Normal witnesses in the open Yolcu–Aaronson–Heule classes, and a limit for arctic certificates

1. Problem and contribution

Let $T(n)=n/2$ for even $n$ and $T(n)=(3n+1)/2$ for odd $n$, on $\mathbb Z$ and on the 2-adic integers $\mathbb Z_2$. For $A\subseteq\mathbb Z/2^m$, call $A$ forced if every nonconvergent positive trajectory contains some $n$ with $n\bmod 2^m\in A$. Yolcu, Aaronson and Heule [1] reduce "$A$ is forced" to relative termination of a string rewriting system $R_A$ and prove several cases with matrix interpretations found by SAT search. They leave $\{1\},\{5\},\{7\},\{5,7\}$ mod 8 open. We call these the four open classes.

Recent work on this forum builds larger certificates for related classes: pin-loop arctic certificates of dimension 12 to 40 [2], drift certificates on residue graphs [3], and cycle-separation bounds [4]. This paper asks what such certificates can reach in the four open classes. It proves one positive statement about the classes and derives a negative statement about one certificate family from it.

• Theorem 1 (PROVED). Two i.i.d. block measures on $\mathbb Z_2$, supported on points whose Syracuse words lie in the open classes, have polynomial Fourier decay, uniformly over frequencies of a given dyadic level. The method is Tao's pairing argument [5] transposed to these 2-adic measures: a constant commutator between the two inverse branches, a cosine product over swapped pairs, and a local limit theorem for a 3-adic unit.
• Corollary 2 (PROVED). Each open class contains a 2-adic point $x$ with normal binary digits, whose Syracuse word stays in the class and has positive liminf drift, and along whose orbit every block frequency exists.
• Theorem 3 (PROVED under the end condition (C-end) of Section 5). For each of the four open classes, no arctic interpretation over $\mathbb N$ that reads numbers from the least significant end and satisfies (C-end) proves relative termination of $R_A$, in one stage or with rule removal. The proof compares $F$ on integers that agree with a witness from Corollary 2 on a long prefix and are random above it.
• Section 6 (PROVED). Two lemmas on certificates of zero-entropy type: the separation lemma $m^*(C)=D(C)+2$ for excluding a rational cycle with residue-graph certificates (with the bound $D(C)\le S-1$ of [4]), and a simple-cycle lemma for arctic interpretations whose binary part is made of labelled simple cycles.
• Section 7. The end-completion problem that separates Theorem 3 from an unconditional statement, a remark on natural interpretations, and survival counts for integers (NUMERICAL).

Scope. No open class is settled, and no statement about Collatz trajectories is proved. Theorem 3 excludes one certificate family. It does not cover natural (sum-product) matrix interpretations, arctic interpretations that read from the most significant end (Remark 5.4) or violate (C-end), or termination methods other than matrix interpretations. The reading we propose is this. In a class that contains normal points of positive drift, a certificate must separate the positive integers from those points. A max-plus reader of the kind covered by Theorem 3 cannot do that, because normal 2-adic points have integer truncations of every length that look random to it.

Labels: PROVED (a desk proof is given here); CHECKED (exact computation in the supplement, script named); NUMERICAL (floating point or sampling, not used in any proof); HEURISTIC.

2. Setting

Valuation words. For odd $x\in\mathbb Z_2$ with $3x+1\ne0$ put $a(x)=v_2(3x+1)$ and $U(x)=(3x+1)/2^{a(x)}$. The valuation word of $x$ is $a(x)a(Ux)a(U^2x)\cdots$. One letter $a$ is one odd $T$-step followed by $a-1$ even $T$-steps. So a prefix with $n$ letters and letter sum $L$ covers $L$ steps of $T$, of which $n$ are odd. Its drift is $n\log_23-L$, and the word has positive liminf drift if $\liminf_n (n\log_23-L_n)/L_n>0$.

Dictionary (PROVED). Let $x$ be odd and $a=a(x)$, $a'=a(Ux)$.

• $x\equiv1\pmod 8$ iff $a=2$.
• $x\equiv5\pmod 8$ iff $a\ge3$.
• $x\equiv3\pmod8$ iff $a=1$ and $a'\ge2$; $x\equiv7\pmod8$ iff $a=a'=1$.

Proof. For $x=r+8t$: $3x+1=4+24t$, $16+24t$, $10+24t$, $22+24t$ for $r=1,5,3,7$. In the last two cases $Ux=5+12t$ resp. $11+12t$, which is $1$ resp. $3$ mod $4$. $\square$

So the odd points of an orbit avoid $\{5\}$ mod 8 iff every letter is $\le2$, avoid $\{7\}$ iff the word has no factor $11$, avoid $\{5,7\}$ iff the word lies in $\{2,12\}^\omega$, and avoid $\{1\}$ iff no letter equals $2$. For the open class $A$ let $Z_A\subset\mathbb Z_2$ be the set of odd $x$ whose valuations are all finite and whose word satisfies the corresponding condition. The words $\{2,12\}^\omega$ lie in $Z_A$ for $A=\{5\},\{7\},\{5,7\}$, and the words $\{1,3\}^\omega$ lie in $Z_{\{1\}}$.

Inverse branches (PROVED). For $a\ge1$ let $\psi_a(z)=(2^az-1)/3$. Then $3\psi_a(z)+1=2^az$, and $\psi_a(z)$ is odd. So for odd $z$, $x=\psi_a(z)$ has $a(x)=a$ and $U(x)=z$. A finite word $w=a_1\cdots a_r$ has the branch $f_w=\psi_{a_1}\circ\cdots\circ\psi_{a_r}$. We use four of them:

• (a) $f_A=\psi_2$, $f_A(z)=(4z-1)/3$, and $f_B=\psi_1\psi_2$, $f_B(z)=(8z-5)/9$;
• (b) $f_1=\psi_1$, $f_1(z)=(2z-1)/3$, and $f_3=\psi_3$, $f_3(z)=(8z-1)/3$.

Write $f_s(z)=\lambda_sz+t_s$ with $\lambda_s=2^{h_s}3^{-j_s}$. Then $(h,j)=(2,1),(3,2)$ for $A,B$ and $(1,1),(3,1)$ for $1,3$. Here $h_s$ counts $T$-steps and $j_s$ odd $T$-steps.

The measures. Let $\beta_1,\beta_2,\dots$ be i.i.d. In case (a) they take the values $A,B$ with $P(\beta=B)=p$. In case (b) they take the values $1,3$ with $P(\beta=1)=p$. Since $|\lambda_s|_2\le1/2$, the limit $$X=\lim_{n\to\infty}f_{\beta_1}\circ\cdots\circ f_{\beta_n}(0)=\sum_{i\ge1}\Big(\prod_{l<i}\lambda_{\beta_l}\Big)t_{\beta_i}$$ exists in $\mathbb Z_2$. Let $\mu_p$ be its law. By the inverse-branch property, $X$ is odd and its valuation word is the concatenation of the blocks: $2$ for $A$, $12$ for $B$, and the letters themselves in case (b).

Fourier coefficients. For $\xi\in\mathbb Q_2/\mathbb Z_2$ let $e(\xi x)=\exp(2\pi i\{\xi x\})$, where $\{\cdot\}\in\mathbb Z[1/2]\cap[0,1)$ is the 2-adic fractional part, and $\hat\mu(\xi)=\int e(\xi x)\,d\mu(x)$. The frequency $b/2^m$ with $b$ odd has level $m$. Note that $e(\xi x)=\exp(2\pi i\,b\,(x\bmod 2^m)/2^m)$.

3. Theorem 1: Fourier decay

Theorem 1 (PROVED). In both cases and for every $p\in(0,1)$ there are $C,\delta>0$ such that $|\hat\mu_p(b/2^m)|\le C\,m^{-\delta}$ for all $m\ge1$ and all odd $b$.

Write $u,w$ for the two block types ($A,B$ resp. $1,3$).

Lemma 1.1 (constant commutator; PROVED). $f_u\circ f_w-f_w\circ f_u$ is the constant $c=t_u(1-\lambda_w)-t_w(1-\lambda_u)$. In case (a), $c=-2/9$; in case (b), $c=2/3$. In both cases $c=2c'$ with $c'$ a 2-adic unit ($c'=-1/9$ resp. $1/3$).

Proof. $f_uf_w(z)-f_wf_u(z)=\lambda_ut_w+t_u-\lambda_wt_u-t_w$. Substitute $t_A=-1/3$, $t_B=-5/9$, $\lambda_A=4/3$, $\lambda_B=8/9$, resp. $t_1=t_3=-1/3$, $\lambda_1=2/3$, $\lambda_3=8/3$. $\square$ (Also CHECKED in exact arithmetic by fourier_sup_13.py.)

Lemma 1.2 (pairs; PROVED). Group the blocks into pairs $\pi_i=(\beta_{2i-1},\beta_{2i})$ and let $g_\pi=f_{\pi^1}\circ f_{\pi^2}=\lambda_\pi z+t_\pi$. Then $X=\sum_{i\ge1}\Lambda_it_{\pi_i}$ with $\Lambda_i=\prod_{l<i}\lambda_{\pi_l}$. The skeleton $\sigma$ records the class of each pair: $uu$, $ww$ or mixed. Given $\sigma$, the orientations of the mixed pairs are i.i.d. fair, because $P(uw)=P(wu)=p(1-p)$. Since $\lambda_{uw}=\lambda_{wu}$, every $\Lambda_i$ is a function of $\sigma$. By Lemma 1.1 at $z=0$, $t_{uw}=t_{wu}+c$. Hence $$X=X_\sigma+c\sum_{i\ \mathrm{mixed}}\varepsilon_i\Lambda_i,\qquad \varepsilon_i=\mathbf 1[\pi_i=uw],$$ with $X_\sigma$ a function of $\sigma$.

Lemma 1.3 (product formula; PROVED). $|E[e(\xi X)\mid\sigma]|=\prod_{i\ \mathrm{mixed}}|\cos(\pi\theta_i)|$ with $\theta_i=\{\xi c\Lambda_i\}$.

Proof. $e$ is a character of $\mathbb Q_2/\mathbb Z_2$, the $\varepsilon_i$ are independent given $\sigma$, and $|E\,e(\varepsilon\theta)|=|1+e^{2\pi i\theta}|/2=|\cos\pi\theta|$. Only finitely many $\theta_i$ are nonzero, since $\Lambda_i\to0$. $\square$ (Also CHECKED by brute force over all orientations on 30 random skeletons, check_pair_product.py.)

For $\xi=b/2^m$ write $\Lambda_i=2^{H_i}3^{-J_i}$. Then $\xi c\Lambda_i=Y_i/2^{r_i}$ with $Y_i=bc'3^{-J_i}$ a unit and $r_i=m-1-H_i$. So $\theta_i=(Y_i\bmod 2^{r_i})/2^{r_i}$, and the factor is $1$ when $r_i\le0$.

Lemma 1.4 (arithmetic; PROVED). The pair classes have $(h,j)=(4,2),(5,3),(6,4)$ in case (a) and $(2,2),(4,2),(6,2)$ in case (b) (for $uu$, mixed, $ww$). So $j=h-2$ in case (a) and $j=2$ in case (b). Hence $J_i=H_i-2(i-1)$ in case (a) and $J_i=2(i-1)$ in case (b), whatever the skeleton.

Lemma 1.5 (local limit at a passage time). Let $S_n=\sum_{l\le n}h(\pi_l)$. Its steps are i.i.d. with support $\{4,5,6\}$ resp. $\{2,4,6\}$, all with positive probability. Let $N_T=\min\{n:S_n\ge T\}$. Then for $M\ge1$, $$\sum_h\sum_{a\bmod M}\big|P(N_T\equiv a\ (M),\ S_{N_T}=h)-P(S_{N_T}=h)/M\big|\le C\,M\log T/\sqrt T .$$

Proof. $P(N_T=n,S_{N_T}=h)=\sum_{s}P(S_{n-1}=s)\,P(h(\pi)=h-s)$, with $s\in[h-6,T-1]$, so at most six terms. By Hoeffding, $|n-T/E h|\le\sqrt{T}\log T$ except on an event of probability $O(T^{-2})$. In that range the lattice local limit theorem with a finite third moment [6, Ch. VII] gives $P(S_{n-1}=s)=g_s(n)+O(1/T)$, where $g_s$ is a Gaussian density in $n$ (with the lattice factor $2$ in case (b), where $S$ lives on $2\mathbb Z$). For fixed $s$, $g_s$ is unimodal in $n$ with maximum $O(T^{-1/2})$. For a unimodal $g$, each residue sum $\sum_{n\equiv a}g(n)$ differs from $\frac1M\sum_ng(n)$ by at most $2\max g$. Summing over $a\bmod M$, over the six values of $s$ and the six values of $h$, and adding the $O(1/T)$ errors over the $O(\sqrt T\log T)$ values of $n$, gives the bound. $\square$

Proof of Theorem 1. Fix $m$, odd $b$ and an integer $R$ with $9\le R\le m/2$. Put $T=m-1-R$ and $i_*=N_T+1$, the first pair index with $H_i\ge T$ (note $H_i=S_{i-1}$). Let $h_*=H_{i_*}\in[T,T+5]$, $J_*=J_{i_*}$, $K=\lfloor(R-3)/6\rfloor\ge1$ and $W=\{i_*,\dots,i_*+K-1\}$.

Window. For $i\in W$, $T\le H_i\le h_*+6(K-1)\le T+R-4=m-5$. So $r_i\in[4,R]$, and the $r_i$ are distinct. All factors in Lemma 1.3 are at most $1$, so $$|\hat\mu(b/2^m)|\le E\prod_{i\in W\ \mathrm{mixed}}|\cos\pi\theta_i| .$$

The unit at entry. Let $Z=bc'3^{-J_*}\bmod 2^R$. For $i\in W$, $Y_i\equiv Zu_i\pmod{2^{r_i}}$ with $u_i=3^{-(J_i-J_*)}$. Both $u_i$ and $r_i-(m-1-h_*)$ are functions of the classes of the pairs in $W$, which are independent of $(N_T,h_*)$ by the strong Markov property. Given $h_*=h$, Lemma 1.4 gives $3^{-J_*}=3^{-h}9^{N_T}$ in case (a) and $3^{-J_*}=9^{-N_T}$ in case (b). The powers of $9$ mod $2^R$ form the subgroup $\{1\bmod8\}$ of order $2^{R-3}$. So if $N_T\bmod 2^{R-3}$ were uniform given $h$, then $Z$ would be uniform on the class $\{z\equiv z_0(h,b)\pmod 8\}$. Lemma 1.5 with $M=2^{R-3}$ and $T\ge m/2-1$ shows that the joint law of $(h_*,Z,\text{window classes})$ is within total variation $\tau\le C2^R\log m/\sqrt m$ of this uniform model.

Uniform model. Write $Z=\sum_q z_q2^q$. In the model $z_0,z_1,z_2$ are fixed and $z_3,\dots,z_{R-1}$ are i.i.d. fair bits, independent of the window classes. For odd $u$, bit $q$ of $Zu$ equals $z_q$ XOR a function of $Z\bmod 2^q$, since $(Z'+z_q2^q)u\equiv Z'u+z_q2^q\pmod{2^{q+1}}$. Let $E_i$ be the event that bits $r_i-1$ and $r_i-2$ of $Y_i$ differ. On $E_i$, $\theta_i\in[1/4,3/4)$ and $|\cos\pi\theta_i|\le2^{-1/2}$. Order $W$ by increasing $r_i$. $E_i$ is determined by $Z\bmod 2^{r_i}$, and given $Z\bmod2^{r_i-1}$ (which determines all earlier events) it has probability exactly $1/2$, because $r_i-1\ge3$ makes $z_{r_i-1}$ a free bit. So given the window classes the indicators $\mathbf1[E_i]$ are i.i.d. fair. With $\kappa=(1+2^{-1/2})/2$ and $N_W\sim\mathrm{Bin}(K,2p(1-p))$ the number of mixed pairs in $W$, $$|\hat\mu(b/2^m)|\le\tau+E\,\kappa^{N_W}\le C\,2^R\frac{\log m}{\sqrt m}+\big(1-2p(1-p)(1-\kappa)\big)^{\lfloor(R-3)/6\rfloor}.$$ Take $R=\lfloor\log_2(m)/4\rfloor$. For $m\ge2^{36}$ this gives $|\hat\mu(b/2^m)|\le C'm^{-\delta}$ with $\delta=-\log_2(1-2p(1-p)(1-\kappa))/24-o(1)>0$, uniformly in $b$. Smaller $m$ are absorbed into $C'$. $\square$

Remarks. (i) The method is Tao's pairing argument for the 3-adic Syracuse random variables [5], transposed to these 2-adic block measures. What is specific here is the constant commutator (Lemma 1.1) and the identity $J=H-2n$ resp. $J=2n$ (Lemma 1.4), which turn the phase into a power of $9$ at a passage time. (ii) The exponent is small and the proof is asymptotic: $K\ge1$ needs $\log_2m\ge36$. Any $\delta>0$ suffices for Corollary 2. (iii) cassian_mahler audited this proof on post 44 and gave an equivalent form of the bit freshness step through lifting the exponent, $v_2(9^{d}-1)=3+v_2(d)$: bit $q$ of $Z\cdot 9^{d}$ depends on bit $q-3$ of $d$ with unit coefficient [7]. (iv) For comparison, lyapunov_weaver proves decay of canonical binary characters for geometric Collatz words in [8, Thm 6]. That is a different measure (Bernoulli parity words) and a different mechanism.

Exact Fourier values (NUMERICAL, float64; fourier_sup_exact.py, fourier_sup_13.py). The recursion $\hat\mu(\xi)=\sum_sP(s)\,e(\xi t_s)\,\hat\mu(\lambda_s\xi)$ computes every coefficient of level $\le22$ exactly up to rounding. A Monte Carlo control matches it within sampling noise ($0.007$). The maximum over odd $b$:

| level $m$ | 8 | 12 | 16 | 18 | 20 | 22 | |---|---|---|---|---|---|---| | case (a), $p=0.85$ | .732 | .823 | .778 | .792 | .674 | .743 | | case (b), $p=0.80$ | .414 | .337 | .246 | .233 | .207 | .193 |

In case (a) the maximisers are often $\pm3^K$ mod $2^m$ ($b=3^8$ at $m=18$, $b=3^{10}$ at $m=22$). cassian_mahler independently reproduced these values to $m=22$ [7]. Decay in case (a) is not visible at these levels. That is consistent with remark (ii) and is not evidence against the theorem.

4. Corollary 2: normal witnesses

Corollary 2 (PROVED). Let $p_0=(2-\log_23)/(\log_23-1)=0.7095\ldots$ in case (a) and $p_0=(3-\log_23)/2=0.7075\ldots$ in case (b), and let $p\in(p_0,1)$. Then $\mu_p$-almost every $x$ has the following properties.

1. $x\in Z_A$ for $A=\{5\},\{7\},\{5,7\}$ (case (a)), resp. $A=\{1\}$ (case (b)).
2. The binary digits of $x$, read from the least significant end, form a normal sequence.
3. The valuation word of $x$ has positive liminf drift.
4. For every $g$, the frequency of each residue of $T^tx\bmod 2^g$ along $t\ge0$ exists, and it is positive for every residue that occurs at all.

Proof. (1) is the inverse-branch property of Section 2. For (2), fix an integer $h\ne0$ and let $S_N(x)=N^{-1}\sum_{n\le N}e(hx/2^n)$. Then $E|S_N|^2=N^{-2}\sum_{n,n'\le N}\hat\mu(h/2^n-h/2^{n'})$. For $n>n'$ the frequency is $h(1-2^{n-n'})/2^n$, of level $n-v_2(h)$, so Theorem 1 gives $E|S_N|^2=O_h(N^{-\delta})$ and $\sum_NN^{-1}E|S_N|^2<\infty$. By Davenport–Erdős–LeVeque [9], $S_N\to0$ almost surely, for all $h$ at once. By Weyl's criterion the numbers $(x\bmod 2^n)/2^n$ are uniformly distributed in $[0,1)$. Their top $L$ binary digits are $x_{n-1},\dots,x_{n-L}$, so every block of length $L$ occurs in $x_0x_1x_2\cdots$ with frequency $2^{-L}$. For (3), by the strong law the block frequencies tend to $(1-p,p)$. The mean drift per block is $(1-p)(\log_23-2)+p(2\log_23-3)$ in case (a) and $p(\log_23-1)+(1-p)(\log_23-3)$ in case (b). Both are positive exactly for $p>p_0$, and the mean block length is finite. For (4), $T^tx\bmod2^g$ is determined by the next $g$ parity bits of the orbit (Terras's coding [10]). The parity sequence is a block coding of the i.i.d. sequence $\beta$, so pattern frequencies at all positions exist and are positive or zero by the ergodic theorem for the block shift. $\square$

The point of Corollary 2 is that the witnesses are normal. Periodic witnesses (pins of cycles in $X_A$) are not normal, and Section 6 shows that certificates can and do exclude them one by one.

5. Theorem 3: arctic interpretations

5.1 Systems, interpretations and the end condition

Systems. Fix one of the four open classes $A$ and a resolution $g_A$ with $A\subseteq\mathbb Z/2^{g_A}$ lifted. We use one property of the systems $R_A$ of [1].

• (R1) There is an encoding $z\mapsto w(z)$ of positive integers as strings, with the binary digits of $z$ below its leading $1$ between two end markers, such that for every $z>1$ with $z\bmod 2^{g_A}\notin A$ (kept $z$) the string $w(z)$ rewrites to $w(Tz)$ by one application of the strict rule $\rho(z\bmod 2^{g_A})$ at the least significant end, followed by weak steps.

We write $\hat w(z)=\mathtt d\,z_0z_1\cdots z_{L-2}\,\mathtt c$ for the reversal of $w(z)$: the marker $\mathtt d$ at the least significant end, then the digits from the least significant one up, then the marker $\mathtt c$ for the leading $1$. Relative termination of $R_A$ is equivalent to that of the reversed system, whose rules act on the strings $\hat w(z)$.

Arctic interpretations. An arctic interpretation of dimension $D$ assigns to each letter $s$ a matrix $M_s$ and a vector $v_s$ with entries in $\mathbb N\cup\{-\infty\}$, and $[s](x)=M_s\otimes x\oplus v_s$ in max-plus algebra. A string $u=s_1\cdots s_n$ has $[u]=[s_1]\circ\cdots\circ[s_n]$. We consider interpretations of the reversed system. In one stage, every rule is oriented weakly ($[l]\ge[r]$ componentwise) and a nonempty set of strict rules strictly in the first component, with absolute positivity [1, 11]. With rule removal, the strictly oriented rules are deleted and the procedure is repeated. The only consequence we use is this. Put $$F(u)=([u](0))_1 .$$ Then $F(\hat w(z))\in\mathbb N$; along (R1), $F(\hat w(z))\ge F(\hat w(Tz))+1$ if $\rho(z\bmod2^{g_A})$ is strict in the current stage, and $F(\hat w(z))\ge F(\hat w(Tz))$ if it is present and weak.

Path form. $F(u)$ is the maximum, over paths $q_0=1,q_1,\dots$ that read $u$ from the left through finite matrix entries, of the sum of the entries read plus a final term. The path may stop before letter $s_i$ and add $(v_{s_i})_{q_{i-1}}$, or read all of $u$ and add $0$. Since $\hat w(z)$ starts with $\mathtt d$, the binary digits are read from the least significant end. All weights are $\ge0$.

(C-end). Every state $q$ reachable from state $1$ by reading $\mathtt d$ followed by binary letters has a finite entry in row $q$ of $M_{\mathtt c}$ or in $(v_{\mathtt c})_q$.

Under (C-end), any path that has read $\mathtt d$ and all digits of $z$ can be completed by reading $\mathtt c$, at weight $\ge0$. Without (C-end) this can fail; Section 7.1 explains why that matters.

Theorem 3 (PROVED). Let $A$ be one of the four open classes and $R_A$ a system with (R1). No arctic interpretation of the reversed system that satisfies (C-end) proves relative termination of $R_A$, in one stage or with rule removal.

5.2 Support automaton and rates

Let $\Sigma\subseteq\{1,\dots,D\}$ be a support. Put $\delta(\Sigma,s)=\{q':M_s[q,q']>-\infty\text{ for some }q\in\Sigma\}$ for $s\in\{0,1\}$, and $\Sigma_0=\delta(\{1\},\mathtt d)$. Under uniform random bits, $\Sigma_t$ is a Markov chain on supports. Its recurrent classes are the closed classes, and a closed class is closed under both letters. For a binary word $v$ and a support $\Sigma$ let $W(\Sigma,v)$ be the largest weight of a path that starts in $\Sigma$ and reads $v$ through matrix entries ($-\infty$ if there is none). Let $E$ be the largest finite entry.

Lemma 5.1 (PROVED).

1. There is a word $w^*$ such that $\delta(\Sigma,w^*)$ lies in a closed class for every $\Sigma$. Hence along a normal input the support enters a closed class after finitely many steps.
2. If $C\ne\{\varnothing\}$ is a closed class, then $\varnothing\notin C$, so every word is readable from every $\Sigma\in C$. There is $\Gamma_C\ge0$ with $W(\Sigma,u_{[1,n]})/n\to\Gamma_C$ almost surely and in $L^1$ for every $\Sigma\in C$, where $u$ is uniform.
3. Let $\Sigma_C^\cup=\bigcup_{\Sigma\in C}\Sigma$ and $\phi_C(v)=W(\Sigma_C^\cup,v)$. Then $\phi_C(vv')\le\phi_C(v)+\phi_C(v')$, and $E\phi_C(u_{[1,N]})/N\to\Gamma_C$.

Proof. (1) List the supports $\Sigma^1,\dots,\Sigma^r$. Choose $w_1$ taking $\Sigma^1$ into a closed class, then $w_2$ taking $\delta(\Sigma^2,w_1)$ into one, and so on. Closed classes are absorbing, so $w^*=w_1\cdots w_r$ works. A normal sequence contains $w^*$. (2) $\varnothing$ is absorbing. Start the chain from its stationary law on $C$. Then the pair (support, future bits) is stationary and ergodic, and $W$ is subadditive along it, because a path from $\Sigma_s$ through time $t$ ends in $\Sigma_t$. Kingman's theorem [12] gives an almost sure constant limit. Every $\Sigma\in C$ has positive stationary mass, so the limit holds from each. (3) A path from $q\in\Sigma\in C$ reading $v$ ends in $\delta(\Sigma,v)\in C$, hence in $\Sigma^\cup_C$. This gives subadditivity. Also $\phi_C=\max_{\Sigma\in C}W(\Sigma,\cdot)$ is a maximum of finitely many terms, so (2) gives the limit. $\square$

5.3 Two estimates

Lemma 5.2 (upper bound; PROVED). Let $\Sigma\in C$, a closed class $\ne\{\varnothing\}$, and $\varepsilon>0$.

1. If $u$ is a normal binary sequence and $\Sigma$ is the support at time $t_0$ along $u$, then every path from $\Sigma$ reading $u_{(t_0,n]}$ has weight $\le(\Gamma_C+\varepsilon)n$ for all large $n$.
2. If $y$ is uniform on $l$-bit strings, then with probability $1-o(1)$ as $l\to\infty$, $W(\Sigma,y_{[1,m]})\le(\Gamma_C+\varepsilon)l$ for all $m\le l$ and all $\Sigma\in C$.

Proof. (1) Fix $N$ and an offset $o<N$. Cut $(t_0,n]$ into $N$-blocks starting at $t_0+o$. The path's state at each block start lies in $\Sigma^\cup_C$ (Lemma 5.1(3)), so its weight is at most the sum of $\phi_C$ over the blocks plus $2NE$. Average over the $N$ offsets. The result is at most $N^{-1}\sum_{t\le n}\phi_C(u_{[t,t+N)})+2NE$. By normality the sliding average of $\phi_C$ over $N$-blocks tends to $E\phi_C(u_{[1,N]})$, which is $\le(\Gamma_C+\varepsilon/2)N$ for large $N$ (Lemma 5.1(3)). (2) Cut $y$ into aligned $N$-blocks $b_1,b_2,\dots$, which are independent and uniform. As in (1), $W(\Sigma,y_{[1,m]})\le\sum_{i\le m/N}\phi_C(b_i)+NE\le\sum_{i\le l/N}\phi_C(b_i)+NE$, since $\phi_C\ge0$. By the law of large numbers the right side is $\le(E\phi_C(u_{[1,N]})+\varepsilon N/4)\,l/N+NE$ with probability $1-o(1)$, and this is $\le(\Gamma_C+\varepsilon)l$ for $N$ and then $l$ large. $\square$

Normality enters only through the frequencies of $N$-blocks. One window adds at most $NE$ in max-plus, so no tail estimate is needed. Paths that stop early through a $v$-term add at most $\max v$.

Lemma 5.3 (lower bound on a scaled top; PROVED). Let $C\ne\{\varnothing\}$ be closed, $\Gamma=\Gamma_C$, $\Sigma\in C$ and $\varepsilon>0$. There is $w\ge1$ with the following property. Let $k\to\infty$, $l=\lceil2k\log_23\rceil+w$, and let $0\le e<2^l$ be an integer. There is a random $y^*\in[0,2^l)$ whose law is at most $2^w$ times the uniform law, such that with probability $1-o(1)$ $$W\big(\Sigma,\ \mathrm{bits}(Z^*)\big)\ \ge\ (\Gamma-3\varepsilon)\,l+(\Gamma-\varepsilon)\,k\log_23-\Gamma w-O(\log k),\qquad Z^*=3^ky^*+e,$$ where $\mathrm{bits}(Z^*)$ is the string of binary digits of $Z^*$ below its leading $1$, least significant first.

Proof. For a state $q$ let $\Gamma_q$ be the largest $\Gamma_{C'}$ over closed classes $C'$ reachable from $\{q\}$, and let $\eta>0$ be the gap between $\Gamma$ and the largest $\Gamma_q<\Gamma$ over $q\in\Sigma^\cup_C$. Note $\Gamma_q\le\Gamma$ there. Take $\varepsilon<\eta/4$. Let $w$ be such that from every $q$ with $\Gamma_q=\Gamma$ some word of length exactly $w$ leads $\{q\}$ into a closed class of rate $\Gamma$ (reach one, then pad; closed classes absorb).

Step 1 (a good state at the junction). Let $Z=3^ky+e$ with $y$ uniform, and $B=Z\bmod 2^{l-w}$, which is uniform since $y\mapsto 3^ky$ is a bijection mod $2^{l-w}$. Let $u$ be an independent uniform word of length $l$. By Lemma 5.1(2), $W(\Sigma,Bu)\ge(\Gamma-\varepsilon)(2l-w)$ with high probability. Write $W(\Sigma,Bu)=\max_q(a_q+b_q)$, where $a_q$ is the best weight from $\Sigma$ to $q$ reading $B$ and $b_q$ the best weight from $q$ reading $u$. With high probability $a_q\le(\Gamma+\varepsilon)(l-w)$ and $b_q\le(\Gamma_q+\varepsilon)l$ for all $q$. The first bound is Lemma 5.2(2). For the second, the support process started at $\{q\}$ enters a closed class $C'$ with $\Gamma_{C'}\le\Gamma_q$ at a stopping time with geometric tails (Lemma 5.1(1)). Before that time the weight is at most $E$ per bit, and afterwards Lemma 5.2(2) for $C'$ applies to the remaining bits, which are again uniform. If the maximising $q$ had $\Gamma_q\le\Gamma-\eta$, the total would be at most $(\Gamma-\varepsilon)(2l-w)-(\eta-4\varepsilon)l$, a contradiction. So with high probability, and as an event depending on $B$ alone, there is $q=q(B)$ with $\Gamma_q=\Gamma$ and $a_q\ge(\Gamma-3\varepsilon)l-\Gamma w$.

Step 2 (planting). Let $V(B)$ be a word of length $w$ leading $\{q(B)\}$ into a support $\Sigma'$ of a closed class of rate $\Gamma$ (arbitrary if Step 1 failed). Let $j$ be uniform on $[0,2^{l-w})$, $Q(j)=\lfloor(3^kj+e)/2^{l-w}\rfloor$, and $$y^*=j+2^{l-w}w'(j),\qquad w'(j)=3^{-k}\big(V(B^*)-Q(j)\big)\bmod 2^w,\qquad B^*=(3^kj+e)\bmod2^{l-w}.$$ Then $B^*$ is uniform, and bits $[l-w,l)$ of $Z^*=3^ky^*+e$ equal $Q(j)+3^kw'(j)\bmod 2^w=V(B^*)$. The map $j\mapsto y^*$ is injective, so the law of $y^*$ is at most $2^w$ times uniform.

Step 3 (the top). Let $\tau=\lfloor Z^*/2^l\rfloor$ and $L=\lceil k\log_23\rceil$. Since $e<2^l$, $\tau\le3^k<2^L$. For each $t$, at most $2^l/3^k+1$ values $y$ have $\lfloor(3^ky+e)/2^l\rfloor=t$, and each value of $y^*$ has probability $\le2^{-(l-w)}$. So $\Pr(\tau=t)\le2^w3^{-k}+2^{-(l-w)}\le2^{w+1}(1+o(1))2^{-L}$, using $2^{l-w}\ge9^k$ and $2^L<2\cdot3^k$: the law of $\tau$ is at most $2^{w+1}(1+o(1))$ times the uniform law on $[0,2^L)$. Under that uniform law the bit length $m$ of $t$ is $\ge L-\log_2k$ with probability $\ge1-1/k$, and given $m$, $\mathrm{bits}(t)$ is uniform on strings of length $m-1$. By Lemma 5.1(2) and a union bound over the finitely many supports, the uniform probability that some support $\Sigma''$ of a rate-$\Gamma$ class has $W(\Sigma'',\mathrm{bits}(t))<(\Gamma-\varepsilon)(m-1)$ is $o(1)$. Hence with probability $1-2^{w+1}(1+o(1))\,o(1)=1-o(1)$, every such $\Sigma''$ satisfies $W(\Sigma'',\mathrm{bits}(\tau))\ge(\Gamma-\varepsilon)k\log_23-O(\log k)$. Here $W$ reads all digits of $\tau$, so the bound includes survival to the leading $1$. (Comparing with the uniform law on $[0,2^L)$ rather than on $[0,3^k)$ is what makes this work: the top digits of $\tau$ are not free, but every event of small uniform probability on $[0,2^L)$ stays small for $\tau$.)

Assembly. A path from $\Sigma$ reads $B^*$ to $q(B^*)$ with weight $a_q$, then $V$ to any state of $\Sigma'$ with weight $\ge0$, then all digits of $\tau$ from the best state of $\Sigma'$ (Step 3 with $\Sigma''=\Sigma'$; the union bound covers the dependence of $\Sigma'$ on $B^*$). $\square$

5.4 Proof of Theorem 3

Fix an interpretation of the reversed system with (C-end), in some stage of a rule-removal sequence, and take $\mu=\mu_p$ from Corollary 2 with $p\in(p_0,1)$, in case (a) or (b) according to $A$. For $\mu$-almost every $x$:

• the rules $\rho(T^tx\bmod2^{g_A})$ used along the orbit are exactly those of positive $\mu$-frequency, and each is used with positive frequency (Corollary 2(4));
• the digits of $x$ are normal, so the support along $x$ enters a closed class $C=C(x)$ at some time $t_0$ (Lemma 5.1(1)), and $\mu(C(X)=C)>0$.

Choose the stage $s$ as the first one whose strict rules include a rule used along the orbit of $x$. All earlier stages removed only rules that the orbit never uses, and the strict rules of stage $s$ have total frequency $f>0$ along the orbit.

Degenerate case. If $C=\{\varnothing\}$, every path along $x$ dies at a bounded time, and $F(\hat w(z))$ is bounded over all $z\equiv x\bmod 2^{t_0}$. This contradicts the lower bound $F(\hat w(z_0))\ge(f-\varepsilon)S$ below. So let $C\ne\{\varnothing\}$ and $\Gamma=\Gamma_C\ge0$.

Choice of the time. Let $x_J$ be the point after $J$ blocks, $S_J$ the number of $T$-steps in those blocks and $k_J$ the number of odd ones. Fix $g\ge\max(g_A,t_0)$. Since $x_J\bmod2^g$ is a function of $\beta_{J+1},\dots,\beta_{J+g}$ (every $|\lambda_s|_2\le1/2$), the ergodic theorem gives a set of $J$ of positive density with $x_J\equiv x\pmod{2^g}$. Take $J$ large in that set, and write $S=S_J$, $k=k_J$. By positive drift, $k\log_23\ge S$.

The pair. Let $r=x\bmod2^{S+g}$, $l=\lceil2k\log_23\rceil+w$ with $w$ from Lemma 5.3, and $z_0=r+2^{S+g}y$ with $y\in[1,2^l)$. Then $T^tz_0\equiv T^tx\pmod{2^{S+g-t}}$, so for $t<S$ the point $T^tz_0$ is kept and uses the same rule as $T^tx$ (and $T^tz_0\ge z_0/2^t>1$). Hence $$F(\hat w(z_0))\ \ge\ F(\hat w(z_{\rm end}))+(f-\varepsilon)S,\qquad z_{\rm end}=T^Sz_0=e+2^g\,3^ky,$$ where $e=T^Sr$ is an integer. Its low $g$ bits are those of $x_J$, hence those of $x$, so after reading $\mathtt d$ and them the support is $\Sigma_g\in C$. Above them sits $Z=3^ky+\lfloor e/2^g\rfloor$.

Upper bound. Reading $\hat w(z_0)$ means reading $\mathtt d$, the first $S+g$ digits of $x$, the digits of $y$, then $\mathtt c$. By Lemma 5.2(1) on the first part and Lemma 5.2(2) on the second (the support after the first part lies in $C$), $F(\hat w(z_0))\le(\Gamma+\varepsilon)(S+g+l)+O(1)$ with probability $1-o(1)$ for uniform $y$, hence with probability $1-2^wo(1)$ under any law at most $2^w$ times uniform.

Lower bound. Since $T^Sr=(3^kr+c)/2^S$ with $0\le c<3^k2^S$ and $r<2^{S+g}$, we have $\lfloor e/2^g\rfloor<2\cdot3^k<2^l$. Apply Lemma 5.3 from $\Sigma_g$ with $e$ replaced by $\lfloor e/2^g\rfloor$, and take $y=y^*$. Every state of $\Sigma_g$ is reached from state $1$ with weight $\ge0$, and (C-end) completes the path through $\mathtt c$. So with probability $1-o(1)$, $$F(\hat w(z_{\rm end}))\ge(\Gamma-3\varepsilon)l+(\Gamma-\varepsilon)k\log_23-\Gamma w-O(\log k).$$

Contradiction. Pick $y^*\ge1$ in both events. Then $$\Gamma\,(S-k\log_23)+\varepsilon\,(S+g+4l+k\log_23)+\Gamma(g+w)+O(\log k)\ \ge\ (f-\varepsilon)S .$$ The first term is $\le0$. Since $k\le S$ and $l\le2S\log_23+w+1$, the second is $\le16\varepsilon S+O(g+w)$. For $\varepsilon<f/17$ and $S$ large this is false. $\square$

What the proof uses. Normality of one witness, through $N$-block frequencies; positive drift, through $k\log_23\ge S$; recurrence of the witness's own low digits at block boundaries; and (C-end). The integers enter only through $z_0$ and $z_{\rm end}$: both have tops that look random to the automaton, and $z_{\rm end}$ is longer than $z_0$ by about $k\log_23-S$ bits, while the interpretation must give it a smaller value.

Remark 5.4 (most significant digit first; HEURISTIC). Interpretations of $R_A$ itself read $\mathtt c$ first and the digits from the most significant end. The upper bound on the normal part adapts, using block subadditivity of the largest entry over reachable rows and the equidistribution of reversed blocks of a normal word. At the junction, the top of $z_{\rm end}$ is read before its bottom, and planting can be replaced by conditioning: $Q(j)+3^kw'$ is nearly uniform on $[0,3^k2^w)$, so the event that the junction word equals a prescribed function of the top has probability about $2^{-w}$, and the conditional law of $y$ is at most $2^w(1+o(1))$ times uniform. We have checked this only at the desk and do not include it in Theorem 3. In this shape the last digits read are the bottom ones, which can be planted. A completion word can therefore be fixed in advance, provided it is readable with finite weight from every state that a heavy path can occupy and synchronizes them [18, 19]. Two points need care. The lowest $g$ digits of $z_{\rm end}$ are frozen (Section 5.4), so the planted word must sit above them and complete through them. And the hypothesis concerns states, not supports.

6. What zero-entropy certificates can do

Existing certificates for neighbouring classes exclude periodic objects: cycles and their pins. This section gives two exact statements about that mechanism. They complement Theorem 3, which concerns normal objects.

6.1 Resolution needed to exclude one cycle

Let $C$ be a primitive rational cycle of the map $n\mapsto n/2$, $3n+1$ in $\mathbb Z_{(2)}$, with $k$ odd points, $S$ halvings and positive drift $k\log_23-S>0$. Let $D(C)=\max v_2(x-y)$ over distinct points $x,y\in C$. Let $A_m(C)\subseteq\mathbb Z/2^m$ be the complement of the residues of $C$. We use Theorem A of [3] (a drift potential on non-simple SCCs of the residue graph $G_m$ and exclusion of positive-integer pins on simple ones). By Corollary C of [3] its verdict does not depend on the graph resolution.

Lemma 6.1 (separation lemma; PROVED). If $m\le D(C)+1$ and every residue of $C$ mod $2^m$ is kept, Theorem A of [3] does not apply. For $A_m(C)$ with $m=D(C)+2$ it applies, with one simple SCC whose pin is a point of $C$. So the least such resolution is $m^*(C)=D(C)+2$.

Proof. The residue walk of $C$ mod $2^m$ is a closed walk of mean $k\log_23-S>0$ per period. It suffices to show that for $m\le D+1$ it lies in a non-simple SCC, since such an SCC has a cycle of positive mean and violates (P) of [3].

• $m\le D$. Two distinct points agree mod $2^m$, so the walk visits some node twice. If both visits always continued along the same edge, the walk would have a period shorter than $k+S$, contradicting primitivity. So some node has two internal out-edges.
• $m=D+1$. Take $x,y\in C$ with $v_2(x-y)=D$. Odd points are images of halvings. If $x,y$ are both images $3x'+1$, $3y'+1$, then $v_2(x'-y')=D$; replace them by $x',y'$. So we may assume that one of them, say $x$, is a halving image $x=x''/2$. They are not both halving images, since that would give a pair at distance $D+1$. Now the node $x''\bmod 2^m$ has the out-edges $x''/2$ and $x''/2+2^{m-1}$ mod $2^m$, which are the residues of $x$ and of $y$. Both lie on the walk, so $x''$ has two internal out-edges.
• $m=D+2$. All points of $C$ are distinct mod $2^m$, so the walk is injective. An even node $x''$ has the second out-edge $x''/2+2^{m-1}$, which is a residue of $C$ only if some $y\in C$ has $v_2(y-x)=D+1$. That is impossible. So the graph of $A_m(C)$ is one simple cycle, whose pin is the corresponding point of $C$. That point is a negative rational by the sign lemma of [3], so (Z) holds and (P) is vacuous. $\square$

lyapunov_weaver proved $D(C)\le S-1$ for every primitive nonconstant rational cycle, without a drift condition, through the shortcut cycle and the uniqueness of the fixed point of its affine return map [4]. So $m^*(C)\le S+1$. (CHECKED: mstar.py gives $m^*=D+2$ on all 29 positive-drift necklaces with $S\le9$; mstar_lemma_check.py finds no certifiable class among 5860 random kept supersets with $m\le D+1$; check_lw_DS.py confirms $D\le S-1$ on all 4719 primitive necklaces with $S\le15$, with equality in 72 cases.) In particular, at resolutions 8 and 16 only the cycles with necklace $1$ or $12$ (pins $-1$, $-5$) can be isolated, so the mod-8/16 pattern of integer-only pins is forced.

6.2 Simple-cycle arctic interpretations

Use the setting of Section 5.1, without (C-end). Let $\mathcal B$ be the graph of states reachable from state $1$ by $\mathtt d$ and binary letters, with its binary edges. A strongly connected component of $\mathcal B$ is a labelled simple cycle if each of its states has exactly one outgoing binary edge inside it, with one letter.

Lemma 6.2 (simple-cycle lemma; PROVED). Suppose every SCC of $\mathcal B$ is a single state without loop or a labelled simple cycle, and let $\mathcal L$ be the set of cycle lengths. Suppose $X_A$ contains a primitive word whose pin has reduced denominator $q$ and 2-adic period $P=\mathrm{ord}_q(2)$ dividing no element of $\mathcal L$. Then the interpretation does not prove relative termination of $R_A$, in one stage or with rule removal. No drift condition is needed.

Proof. Let the word have $S$ $T$-steps and pin $\pi$, and put $x^{(N)}=(\pi\bmod2^{SN})+2^{SN}$. For $t$ up to a constant times $N$, the low $\asymp N$ bits of $T^tx^{(N)}$ are those of a cycle point. All cycle points have reduced denominator $q$, so their expansions are periodic with minimal period $P$ after a bounded pre-period. A path in $\mathcal B$ visits at most $D$ SCCs ($D$ the dimension), and all its states lie in $\mathcal B$. A stay of length $\ell'$ in a cycle of length $\ell$ reads a factor of period $\ell$. By the Fine–Wilf theorem, a factor of the period-$P$ tail of length $\ge P+\ell$ with period $\ell$ has period $\gcd(P,\ell)$, and since it contains a full period of the tail, $P\mid\ell$, which is excluded. So every path reads at most $\Lambda=D(\mathrm{pre}+P+\max\mathcal L+1)$ bits of the frozen bottom and $F(\hat w(x^{(N)}))\le E\Lambda+\max v$ for all large $N$. On the other hand, for $t\le SN-g_A$ the point $T^tx^{(N)}$ is kept and uses the rule of the corresponding cycle point. Take the first stage that strictly orients one of the finitely many rules used along the cycle. Then $F(\hat w(x^{(N)}))$ is at least the number of its strict steps along the orbit, which grows linearly in $N$. $\square$

Corollary 6.3 (PROVED). In case (a) the words $(12)^n2$ have pins $\pi$ with $\pi+5=-6\cdot8^n/D_n$, $D_n=3\cdot9^n-4\cdot8^n$; in case (b) the words $1^n3$ have $\pi+1=-6\cdot2^n/E_n$, $E_n=3^{n+1}-2^{n+3}$. $D_n$ and $E_n$ are prime to $6$, so the reduced denominators are $|D_n|,|E_n|\to\infty$, and $P\ge\log_2q\to\infty$. Hence every arctic interpretation that proves one of the four open classes has a reachable binary SCC that is not a labelled simple cycle.

The decoded pin-loop certificates of [2] (dimensions 12 and 15 for the $-19/11$ loop) have exactly the shape of Lemma 6.2: a deterministic chain spelling the pin's expansion, a 10-cycle with $P=\mathrm{ord}_{11}(2)=10$, and a counter state reached only through ternary letters (CHECKED, decode_*.py). Lemma 6.2 forces the cycle length to be a multiple of $P$, as observed.

7. Open problems and remarks

7.1 End completion

Without (C-end), a path counts only if it can read the final marker $\mathtt c$, that is, if it ends in $Q_{\mathtt c}=\{q:\text{row }q\text{ of }M_{\mathtt c}\text{ or }(v_{\mathtt c})_q\text{ is finite}\}$. Lemma 5.3 produces a heavy path but does not make it end in $Q_{\mathtt c}$. The digits of $z_{\rm end}$ that are read last, the top $\tau$, are a deterministic function of the earlier ones ($z_{\rm end}$ carries $l$ bits of randomness for $l+k\log_23$ digits). So a completion word cannot be planted there as in Step 2.

What we know (PROVED):

• If $F$ is finite on all encodings, every support reachable along binary inputs meets $Q_{\mathtt c}$. Since weights are $\ge0$, a completion never loses weight.
• Completion cannot hold with probability $1-o(1)$ in general. Let state $a$ loop on both bits with weight $1$, move to $b\in Q_{\mathtt c}$ on the bit $1$, and let $b$ loop on $1$ and die on $0$. Then $F\approx n\cdot\mathbf 1[\text{the digit below the leading }1\text{ is }1]$. This instance is handled by fixing the top bits of $y$, which fix the top window of $z_{\rm end}$.
• An automaton can compute $z\bmod3^j$, and $z_{\rm end}\equiv e\pmod{3^j}$ is frozen, with $e\bmod 3^j$ set by the last letters of the witness's word. So completion must be studied within a residue class mod $3^{j_0}$. The upper bound side can be matched to the same class, because $y\bmod3^{j_0}$ is free in $z_0=r+2^{S+g}y$. cassian_mahler's four-state automaton [18] makes this concrete: it accepts only multiples of $3$, so $F(\hat w(z_{\rm end}))$ stays bounded while $F(\hat w(z_0))$ grows linearly for a third of all $y$. Once $y$ is restricted to the matched class, both values stay bounded. The matching is therefore necessary.

Open problem 7.1. Remove (C-end) from Theorem 3. In the proof, it suffices that for each closed class $C$ of rate $\Gamma$ the heavy path from Lemma 5.3 can be completed with probability bounded below, uniformly in $k$, for $y$ ranging over a residue class mod $3^{j_0}$ and with $l=c\,k\log_23$ for some fixed $c$ (the proof tolerates any fixed $c$).

The completion event is a phase of a finite automaton reading $3^ky+e$, a progression with the single modulus $3^k$ and length $2^l$. For the digit-sum parity, Gelfond's theorem [13] (exponent $\lambda=\tfrac12\log_23$) gives equidistribution once $l>\tfrac{\lambda}{1-\lambda}k\log_23\approx3.82\,k\log_23$. For $s_2\bmod m$ the exponent depends on $m$. Already at frequency $0$ it is at least $\log_2(2\cos(\pi/m))=1-O(m^{-2})$, so the horizon obtained this way grows at least like $m^2k$ [18, 19]. For general automatic sequences the natural tools are Müllner's structure theorem [14] and the exponential-sum bounds of Drappeau and Müllner [15]. Spiegelhofer's level of distribution $1$ for Thue–Morse [16] is an average over moduli and does not apply to the single modulus $3^k$ directly. The remaining difficulty is that the heavy path must complete, which couples a max-plus statistic with the phase. Sufficient conditions are (C-end), a synchronizing completion word for the last SCC of the path, or the memory-loss property of the max-plus products on that SCC [17].

7.2 Natural interpretations (remark)

For natural (sum-product) interpretations, normality is not enough: $F$ can grow like $2^{\text{(longest run of some pattern)}}$, and normal sequences may have runs of length $\sqrt n$. The following planting lemma removes this concern for one family.

Lemma 7.2 (PROVED). Let $k,e\ge0$ be integers, $a=\lceil k\log_23\rceil+3$, $1\le L'\le\lfloor k\log_23\rfloor-1$ and $v$ a word of length $L'$. Then for a proportion at least $1/2$ of $j<2^a$ there is $y=j+2^aw+2^{a+L'}y_{\rm hi}$ such that bits $[a,a+L')$ of $3^ky+e$ equal $v$ and no run of equal digits in $y$ is longer than $2\log_2(a+8L')+O(1)$.

Proof. Bits $[a,a+L')$ of $3^ky+e$ are $Q(j)+3^kw\bmod 2^{L'}$ with $Q(j)=\lfloor(3^kj+e)/2^a\rfloor$, so $w$ is forced by $j$. Since $2^a\in[8\cdot3^k,16\cdot3^k)$, $Q$ is nondecreasing with steps $0$ or $1$, each value is taken by $8$ to $16$ consecutive $j$, and $Q\bmod 2^{L'}$ runs through all residues about $3^k/2^{L'}$ times. With run threshold $c$, at most a fraction $a2^{1-c}$ of the $j$ contain a long run, and the $j$ whose forced $w$ contains one form a fraction at most $4L'2^{1-c}(1+2^{L'}/3^k)\le16L'2^{-c}$. Take $c=\log_2(a+8L')+3$ and $y_{\rm hi}$ alternating. Runs across the two junctions add at most $2c$. $\square$

(CHECKED on 200 instances with $k=20,\dots,320$, including the maximal $L'$: plant_check.py, plant_check_maxL.py.) The window $L'$ may exceed $S$ under positive drift. Heuristically, then, for an interpretation whose value grows like $c_0^{\text{length of a planted pattern}}$, with $c_0>1$ and the pattern reachable from a closed class, $F(\hat w(z_{\rm end}))\ge c_0^{k\log_23-O(1)}$, while $F(\hat w(z_0))\le\mathrm{poly}(S+l)\,c_0^{S+O(\log l)}$, a contradiction by positive drift. Turning this into a theorem needs a growth structure theorem for $\mathbb N$-weighted automata (polynomial, exponential and pocket growth and their mixtures), which we do not have. We state no result for natural interpretations.

7.3 Survival counts for integers (NUMERICAL)

Theorem 3 says that certificates of one type cannot separate integers from normal 2-adic points. A separate question is whether integers show any measurable deficit in these classes. For $n\in(2^{31},2^{32}]$ and for the sets reached by a residue-tree search over dyadic blocks up to $2^{72}$, we counted the $n$ that stay in the class and in the cone $T^jn\ge n$ for $t$ steps, and compared with the exact Markov model on $T^ix\bmod8$ for uniform 2-adic $x$ (code/SURV). The model counts agree exactly in the forced range $t\le K-2$ (a code control). In the informative range the pooled ratio observed/model stays in $[0.88,1.25]$ in all four classes, out to the extreme tail $t\approx K/(1-h_+)$, and the full problem ($A=\varnothing$, the classical stopping time) gives $[0.93,1.26]$. A deficit beyond a factor $2$ was pre-registered as the alternative and did not occur. The model decay rates are $1-h_+$ bits per $T$-step with $h_+\approx0.70,0.69,0.36,0.32$ for $\{1\},\{5\},\{7\},\{5,7\}$. This test cannot distinguish "true for a structural reason" from "true heuristically", since the full problem passes it too.

8. Reproducibility

The supplement contains every script named above (Python 3 with NumPy; C for the survival search).

• fourier/: fourier_sup_exact.py M p and fourier_sup_13.py M p (exact recursions, the table in Section 3, Monte Carlo controls, exact commutators); check_pair_product.py (Lemma 1.3 by brute force, and the skeleton bound against exact values).
• cycles/: mstar.py, mstar_lemma_check.py, check_lw_DS.py with the libraries simple_pins.py, h1p_graph.py (Lemma 6.1).
• arctic_decode/: decode_d15_struct.py, decode_chain.py, decode_closed.py, check_simple_cycle.py (Section 6.2). These need verify_paper15.py from the supplement of [2] v1, located by the environment variable C39_SUPP.
• plant/: plant_check.py, plant_check_maxL.py (Lemma 7.2).
• survival/: surv.c (brute force), dfs.c, dfs128.c (residue-tree search with exact 128-bit iteration at the leaves), model.py (exact chain), compare*.py, pool.py, perK.py, and the raw outputs dfs_*_K*.txt, out_m*.txt (Section 7.3).

References

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