.
• (S4), the long-run tail [2, Lemma 12.23(i), Propositions 12.24, 12.26], uses only Key(r), uniformly over the prefix, and the consistency of residues under extension. Groups of $n_1+n_2$ blocks have between $8(n_1+n_2)$ and $11(n_1+n_2)$ digits, so the charging of [2, Lemma 12.25(ii)] is unchanged.
• Free-bit and end-window estimates (V1)–(V4) [2, Lemma 12.16(ii), Proposition 12.28] use uniform free bits and the analogue of [2, Lemma 6.4]. That analogue says the low $b$ bits of $h=\lfloor x_1/2^{n'}\rfloor$ are within $\ell^1$ distance $3\cdot2^{-\Delta}+2^{s'+2}3^A2^{-n'}$ of uniform when $2^{b+\Delta+K_\tau}\le3^A$. Its proof uses only the definition of $t$, $c_B<3^A$ and oddness of $3^A$.
• Rule uses. Each dynamic rule is used at least $k$ times along $k$ appended blocks [2, §7, after Proposition 7.2]. With probability tending to one, each left-end rule is used at least $c_Lk$ times [2, Proposition B.9 via §7]. For the carry rules, the first step of $x_0$ is an $a$ step. It sweeps a ternary $0$ across $\mathrm{bin}'(x_0/4)$, which contains $[\upsilon]_F$. From any carry, two input zeros reset the carry to $0$, and then $00,01,11$ produce carry $0,1,2$. So each occurrence of $00\,w_5(d)\,b$ in the reversed free bits uses the carry rule $bd\to d'b'$. For uniform free bits, each of these six words occurs at least $F/128$ times outside an event of probability tending to $0$.
• The analogue of [2, Corollary 12.18]. For $F$ large, every set of at least $2^{F-L_c}$ free words contains one with at least $c_Uk$ uses of all eleven rules, where $c_U=\min(1,c_L,1/128)$. This needs $F\ge k$, $n'>m$, and a block word with the left-end event.

10.5 Proof of Theorem 2

Use the notation of [2, §12.9], with $L$, $\mathcal K_S$ and the inputs of §10.4.

Case A: some $c\in\mathcal K_S$ has $C_d(c)=0$ for all $d\ge1$. Follow [2, Lemma 12.30]. Choose $k$ with $V_0(s(c))<c_Uk$ and fix a block word with the left-end event. Choose $F\ge L_c+m+k+K_1+1$ in the cyclic class of the state $d$ given by Lemma 10.3(v), and large enough for the analogue of [2, Corollary 12.18] in §10.4. Among the at least $2^{F-L_c}$ free words that realize $c$, choose one with at least $c_Uk$ uses of every rule. The configuration of $x_0$ is $c$, so $x_0\bmod M\in S_K$ by Lemma 10.3(ii′). Hence $3\nmid x_0$ and $L(x_0)=V(x_0)$, and [2, Lemma 12.22(iii)] for $L$ gives $L(x_0)=V_0(s(c))$. Therefore $V(x_0)<c_Uk\le\#_\rho(x_0,p)$ for all eleven $\rho$, with $x_0\ge K_1$.

Case B: otherwise. We show that Case B cannot occur. Run the proof of [2, Proposition 12.31] with these changes; everything not listed is unchanged.

• Step 0 takes the lexicographic minimum $(d_{\rm lo},C_{\rm lo})$ over $\mathcal K_S$.
• Step 1 fixes its constants in the same order, with $L_c$ and $P_{\rm cyc}$ from Lemma 10.3(v). In particular $\eta_u=2^{-(L_c+4)}$ and $\Delta=L_c+6$ are chosen after $L_c$.
• Step 2 draws the block word from the intersection of (S1)–(S4), which has probability at least $c_\tau/4$.
• Step 3 gives a good set of $2^{F-L_c}=16\eta_u2^F$ free words realizing $c_{\rm lo}$. The bad sets (V1)–(V4) have at most $5.75\,\eta_u2^F$ elements together.
• Step 4 additionally uses that $k$ is chosen after $K$, so the block word has at least $\lceil K/5\rceil$ blocks.
• Step 5: [2, Proposition 12.27(iii)] gives $L(x_0)\le(C_{\rm lo}+\varepsilon)\ell'(x_0)^{d_{\rm lo}}$.
• Step 6: $\mathrm{bin}'(x_1)$ begins with $w_{\rm top}$ by (S1), and $x_1\bmod M\in S_K$ by Lemma 10.2. So the configuration $c_1$ of $x_1$ lies in $\mathcal K_S$ by Lemma 10.3(ii′), and [2, Proposition 12.28(iii)] gives $(C_{d_1}(c_1)-\varepsilon)\ell'(x_1)^{d_1}\le L(x_1)$.
• Step 7: by Step 0 and $\ell'(x_1)\ge(1+c_e/34)\ell'(x_0)$, [2, Lemma 12.29(i)] gives $L(x_0)<L(x_1)$.
• Step 8 replaces (M3). Both endpoints are accepted, so $L(x_0)=V(x_0)$ and $L(x_1)=V(x_1)$. Weak $H$-decrease along the legal history gives $V(x_1)\le V(x_0)$, a contradiction.

The restriction to $S_K$ is used in both cases. With Nashida's set $\{\mathrm{end}\neq0\}$, Case A could select an unaccepted configuration, where $L=0$ says nothing about $V$. Intermediate points of the history need not be accepted; only $V$ is used along it.

$R_H$. By [2, Lemma 7.4], each rule of $R_H^u$ occurs in $\mathrm{cd}_{R_H}(n)$ at least as often as a corresponding rule of $H$ occurs in $\mathrm{cd}_H(n)$. Summing over the segment gives $\#^{R_H}_{\rho'}(x_0,p)\ge\#_\rho(x_0,p)>V(x_0)$. $\square$

10.6 Reversal, top forms and dependency pairs

Nashida derives parts (ii) and (iii) of [2, Theorem 9.2], and part (i) for $T^{\rm rev}$, from a value-level statement by finite arguments about strings and matrices [2, §§13.1–13.2]. Theorem 2 holds for every natural reader, with no assumption on the shape of the matrices, so the same arguments give the analogues for $H$. Write $D_H$, $A$ and $B$ for the dynamic, carry and left-end rules of $H$, and $H^{\rm rev}$ for the reversed system. By [2, §2.3], the essential dependency-pair SCCs of $H$ and $H^{\rm rev}$ are $P_B=\{/^\#d\to/^\#\mathrm{bin}'(3+d)\}$ with usable rules $D_H\cup A$, and $P^{\rm rev}_H=\{.^\#\mathtt{ff}\to.^\#\mathtt0,\ .^\#\mathtt{ttt}\to.^\#\mathtt{22}\}$ with usable rules $A^{\rm rev}\cup B^{\rm rev}$.

Corollary 4. Let $J$ be a natural-number matrix interpretation of any dimension. (i) Reversal. If $J$ is monotone and weakly orients the eleven rules of $H^{\rm rev}$, it strictly orients none of them. If $J$ is monotone and weakly orients the reversals of the rules of some $R'$ with $R^u_H\subseteq R'\subseteq R_H$, every reversed rule that it strictly orients is $\mathtt0L\to L$ or $\mathtt fL\to L$. (ii) Top forms. If $J$ weakly orients all rules of $H$, it strictly orients none of the three left-end rules, whether or not it is monotone. If $J$ weakly orients all rules of $H^{\rm rev}$, it strictly orients neither $.\mathtt{ff}\to.\mathtt0$ nor $.\mathtt{ttt}\to.\mathtt{22}$, whether or not it is monotone. For $R'$ as in (i), the same holds for $R'$ and the rules $L\mathtt1\to L\mathtt t$, $L\mathtt2\to L\mathtt{tf}$, and for the reversal of $R'$ and the rules $.\mathtt{ff}\to.\mathtt0$, $.\mathtt{ttt}\to.\mathtt{22}$. (iii) Dependency pairs. Let $J$ also interpret the marked letters, monotone or not. If $J$ weakly orients every rule of $D_H\cup A$ and weakly orients every pair of $P_B$ in the first component, it strictly orients no pair of $P_B$ in the first component. The same holds for $P^{\rm rev}_H$ with the usable rules $A^{\rm rev}\cup B^{\rm rev}$.

Proof. (i) Reversing every string of the canonical derivation of $m$ gives a derivation $\mathrm{rev}(\mathrm{can}(m))\to^+\mathrm{rev}(\mathrm{can}(H(m)))$ of $H^{\rm rev}$ that uses $\rho^{\rm rev}$ as often as $\mathrm{cd}_H(m)$ uses $\rho$ [2, Lemma 13.1(i)]. Writing $d$ for the dimension of $J$, put $\check M_s=\hat M_s^{\mathsf T}$. Then $$\Phi(\mathrm{rev}(\mathrm{can}(n)))=(\hat M_{\mathrm{rev}(\mathrm{can}(n))})_{1,d+1}=e_{d+1}^{\mathsf T}\check M_/\,\check M_{\mathrm{bin}'(n)}\,\check M_.\,e_1,$$ which is a natural reader with $N_0=\check M_{\mathtt f}$ and $N_1=\check M_{\mathtt t}$. Weak orientation of the rules of $H^{\rm rev}$ makes it weakly $H$-decreasing. If $J$ is monotone, each use of a strictly oriented $\rho^{\rm rev}$ lowers $\Phi$ by at least $1$ [2, Lemma 13.1(ii)], so $V(n)\ge\#_\rho(n,p)$ on every segment, against Theorem 2. For $R_H$ use $\mathrm{rev}(\mathrm{can}_{R_H}(n))$, with initial row $e_{d+1}^{\mathsf T}\check M_L\check M_{\mathtt t}$, and the $R_H$ counts of Theorem 2. (ii) In $H$ the letter $/$ occurs only in the left-end rules, as the leftmost letter of both sides. So in every string of a canonical derivation $/$ is the leftmost letter, and left-end rules apply only with empty left context. There a strictly oriented rule lowers $\Phi$ by $([\ell](z)-[r](z))_1\ge(v_\ell)_1-(v_r)_1\ge1$ without monotonicity, and weakly oriented rules do not raise it [2, Lemma 13.1(iii)]. Theorem 2 gives the contradiction as in (i). The letter $.$ plays this role in $H^{\rm rev}$ and in the reversal of $R_H$, and $L$ in $R_H$. The two rules $L\mathtt1\to L\mathtt t$ and $L\mathtt2\to L\mathtt{tf}$ are used [2, Lemma 2.3(iii)], so Theorem 2 counts them. (iii) The quadruples $(h,R_{\rm top},P,U)=(/,B,P_B,D_H\cup A)$ and $(.,\{.\mathtt{ff}\to.\mathtt0,.\mathtt{ttt}\to.\mathtt{22}\},P^{\rm rev}_H,A^{\rm rev}\cup B^{\rm rev})$ have the properties assumed in [2, Lemma 13.2]: $R_{\rm top}\cup U$ is the system, and $h$ occurs in each rule of $R_{\rm top}$ exactly once on each side, as the leftmost letter, and not in $U$. The proof of [2, Lemma 13.2] uses only these properties and the identity $\alpha^{[1]}\circ\beta=(\alpha\circ\beta)^{[1]}$. So (iii) follows from (ii), as [2, Theorem 9.2(iii)] follows from [2, Theorem 9.2(ii)]. $\square$

The comparisons of pairs in [2, Proposition 13.3] also follow: entrywise pair orientation implies orientation in the first component; a scalar marked letter $y\mapsto u^{\mathsf T}y+c$ is embedded as $y\mapsto e_1(u^{\mathsf T}y+c)$; and comparison in any fixed component $k$ is moved to the first by replacing only the marked map $\alpha$ with $\alpha^{[k]}(y)=(\alpha(y))_ke_1$. In each case the usable rules, which contain no marked letter, are unchanged. Thus the same statements hold for $P_B$ and $P^{\rm rev}_H$. We do not claim a result about argument filters [2, Proposition 13.4]: its usable rules and its two shrinking cases are computed for $T^{\rm rev}$, and they would have to be recomputed for $H^{\rm rev}$. Nashida develops only rule removal for $R_H$ in [2, §2.3]; its dependency-pair problem remains untreated here as well.

11. Scope

Theorem 1 is a value-level statement. Theorem 2 adds the counts of rule uses, and Corollaries 3 and 4 are its consequences for natural-number matrix interpretations. None of them proves or refutes termination of $H$, of $R_H$ or of the Collatz map. Corollary 3 says that one technique, rule removal with monotone natural-number matrix interpretations used alone, cannot prove termination of $H$, and on $R_H$ can remove only the two unused rules. Corollary 4 gives the same for $H^{\rm rev}$, for relative top termination, and for reduction pairs on the essential dependency-pair problems of $H$ and $H^{\rm rev}$. We do not treat:

• argument filters for the dependency pairs of $H$ and $H^{\rm rev}$ [2, Proposition 13.4], and dependency pairs of $R_H$;
• the typical-degree part of [19, Conjecture 6.2];
• other forms of interpretations [2, Q5(b), (c)];
• proofs in which other techniques act first [2, Q5(d)].

The proofs import statements of [2] by number, read in revision r5. These include Lemmas 11.3, 11.5–11.7, 11.10, Corollary 11.2, Proposition D.5 and Lemma D.6 for Theorem 1, Proposition 7.2, Lemma 7.4, Theorems B.7–B.8 and §§12.1–12.9 for Theorem 2, and Lemmas 13.1–13.2 and Proposition 13.3 for Corollary 4. Many are kernel-checked in [2]. Our adaptations are written proofs only: they are not formalized, and as a whole they have been read only by the two authors. Every step is a finite or asymptotic argument. The computations in the supplement check examples and interface identities and are not used in any proof. Reproduction instructions, expected outputs and licenses are in its README. The five-process exact example audit checks 7,809 full-rule inequalities or equalities, 2,184 inequalities for the history-threshold examples and 18 legal witness histories. A separate exact arithmetic audit checks 2,480 block/source pairs, 40,832 ordinary $H$ steps, 14,880 carry-pattern bounds, 752 top-window checks and 327,680 free-cube points. These finite checks do not certify the asymptotic imports or prove termination.

References

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