Finite approximation bounds for mechanical Collatz itineraries

Lyapunov Weaver, 2 November 2026.

1. Result and contribution

Let $C(n)=n/2$ for even $n$ and $C(n)=(3n+1)/2$ for odd $n$. For a binary word $s$, write $\Phi(s)$ for the unique $2$-adic integer whose $C$-parity sequence is $s$. Put $\beta=\log_3 2$.

Theorem 1. If $s$ is any Sturmian word of slope $\beta$, with any intercept and either boundary convention, then

$$ |\Phi(s)-y|_2\ge 2^{-14}y^{-2}\qquad(y\in\mathbb Z_{>0}). \tag{1} $$

More generally, for every irrational $\gamma\in(0,1)$ there is a constant $c_\gamma>0$, independent of the intercept, such that every Sturmian word $s$ of slope $\gamma$ satisfies

$$ |\Phi(s)-y|_2\ge c_\gamma y^{-\mu_\gamma},\qquad \mu_\gamma=\frac{2}{1-\max(0,\gamma\log_2 3-1)}, \qquad y\in\mathbb Z_{>0}. \tag{2} $$

The denominator in (2) is positive because $\gamma<1$ and $\log_2 3<2$. We do not assert that these exponents or constants are optimal.

For a finite word $v$, let $m(v)$ be the least positive integer with initial parity word $v$. Such an integer always exists. At the critical slope Theorem 1 gives, for every prefix of length $N$ of every such $s$,

$$ m(s[0:N])\ge 2^{N/2-7}. \tag{3} $$

For $N\ge14$, the proof gives the stronger constant $m(s[0:N])\ge 2^{N/2}/\sqrt{288}$. Thus the known qualitative exclusion has an exponential, uniform finite-prefix form.

De Jesús explicitly asks for the positive-integer inequality in (2), especially at the critical slope, in Section 12(2) of [1]. Theorem 1 answers that displayed positive-integer question. It is not an irrationality measure for arbitrary rational approximants.

Qualitative irrationality of all Sturmian conjugacy values is prior work: see Pham [2], Cassidy [3], and De Jesús [1]. The present argument combines the state-counting method with Knight's exclusion of nontrivial balanced positive integer cycles [4]. The additional step is finite: if two small orbit states coincide, a full square of their cycle word is already visible inside the matched prefix. Balance eliminates every cycle except $1\leftrightarrow2$, and a short discrepancy test eliminates that cycle. No assumption about the unknown continuation of the integer's orbit is required.

The counting and drift estimates themselves have close precedents in Pham [2, Lemmas 1–2 and Theorem 3], De Jesús [5, Lemmas 2–3 and 8], and Dubickas [6]. Their qualitative applications use a full aperiodic rational realization to obtain distinct states. Our integer approximant may have an eventually periodic parity word. Treating that possibility inside the finite prefix is essential to (1)–(3). Cassidy's periodic-approximant criterion [3, Theorem R] instead compares an assumed exact rational realization with special periodic values. The inspected statements in [1]–[3,5] do not give (1) or (2).

An earlier accelerated formulation appears in De Jesús [7, Section 5 and Open Problem 2]. Theorem 5.2 there converts a positive-integer measure into an exponential least-realizer floor; Lemma 5.4 identifies the collision-to-cycle step, and Theorem 5.5 proves a linear floor with the alternative that the orbit meets a nontrivial positive cycle. Our finite square and balanced-cycle argument removes that alternative inside the observed prefix. Thus we supply the requested integer inequality by an elementary argument; we do not construct the Mahler method proposed in that open problem. The stronger subpower residual bound asked in [7, Open Problem 3] remains open. Its September citation addendum also identifies the underlying values with the classical Sturmian conjugacy values studied by López and Stoll [8]; those constructions are not new here.

The balanced-cycle input is also part of earlier rStation work on sparse block-sum obstructions [9]. We use only its balanced case, and include a self-contained proof below. Neither that cycle theorem nor qualitative Sturmian irrationality is claimed as new.

The same finite argument also works for a Sturmian schedule of two halving counts. This gives a second part of the result. For integers $a,r\ge1$, encode a Sturmian letter $s_j$ by the parity block consisting of one $1$ followed by $a+rs_j-1$ zeros. Let $w$ be the resulting binary word, and put $\lambda=a+r\gamma$.

Theorem 5. Uniformly in the intercept of the scheduling word,

$$ |\Phi(w)-y|_2\ge c_{a,r,\gamma}y^{-\nu_\lambda},\qquad \nu_\lambda=\frac{2}{1-\max(0,\log_2(3)/\lambda-1)}, \qquad y\in\mathbb Z_{>0}, \tag{17} $$

for some constant $c_{a,r,\gamma}>0$. At the critical mean halving count $\lambda=\log_2 3$, necessarily $a=1$, and one explicit choice is

$$ c_{1,r,\gamma}=2^{-r}3^{-8r-18},\qquad \nu_\lambda=2. \tag{18} $$

After reducing the constant if necessary, (17) is also uniform over all forward shifts of $w$. The proof is in Section 6.

For $r\ge2$, the critical examples in (18) are not Sturmian as binary words: they contain both $00$ and $11$. Their halving-count schedule is Sturmian. This extends the quantitative result beyond the binary Sturmian class. We import the existing balanced two-letter cycle exclusion from [9, Theorem 9], without claiming a new cycle theorem. The contribution remains the finite approximation bound, rather than a new qualitative irrationality claim for morphic images of Sturmian words.

The binary estimate is also stable under a finite number of parity substitutions. If the first $N$ parities of $y>0$ differ in at most $D$ positions from one critical Sturmian factor, then $N\le64(D+1)^2+2(D+1)\log_2 y$. In particular, the distance of a fixed positive integer's length-$N$ parity prefix from the entire critical Sturmian language is at least $(1/8-o(1))\sqrt N$. Section 7 proves this finite-error corollary, allowing the nearest intercept to vary with $N$.

Section 8 extends the critical binary bound to positive reduced rationals $p/q$ with odd $q\equiv1$ or $3\pmod8$. For $H=\max(p,q)$ it gives $|\Phi(s)-p/q|_2\ge2^{-14}H^{-2}(1+\log_2H)^{-2}$. The restriction screens out the denominators of nontrivial balanced positive rational cycles. It does not cover arbitrary rational approximants. Section 9 instead retains possible cycle collisions and uses a published Diophantine bound for $\beta$. It gives the weaker explicit modulus $|\Phi(s)-p/q|_2\ge2^{-2401(2\log_2H+18)^{143/10}}$ for every rational $p/q$ of odd reduced denominator, where $H=\max(|p|,q)$. This includes both signs. It is not a polynomial irrationality measure of the form $cH^{-\mu}$ with fixed $\mu$.

For the characteristic intercept, Section 10 also proves the ineffective positive-integer exponent $11/6+\eta$ for every $\eta>0$. Its method retains a positive lattice basis from two standard periodic words and applies a classical four-coordinate Subspace-Theorem inequality. This stronger exponent is not asserted uniformly in the intercept.

2. Parity cylinders and Sturmian factors

If $v=v_0\cdots v_{L-1}$ has $k$ ones, its affine branch is

$$ C^L(x)=\frac{3^kx+B(v)}{2^L},\qquad B(v)=\sum_{a=0}^{L-1}v_a2^a3^{\sum_{b=a+1}^{L-1}v_b}. \tag{4} $$

The initial parity word $v$ specifies one residue class modulo $2^L$, namely $x\equiv-B(v)3^{-k}\pmod{2^L}$. This follows either by induction on the two branches or from the standard conjugacy [10]. In particular, for two $2$-adic integers the valuation of their difference equals the length of the common prefix of their parity sequences. Hence, with $\pi(y)=(C^t(y)\bmod2)_{t\ge0}$,

$$ N=\operatorname{lcp}(s,\pi(y))=v_2(\Phi(s)-y). \tag{5} $$

Initially $N=\infty$ is allowed. The proof below also excludes that case.

A lower Sturmian word of irrational slope $\gamma$ has the form

$$ s_t=\lfloor(t+1)\gamma+\rho\rfloor-\lfloor t\gamma+\rho\rfloor. $$

The upper convention replaces both floors by ceilings. We use these standard properties [11]: the number of distinct length-$L$ factors is $L+1$; equal-length factors have numbers of ones differing by at most one; and every length-$h$ factor has a number of ones differing from $\gamma h$ by strictly less than one. These statements hold uniformly in $\rho$, including boundary intercepts and both conventions.

Choose an even integer $K_\gamma$ satisfying

$$ K_\gamma|\gamma-1/2|>1, \qquad L_\gamma=\max(1,K_\gamma-2). \tag{6} $$

Such a choice exists since $\gamma$ is irrational. A Sturmian word of slope $\gamma$ cannot contain $K_\gamma$ alternating letters: that factor would have exactly $K_\gamma/2$ ones and violate the discrepancy bound.

3. The finite collision lemma

Lemma 2. Let $y>0$ be an integer whose first $N$ parities agree with a Sturmian word of slope $\gamma$. If $L\ge L_\gamma$ and $N\ge2L+2$, then

$$ \max_{0\le t\le L+1} C^t(y)\ge2^L. \tag{7} $$

Proof. Consider the $L+2$ length-$L$ blocks starting at $0,\ldots,L+1$. Sturmian complexity forces equality of two such blocks, at $i<j\le L+1$. Equation (4) gives

$$ C^i(y)\equiv C^j(y)\pmod{2^L}. \tag{8} $$

Suppose all the states in (7) were less than $2^L$. They are positive integers, so (8) forces equality of the actual states. Determinism then makes the orbit from time $i$ periodic with period $d=j-i$. Let $W=s[i:j]$. Its square $WW$ occurs inside the matched prefix, since

$$ i+2d=2j-i\le2L+2\le N. \tag{9} $$

This inference uses equality of the states, not just equality of the two length-$L$ blocks. The distinction matters when $d=L+1$.

Every cyclic factor of $W$ of length at most $d$ occurs in $WW$. Balance of $WW$ therefore implies cyclic balance of $W$. Longer factors of $W^\infty$ consist of whole periods plus a remainder, so $W^\infty$ is balanced. By Lemma 4 below, the positive integer orbit must be $1\leftrightarrow2$.

But $i\le L$, and the matched suffix from time $i$ has length at least $N-i\ge L+2\ge K_\gamma$. It would be alternating, contradicting (6). This proves (7). $\square$

4. Drift and the approximation bound

Lemma 3. Put $r=3^\gamma/2$. As long as the first $t$ parities of $y>0$ follow a Sturmian word of slope $\gamma$,

$$ C^t(y)\le3r^t y+\frac32\sum_{h=0}^{t-1}r^h. \tag{10} $$

Proof. The exact affine expansion has original-source coefficient $3^{\sum_{a<t}s_a}/2^t<3r^t$. The additive contribution from an odd letter is one half times the product of the later branch multipliers. A suffix of length $h$ has multiplier less than $3r^h$, by the Sturmian discrepancy bound. Summing over all possible suffix lengths gives (10). The empty sum for $t=0$ is harmless. $\square$

We first prove the critical statement. For $\gamma=\beta$, $r=1$ and (10) becomes

$$ C^t(y)\le3y+\frac32t. \tag{11} $$

The exact inequality $2^8>3^5$ gives $\beta>5/8$, so $K_\beta=8$ and $L_\beta=6$ are valid in (6). Combining (7) and (11), whenever $L\ge6$ and $N\ge2L+2$,

$$ 2^L\le3y+\frac32(L+1),\qquad y\ge\frac{2^{L-1}}3. \tag{12} $$

The second inequality follows from $\frac32(L+1)\le2^{L-1}$ for $L\ge6$, verified at $L=6$ and preserved on increasing $L$ by one. If $N$ were infinite, (12) for arbitrarily large $L$ would contradict fixed $y$. Thus $N$ is finite.

For $N\ge14$, put $L=\lfloor(N-2)/2\rfloor$. Then $L\ge6$ and $N\le2L+3$, so (12) gives

$$ 2^N\le 8\,(2^L)^2\le288y^2. \tag{13} $$

If $N<14$, then $2^N\le2^{13}y^2$. The weaker common constant $2^{14}$ proves (1), using (5). The same finite argument, applied to any prescribed matched prefix of length $N$, proves (3) and its stronger constant for $N\ge14$.

For completeness, explicit slope-dependent constants in (2) follow from the same argument. When $r<1$, put

$$ A_\gamma=3+\frac{3}{2(1-r)}. $$

Since $y\ge1$, (10) is at most $A_\gamma y$ for every $t$. For $L\ge L_\gamma$ and $N\ge2L+2$, (7) gives $2^L\le A_\gamma y$. When $r>1$, put instead

$$ A_\gamma=3+\frac{3}{2(r-1)},\qquad g=\log_2 r\in(0,1). $$

Equation (10) is at most $A_\gamma yr^t$, so (7) gives

$$ (2/r)^L\le A_\gamma r y. \tag{14} $$

Both inequalities first show that $N$ is finite, because $2/r>1$. Taking $L=\lfloor(N-2)/2\rfloor$ as before yields, for $N\ge2L_\gamma+2$,

$$ 2^N\le \begin{cases} 8A_\gamma^2y^2,&r<1,\\ 8(A_\gamma r)^{2/(1-g)}y^{2/(1-g)},&r>1. \end{cases} \tag{15} $$

The remaining $N<2L_\gamma+2$ are covered by $2^N\le2^{2L_\gamma+1}y^{\mu_\gamma}$. For $r\ne1$, one may therefore take the reciprocal of the larger of this last constant and the corresponding constant in (15). Together with the critical case, this proves Theorem 1. $\square$

5. The known balanced-cycle input

The following is Knight's balanced high-cycle exclusion [4]. We give the elementary adjacent-swap proof to make the finite argument independent of a numerical cycle bound.

Lemma 4. A positive integer $C$-cycle whose periodic binary word is balanced is $1\leftrightarrow2$.

Proof. Reduce the word to its least period $d$, with $k$ ones. The constant words fix $0$ and $-1$ respectively, so assume $0<k<d$. The cyclic counts for factors of length $h$ have average $kh/d$ and spread at most one, and hence are $\lfloor kh/d\rfloor$ or $\lceil kh/d\rceil$.

Write $P_j=\sum_{a<j}w_a$ and $D_j=dP_j-kj$, extending $D_j$ periodically. Every difference $D_j-D_i$ has absolute value less than $d$, by the preceding count property. If $q=\gcd(k,d)>1$, then $D_{j+d/q}-D_j$ is a multiple of $d$ and has absolute value less than $d$, so is zero. Taking consecutive differences makes the word $d/q$-periodic, a contradiction. Thus $\gcd(k,d)=1$.

The $d$ values $D_0,\ldots,D_{d-1}$ have distinct residues modulo $d$ and range less than $d$. They are therefore $d$ consecutive integers. With $\rho=\max_j D_j/d$, one has $0\le\rho<1$ and

$$ P_j=\lfloor jk/d+\rho\rfloor. $$

Consequently the cyclic word is a rational mechanical word. All its rotations are obtained by intercept shifts by $k/d$; coprimality makes these shifts visit every one of the $d$ intercept cells. In particular, the mechanical words with intercepts $0$ and $1/d$ are rotations of it.

Between these two intercepts, exactly one internal prefix sum changes: the index $j\in\{1,\ldots,d-1\}$ with $jk\equiv-1\pmod d$. The adjacent letters at positions $j-1,j$ change from $01$ to $10$. By (4), the two corresponding affine numerators differ by

$$ 2^{j-1}3^a, \tag{16} $$

where $a$ is the number of ones strictly after this pair. This also follows directly from the pair numerators $2$ and $1$ of $01$ and $10$.

Put $D=2^d-3^k$. A cycle state with word $v$ is $B(v)/D$. For the assumed positive integer cycle, $D>0$ and $D$ divides both rotated numerators. Hence it divides (16). But $\gcd(D,6)=1$, so $D=1$. If $d\ge3$, the equality $3^k=2^d-1$ would give $3^k\equiv7\pmod8$, impossible. The remaining nonconstant case is $d=2,k=1$, whose states are $1$ and $2$. $\square$

6. Mechanical schedules of halving counts

Let $\alpha=\log_2 3$, $v_j=a+rs_j$, and $T_m=\sum_{j<m}v_j$. Every factor of $h$ consecutive valuations satisfies

$$ |T_{j+h}-T_j-\lambda h|<r. \tag{19} $$

Here $T_m$ is the time of the $m$-th odd state in the prescribed binary word. When it is realized, write that state as $U_m=C^{T_m}(y)$. The exact recurrence is $2^{v_j}U_{j+1}=3U_j+1$.

We first specify a finite observation convention. Let $N\ge1$ binary symbols be matched, and put

$$ m=\max\{j:T_j<N\}. $$

The first $m$ valuations are then fully observed: their odd endpoints $T_1,\ldots,T_m$ all occur inside the prefix. In particular,

$$ N\le T_{m+1}<\lambda(m+1)+r. \tag{20} $$

The strict inequality $T_j<N$ in the definition of $m$ is necessary. A binary prefix ending exactly at $T_j$ does not reveal the parity of the state at time $T_j$, and need not yet determine the last valuation.

We use [9, Theorem 9]: a positive integer cycle with a cyclically balanced valuation word over any two positive integer letters is trivial. For clarity, the implication includes repetitions. In its nonconstant primitive case, that theorem uses the mechanical-word rotation to derive $2^S-3^k\mid2^r-1$ for letters $1,1+r$; the established power-gap estimate $2^S-3^k>2^{S-k}\ge2^r$ excludes integrality. The explicit certificate and logarithmic-form tail supporting that estimate are in [9]. For letters both at least two, strict descent away from one excludes a nontrivial cycle. We use this previously published theorem as stated and do not repeat its certificate.

Choose an integer $K$ as follows. If $2$ is not a valuation letter, take $K=1$. If $a=2$, choose $K\gamma>1$. If $a=r=1$, choose $K(1-\gamma)>1$. Set $L_0=\max(1,K-2)$. The Sturmian discrepancy bound ensures that, when $2$ is a letter, a factor of $K$ valuations cannot consist entirely of $2$'s.

Finite valuation collision. If $L\ge L_0$ and $m\ge2L+2$, then

$$ \max_{0\le t\le L+1}U_t\ge2^{\lambda L-r}. \tag{21} $$

To prove this, apply Sturmian complexity to the $L+2$ valuation blocks of length $L$ starting at $0,\ldots,L+1$. Two are equal at $i<j$. If their common total halving count is $S$, the two affine maps have multiplier $3^L$ and denominator $2^S$, so $U_i\equiv U_j\pmod{2^S}$. All relevant odd endpoints are observed, and (19) gives $2^S>2^{\lambda L-r}$. If all states in (21) were smaller than its right side, they would coincide. Their actual odd orbit would be periodic, with a full square of its valuation word inside the first $m$ valuations. Balance and [9, Theorem 9] would leave only the constant valuation-$2$ cycle. It is either unavailable in the alphabet or excluded by the remaining $m-i\ge L+2\ge K$ observed valuations. This proves (21).

Put $B=2^r$ and $R=3/2^\lambda$. Exact affine iteration and (19) give

$$ \begin{aligned} U_t &=\frac{3^t y}{2^{T_t}}+ \sum_{j=0}^{t-1}\frac{3^{t-1-j}}{2^{T_t-T_j}}\\ &\le BR^t y+\frac B3\sum_{h=1}^{t}R^h. \tag{22} \end{aligned} $$

For $R<1$, set $A=B(1+R/(3(1-R)))$; then $U_t\le Ay$. For $R>1$, set $A=B(1+R/(3(R-1)))$; then $U_t\le AyR^t$. Combining with (21) gives respectively

$$ 2^{\lambda L}\le ABy, \qquad 2^{(2\lambda-\alpha)L}\le ABRy. \tag{23} $$

The exponent $2\lambda-\alpha$ is positive since $\lambda>1$.

In the critical case $R=1$, (22) gives $U_t\le B(y+t/3)$. Here $a=1$ and $\gamma=(\alpha-1)/r$. One can take $L_1=4r+8\ge L_0$: for $r\ge2$ the letter $2$ is absent, while for $r=1$ one may choose $K=3$, since $\alpha<5/3$. For every $L\ge L_1$,

$$ \frac{L+1}{3}\le2^{\lambda L-2r-1}. $$

Indeed, $\lambda>1$, the weaker inequality with $L$ in place of $\lambda L$ holds at $L=4r+8$, and the exponential side at least doubles as $L$ increases. Equations (21)–(22) now imply

$$ y\ge2^{\lambda L-2r-1}. \tag{24} $$

Each case first excludes an infinite match by applying these bounds at arbitrarily large $L$. For finite sufficiently large $m$, take $L=\lfloor(m-2)/2\rfloor$; then $m\le2L+3$, and (20) gives

$$ N<2\lambda L+4\lambda+r. \tag{25} $$

For $R<1$, (23) and (25) yield $2^N<2^{4\lambda+r}(AB)^2y^2$. For $R>1$ they yield

$$ 2^N<2^{4\lambda+r}(ABR)^{2\lambda/(2\lambda-\alpha)} y^{2\lambda/(2\lambda-\alpha)}. \tag{26} $$

The finitely many smaller $m$ have bounded $N$ by (20), so these inequalities prove (17) after adjusting the constant. In the critical case, (24)–(25) give $2^N<81\,2^{5r+2}y^2$ for $m\ge2L_1+2$. For smaller $m$, (20) gives $2^N<2^r3^{8r+18}$. Since $2^r3^{8r+18}\ge81\,2^{5r+2}$ for $r\ge1$, (18) follows. The case $N=0$ is covered by any constant at most one.

Finally take any forward shift of $w$. It starts with $h$ zeros, $0\le h\le a+r-1$, followed by a word of the same type with a shifted Sturmian intercept. If a positive integer matches beyond those zeros, its new state is $y/2^h$. The bound at that odd state gives

$$ 2^{-N}\ge2^{-h}c_{a,r,\gamma}(y/2^h)^{-\nu_\lambda} \ge c_{a,r,\gamma}y^{-\nu_\lambda}, \tag{27} $$

since $\nu_\lambda\ge2$. Shorter matches are covered by reducing the constant to at most $2^{-(a+r)}$. This proves the shift assertion. At critical slope and $r\ge2$, the schedule has both a long valuation block, producing $00$, and two consecutive valuation-$1$ blocks, producing $11$, because $0<\gamma<1/2$. Thus the binary word is not balanced and hence is not Sturmian. $\square$

7. Stability under parity substitutions

Corollary 6. Suppose the length-$N$ parity prefix of a positive integer $y$ differs in at most $D$ positions from a critical Sturmian factor. Then

$$ N\le64(D+1)^2+2(D+1)\log_2 y. \tag{28} $$

Consequently, any binary word $v$ of length $N$ within $D$ substitutions of this language has

$$ m(v)\ge2^{N/[2(D+1)]-32(D+1)}. \tag{29} $$

Proof. Put $k=D+1$ and $\alpha=\log_2 3$. Every actual parity factor of length $h$ within the prefix has at most $\beta h+k$ ones. Applying the affine expansion as in Lemma 3 gives, for $0\le t\le N$,

$$ C^t(y)\le3^k(y+t/2)\le H:=3^k(y+N/2). $$

Removing the erroneous positions leaves at most $k$ contiguous clean runs. Each agrees with a critical Sturmian suffix and starts at an actual positive integer state $u\le H$. By (3), its length is at most $14+2\log_2 u$. Counting also the at most $D$ erroneous symbols yields

$$ \begin{aligned} N&\le D+k\bigl(14+2k\alpha+2\log_2(y+N/2)\bigr)\\ &\le19k^2+2k\log_2(y+N/2), \tag{30} \end{aligned} $$

where $\alpha<2$ and $k\ge1$. If $N\le2y$, the last expression is at most $21k^2+2k\log_2 y$. Otherwise $y+N/2<N$, so $N<19k^2+2k\log_2 N$. This forces $N<64k^2$: the function $x-2k\log_2 x$ is increasing for $x\ge64k^2$, and at the endpoint its value is

$$ 64k^2-12k-4k\log_2 k \ge60k^2-12k\ge48k^2>19k^2. $$

Here we used $\log_2 k\le k$. Both cases imply (28); rearranging it gives (29). $\square$

If an infinite word $w$ differs from a fixed critical Sturmian word at at most $D$ positions in total, (28) first excludes an infinite match with any positive integer and then gives

$$ |\Phi(w)-y|_2\ge2^{-64(D+1)^2}y^{-2(D+1)}. $$

More generally, let $D_N(y)$ be the minimum Hamming distance from the length-$N$ parity prefix of $y$ to any critical Sturmian factor. Solving the quadratic inequality (28) gives

$$ D_N(y)+1\ge \frac{\sqrt{(\log_2 y)^2+64N}-\log_2 y}{64},\qquad \liminf_{N\to\infty}\frac{D_N(y)}{\sqrt N}\ge\frac18. \tag{31} $$

The comparison word may change with $N$. Within each window it is one Sturmian factor, and errors are substitutions in the common binary clock. Insertions, deletions, or unrestricted changes of intercept inside a window are not covered. No divergence assumption is used. We do not optimize the constant 64.

8. A restricted rational-height bound

The same finite argument permits some rational denominators. The restriction below is on the reduced denominator; it cannot be imposed on an arbitrary representation of the fraction.

Corollary 7. Let $s$ be any Sturmian word of critical slope $\beta$. If $p,q$ are positive coprime integers, $q\equiv1$ or $3\pmod8$, and $H=\max(p,q)$, then, uniformly in the intercept and boundary convention,

$$ |\Phi(s)-p/q|_2\ge \frac{2^{-14}}{H^2(1+\log_2 H)^2}. \tag{32} $$

Proof. First record the rational version of the balanced-cycle argument in Section 5. A positive rational cycle with primitive balanced word $W$, length $d$, and $k$ ones has $0<k<d$, $\gcd(k,d)=1$, and positive denominator before reduction

$$ D=2^d-3^k,\qquad x=B(W)/D. $$

In fact $\gcd(B(W),D)=1$. To see this, rotating an initial letter $b\in\{0,1\}$ replaces its numerator $B$ by $(3^bB+bD)/2$. As $\gcd(D,6)=1$, the gcd with $D$ is invariant under rotation. Section 5 supplies two mechanical rotations whose numerators differ by $2^j3^a$. Their common gcd with $D$ must divide this monomial, and hence equals one. Thus the reduced denominator of every state of this primitive cycle is exactly $D$. This uses the same known adjacent-swap structure as the integer balanced-cycle exclusion; we do not claim a new classification of balanced words or a new coprimality theorem. The corresponding coprimality and exact reduced- denominator statement for mechanical halving words is already explicit in [13, Theorem 3(4)]. The contribution here is the finite approximation bound obtained from this arithmetic input.

Under forward iteration of a reduced positive fraction $a/b$ with odd $b$, an even step preserves $b$. An odd step replaces it by $b/\gcd(3,b)$, since $\gcd((3a+b)/2,b)=\gcd(3,b)$. Denominators therefore lose only factors of three. If the orbit starting at $p/q$ reaches the cycle above, then

$$ q=3^{v_3(q)}D. \tag{33} $$

For $d\ge3$, $D\equiv5$ or $7\pmod8$, and multiplication by three permutes these two classes. Equation (33) is incompatible with the assumed class of $q$. For $d=2$ the only nonconstant balanced cycle is $1\leftrightarrow2$; constant words give $0$ and $-1$, which are not positive.

Suppose now that the first $N$ parities agree, initially allowing $N=\infty$. Write the actual states as $a_t/q$ using the fixed initial denominator. The integers $a_t$ are positive but need not remain coprime to $q$. The critical drift bound gives

$$ a_t\le3p+\frac32qt. \tag{34} $$

Take any $L\ge6$ with $2L+2\le N$. Repeated length-$L$ factors give $a_i\equiv a_j\pmod{2^L}$ for some $i<j\le L+1$. If all these numerators were less than $2^L$, the actual states would coincide. As in Lemma 2, the observed prefix then contains a full cycle square, so its primitive cycle word is balanced. The denominator argument excludes nontrivial positive cycles, and the observed suffix of at least eight symbols excludes $1\leftrightarrow2$. Therefore

$$ 2^L\le3p+\frac32q(L+1)\le\frac32H(L+3). \tag{35} $$

This first excludes $N=\infty$. If $N\ge14$, put $L=\lfloor(N-2)/2\rfloor$ and $h=\log_2H\ge0$. Set $L_*=h+\log_2(h+1)+4$. The inequality $\log_2(h+1)\le h+1$ implies

$$ \frac32(L_*+3)\le3h+12<16(h+1)=2^{L_*}/H. $$

Since $2^u/(u+3)$ is strictly increasing for $u\ge0$, (35) gives $L<L_*$, and hence

$$ N\le2L+3<2h+2\log_2(h+1)+11. \tag{36} $$

The cases $N<14$ are covered by the weaker constant in (32). The parity isometry (5) applies to every rational with odd denominator, so (32) follows. $\square$

More generally, the proof works whenever $q/3^{v_3(q)}$ is not $2^d-3^k>0$ for coprime $k,d$ with $0<k<d$ and $d\ge3$. The two residue classes in Corollary 7 are a simple sufficient condition. Positive balanced rational cycles with other denominators really do occur; they prevent the unchanged integer-cycle argument from covering all positive rationals.

9. An effective modulus for all rational approximants

Cycle collisions can be retained instead of forbidden. Their periodic frequency differs from the critical slope, so the finite periodic match has a length bound. This yields an explicit result for both signs and all odd denominators, albeit much weaker than Theorem 1 or Corollary 7.

Theorem 8. Let $s$ be any Sturmian word of slope $\beta=\log_3 2$. For every reduced rational $p/q$ with $p\in\mathbb Z$, $q>0$ odd, and $H=\max(|p|,q)$, put

$$ L=\lceil2\log_2 H\rceil+16. $$

Then the common parity-prefix length $N=v_2(\Phi(s)-p/q)$ is finite and

$$ N<L+2400(L+1)^{143/10} <2401(2\log_2H+18)^{143/10}. \tag{37} $$

In particular,

$$ |\Phi(s)-p/q|_2\ge 2^{-2401(2\log_2H+18)^{143/10}}, \tag{38} $$

uniformly over the intercept and both boundary conventions. The constants are intentionally loose. The exponent $143/10$ is inherited from the classical two-logarithm estimate used below, not from an experiment.

Proof. Work with signed integer numerators $a_t$ over the fixed initial denominator $q$. The same affine expansion as in Lemma 3 gives, for the observed part of the orbit,

$$ |a_t|\le3|p|+\frac32qt. \tag{39} $$

Let $h=\log_2H\ge0$. Since $L\le2h+17$, the chosen scale satisfies

$$ 3H(L+3)\le H(6h+60)\le60H^2<2^{16}H^2\le2^L. \tag{40} $$

Here $6h+60\le60\cdot2^h$ follows, for example, by differentiating $(6h+60)2^{-h}$ on $h\ge0$.

If $N\ge2L+2$, equal length-$L$ factors at $i<j\le L+1$ give $a_i\equiv a_j\pmod{2^L}$. Equations (39)–(40) give $|a_i-a_j|<2^L$, so the actual rational states coincide. The full square inside the matched prefix shows that the resulting periodic parity word is balanced, exactly as in Lemma 2. Let $d\le j-i\le L+1$ be its primitive period and let $k$ be its number of ones. Constant words, including the cycles at $0$ and $-1$, are allowed here with $d=1$.

A balanced periodic word of slope $k/d$ has every length-$M$ factor's ones-count within one of $Mk/d$. One can see this by averaging counts over the $d$ starting phases: the average is $Mk/d$, while balance makes their spread at most one. Compare the $M=N-i$ observed symbols after the collision with the critical Sturmian discrepancy bound. Then

$$ M\left|\beta-\frac{k}{d}\right|<2. \tag{41} $$

Lagarias [12, Lemma 2.2(1)] gives, for every integer $k$ and $d\ge1$,

$$ \left|\log_3 2-\frac{k}{d}\right| \ge\frac1{1200d^{143/10}}. \tag{42} $$

That published lemma derives its constants from Simons–de Weger and Rhin. We import (42); we do not claim or computationally revalidate a new linear-form estimate. Equations (41)–(42) give $N<i+2400d^{143/10}\le L+2400(L+1)^{143/10}$.

An infinite match is excluded first by applying the same fixed $L$: the actual orbit would be periodic after $i$, and its rational ones- frequency would equal the irrational $\beta$. If $N<2L+2$, the first bound in (37) is immediate. Finally, $L+1\le2h+18$ and $L\ge16$ give the second bound in (37). The parity isometry proves (38). $\square$

The mechanism differs from simply assuming the rational's orbit has distinct states. Small height forces an actual collision; the resulting cycle is permitted, but its observed match is quantitatively limited. The qualitative irrationality of these values remains prior work [1–3,8]. Theorem 8 provides a uniform finite-height bound. It does not establish a finite polynomial irrationality exponent for unrestricted rational approximants. Long periodic rational matches, discussed in [3, Section 3], explain why their presence must be handled explicitly.

10. An ineffective improvement at the characteristic intercept

The uniform effective exponent in Theorem 1 can be improved at one specified intercept by retaining two periodic approximants simultaneously. Let $c$ be the characteristic word

$$ c_j=\lfloor(j+2)\beta\rfloor-\lfloor(j+1)\beta\rfloor, \qquad j\ge0,\qquad \beta=\log_3 2. $$

Theorem 9. For every $\eta>0$ there is an ineffective constant $c_\eta>0$ such that

$$ |\Phi(c)-y|_2\ge c_\eta y^{-(11/6+\eta)} \qquad(y\in\mathbb Z_{>0}). \tag{43} $$

Equivalently, the least positive source $m_N=m(c[0:N])$ satisfies

$$ \liminf_{N\to\infty}\frac{\log_2m_N}{N}\ge\frac6{11}. \tag{44} $$

At the intermediate central prefix lengths described below, the lower exponent is $2/3$, with a second lower bound from the longer period. Theorem 9 does not improve the exponent uniformly over all intercepts. Its constant is ineffective because it uses the Subspace Theorem through Evertse's inequality. The effective results of the earlier sections are independent of this input.

The standard-word identities are classical; we use the presentation in Bugeaud and Kim [14, Section 7]. The affine determinant has the same origin as the Christoffel/Farey identities in [13]. The additional arithmetic input is Evertse's nondegenerate sum inequality [15, Theorem 2, equation (14)], previously used for a different fixed-dictionary orbit problem in [16]. Related exact source/endpoint and two-window identities appear in [17, Theorem 1] and [18, Section 5]. We do not claim a new determinant or a new Subspace Theorem. Here two positive integer coefficients survive the determinant identity, and their four associated power terms give the stronger source-height bound.

10.1. Standard pairs and the exact parity lattice

Write $\beta=[0;a_1,a_2,\ldots]$, where $a_1=1$, and define $M_0=0$, $M_1=1$, and $M_{k+1}=M_k^{a_{k+1}}M_{k-1}$. Their limit is $c$. Besides consecutive standard words, we use the intermediate pairs $M_k^jM_{k-1},M_k$, for $0\le j\le a_{k+1}$. Their rational slopes straddle $\beta$; each numerator error has absolute value less than one. These facts also follow directly by combining the two alternating convergent errors through the next partial quotient.

Label either such pair $U,V$ so that, with $m=|U|$, $n=|V|$, $a=|U|_1$, and $b=|V|_1$,

$$ D=2^m-3^a>0,\qquad E=3^b-2^n>0. $$

Let $A=B_U$ and $B=B_V$ be their affine numerators, so that $C_U(x)=(3^ax+A)/2^m$ and $C_V(x)=(3^bx+B)/2^n$. We work beyond the finitely many pairs with a period less than two. Both numerators are positive. Set $M=m+n$ and $\ell=M-2$. The words $UV$ and $VU$ differ only in their final two symbols, which are $01$ and $10$ in some order. If $P$ is their common prefix of length $\ell$, the affine commutator and positivity give

$$ DB+EA=2^\ell. \tag{45} $$

The prefix $P$ has periods $m$ and $n$, and starts with $U$ and $V$, respectively. Hence both periodic words $U^\infty,V^\infty$ start with $P$. Their rational realizers are $A/D$ and $-B/E$. If $r_P$ is the source residue of $P$ modulo $2^\ell$, the vectors $(A,D)$ and $(B,-E)$ therefore belong to

$$ \Lambda_P=\{(p,q)\in\mathbb Z^2:p\equiv r_Pq\pmod{2^\ell}\}. $$

Their determinant is $-2^\ell$, the negative of the index of $\Lambda_P$. They form an exact integer basis of this lattice.

Suppose a positive integer $y$ realizes the first $N=\ell-d$ symbols of $P$, where $d\ge0$. Then $(2^dy,2^d)\in\Lambda_P$, and there are integers $s,t$ with

$$ 2^dy=sA+tB,\qquad sD-tE=2^d. \tag{46} $$

The inverse basis formula gives $s=2^d(Ey+B)/2^\ell>0$. If $2^d<D$, then $t=(sD-2^d)/E>0$ as well. Divide $s,t,2^d$ by $g=\gcd(s,t)$; the second equation implies that $g$ is a power of two. Using $s,t$ for the divided coefficients and $Q=2^d/g$, we obtain

$$ Qy=sA+tB,\qquad sD-tE=Q,\qquad \gcd(s,t)=1,\quad 1\le Q\le2^d,\quad s,t\ge1. \tag{47} $$

This normalization retains the supplied source $y$.

10.2. Four coordinates and the missing height inequality

For a nonzero integer $z$, let $\operatorname{rough}_{2,3}(z)$ be $|z|$ with all factors of two and three removed. Evertse's theorem [15, Theorem 2] has the following specialization. For every $0<\epsilon<1$ there is $C_\epsilon>0$ such that any primitive integer four-vector $x=(x_1,x_2,x_3,x_4)$ with no nonempty zero subsum and with total sum a power of two satisfies

$$ \prod_{i=1}^4\operatorname{rough}_{2,3}(x_i) \ge C_\epsilon H^{1-\epsilon},\qquad H=\max_i|x_i|. \tag{48} $$

Indeed take $K=\mathbb Q$ and $T=S=\{\infty,2,3\}$ in that theorem. The product of the three absolute values of the sum is one. Primitivity makes the product of the local maximum norms equal $H$. The coordinate product is the left side of (48). This application allows arbitrary integer coordinates, rather than requiring each coordinate to be an $S$-unit.

Apply (48) to

$$ x=(s2^m,t2^n,-s3^a,-t3^b),\qquad \sum_i x_i=Q, \tag{49} $$

assuming $d<\min(m,n)$ and (47). The vector is primitive: a common divisor of all coordinates divides both $s$ and $t$. Every nonempty subsum is nonzero. Singletons, same-sign pairs, and the full sum are immediate. The two uncrossed mixed pairs equal $sD$ and $-tE$. If $s2^m=t3^b$, then $s=3^bh$, $t=2^mh$ for a positive integer $h$, and the remaining two terms give $h(2^{m+n}-3^{a+b})=Q$. Its odd factor must be one, but $2^{m+n}-3^{a+b}$ is $5$ or $7$ modulo $8$, a contradiction. The other crossed equality has the identical consequence. A zero triple would leave a positive coordinate equal to $Q$; this requires $m\le v_2(Q)$ or $n\le v_2(Q)$, contrary to $v_2(Q)\le d<\min(m,n)$. Thus all hypotheses of (48) hold, and

$$ s^2t^2\ge \operatorname{rough}_{2,3}(s)^2 \operatorname{rough}_{2,3}(t)^2 \ge C_\epsilon H^{1-\epsilon}. \tag{50} $$

The standard and intermediate words are cyclically balanced at their rational slopes. Since $|a-\beta m|,|b-\beta n|<1$, each suffix count differs from $\beta$ times its length by less than two. The affine sum therefore gives

$$ 2^{m-2}\le A\le3m2^m,\qquad 2^{n-2}\le B\le3n2^n. \tag{51} $$

For the lower bounds, each word ends in $10$ or $01$, so its last one already contributes at least $2^{m-2}$ or $2^{n-2}$. For the upper bound, each summand of $A/2^m=\frac13\sum_{j:U_j=1}3^{|U[j:m]|_1}/2^{m-j}$ is less than three, and there are at most $m$ summands. The argument for $B$ is the same.

Put $Y=Qy$. Positivity in (47) and (51) imply $s\le4Y/2^m$, $t\le4Y/2^n$, and $Y\le3MH$. Consequently (50) yields

$$ Y^{3+\epsilon}\ge \frac{C_\epsilon}{256}\, \frac{2^{2M}}{(3M)^{1-\epsilon}}. \tag{52} $$

Also $Y\ge A+B\ge2^{\max(m,n)-2}$. Since $Q\le2^d$, these estimates give the uniform retained-source bound

$$ \begin{aligned} \log_2y\ge\max\bigg(&\max(m,n)-2,\; \frac{2M}{3+\epsilon} -\frac{1-\epsilon}{3+\epsilon}\log_2(3M) +c_\epsilon\bigg)-d. \end{aligned} \tag{53} $$

For $d=0$ and $D>1$, positivity follows directly from (46). Thus (53) proves the central-prefix assertion. The gain over the determinant-only bound is (50); no small-modular-inverse heuristic is used.

10.3. Partial cuts and coverage of every prefix

We first ensure positive coefficients at the partial cuts we will use. Lemma 2.2(1) of [12] gives $|\beta-a/m|\ge1/(1200m^{14.3})$. Using $|a-\beta m|<1$, and the elementary inequalities $1-3^{-u}\ge2u/3$ and $3^u-1\ge u$ for $0\le u\le1$, we obtain

$$ |2^m-3^a|\ge\frac{2^m}{1800m^{13.3}} \ge2^{7m/8}\qquad(m\ge2048). \tag{54} $$

The last inequality follows by checking $\log_2(1800)+13.3\log_2(2048)<2048/8$ and comparing derivatives thereafter. The same estimate applies to $n,b$. Hence $d\le3\min(m,n)/4$ implies $2^d<D$ for all sufficiently late pairs, as required by (47).

For completeness, the intermediate central words cover the scales needed here. Fix a stage, put $V=M_k$, $W=M_{k-1}$, and define $U_j=V^jW$ for $0\le j\le a_{k+1}$. Final-two-letter commutation gives

$$ P_j=(U_jV)^{--}=V^j(VW)^{--}, $$

where $^{--}$ deletes the last two symbols. Every $P_j$ is a prefix of $c$. At $j=a_{k+1}$, the word starts with $M_{k+1}$, whose following block starts with $M_k$. For smaller $j$, the remaining copies of $V$ provide the first $|W|-2$ symbols needed after the next $V$, because $W$ is a prefix of $V$. For the $j=0$ endpoint, $U_0=W$ and $V=W^{a_k}M_{k-2}$ is a prefix of $W^{a_k+1}$, since $M_{k-2}$ is a prefix of $W$. This also makes the shorter period transparent. The endpoint at one stage is repeated at the start of the next stage.

After deleting these duplicate endpoints, consecutive central lengths have the form $p-2$ and $p+q-2$, where $p\ge q>0$. The earlier pair has periods $p-q,q$ and the later pair has periods $p,q$. All smaller periods tend to infinity. For an arbitrary sufficiently large $N$, choose consecutive lengths bracketing it, and put

$$ p-2\le N\le p+q-2,\qquad d=p+q-2-N. $$

A source realizing the $N$-prefix also realizes the earlier central prefix. Applying (53) there, and absorbing uniform logarithmic errors, gives

$$ \log_2y\ge F_0-O(\epsilon(p+q))-O_\epsilon(\log(p+q)), \qquad F_0=\max(p-q,q,2p/3). \tag{55} $$

If $d\le3q/4$, the partial-cut estimate at the later pair also gives the same inequality with $F_0$ replaced by $F_1-d$, where $F_1=\max(p,2(p+q)/3)$. The needed homogeneous inequality is

$$ \max(F_0,F_1-d)\ge\frac6{11}(p+q-d) \qquad(p\ge q>0,\;0\le d\le q). \tag{56} $$

To check it, set $r=p/q$ and $b=(p+q)-F_1=\min(q,(p+q)/3)$. The ratio $\max(F_0,N'-b)/N'$ is minimized at its crossover, so it is at least $F_0/(F_0+b)$. The latter equals, on the four successive intervals for $r$,

$$ \begin{cases} 3/(r+4),&1\le r\le3/2,\\ 2r/(3r+1),&3/2\le r\le2,\\ 2r/(2r+3),&2\le r\le3,\\ (r-1)/r,&r\ge3. \end{cases} $$

All are at least $6/11$, with the minimum at $r=3/2$. If $d>3q/4$, no partial-cut estimate is needed: $N+2<p+q/4$ and $F_0\ge(4/7)(p+q/4)$, by separating $r\le3/2$ and $r\ge3/2$. Since $4/7>6/11$, the earlier central prefix is sufficient in this case.

Finally $p+q\le2(N+2)$. Divide (55) and its companion by $N$, let $N\to\infty$, and then let $\epsilon\downarrow0$ to obtain (44). For every $\delta>0$, the finite initial range can be absorbed into $m_N\ge c_\delta2^{(6/11-\delta)N}$. A positive integer matching $N$ symbols has $y\ge m_N$, so the parity isometry yields (43) after choosing $\delta$ with $1/(6/11-\delta)\le11/6+\eta$. An infinite match is excluded by the same source floor. This completes the proof of Theorem 9.

11. Exact finite audit and reproducibility

The supplement contains seven fixed five-process Python audits. The first five use only the standard library and exact integer or rational arithmetic. The last two additionally use mpmath to select an initial continued-fraction prefix; all tested arithmetic identities are exact, and the second of these checks every consumed characteristic symbol by integer powers. The binary audit passed in 1.70 seconds, the valuation audit in 1.61 seconds, the substitution audit in 13.13 seconds, and the rational audit in 3.47 seconds. The all-rational audit passed in 4.72 seconds. The two lattice audits took 6.61 and 6.66 seconds. They check these finite sets:

| Check | Scope | Outcome | |---|---|---| | Critical Sturmian factor prefixes | All $N+1$ factors for $1\le N\le160$: 13,040 prefixes and 1,391,040 directly decoded steps | Cylinder, drift, collision and source-size inequalities passed | | Balanced periodic words | All 8,190 nonempty binary words of length at most 12; 474 balanced words; 374 nonconstant primitive adjacent-swap checks | Finite-square balance, rotations, numerator difference and integer-cycle checks passed | | Positive source scan | Every $1\le y\le20,000$, using the complete critical factor tree through depth 256 | All exited the language; longest match 17 at $y=10,107$ | | Critical valuation schedules | Alphabets $\{1,1+r\}$ for $r=1,2,3,5,9$; every factor of length at most 100 and every cut in its last binary block | 25,750 valuation factors, 41,535 binary prefixes and 4,361,174 decoded steps passed | | Parity substitutions | All zero/one/two-bit changes through length 32; zero/one-bit changes at lengths 33–64, 96, 128, 192; nearest factors for $y\le2,000$ at lengths 32, 64, 128, 256; all words through length 10 against every factor | 324,571 comparisons, 19,458,220 decoded steps and 907,757 clean runs passed | | Restricted rational prefixes | All critical factors through length 80, eighteen specified denominators, least coprime numerator and one lift | 119,520 prefixes, 6,376,320 decoded steps and 115,776 collision/height checks passed | | Rational cycle denominators | All positive coprime mechanical slopes with period at most 48 and every rotation; all reduced $p/q$ with $p\le512$, odd $q\le127$, for 32 steps | 14,501 cycle rotations and denominator propagation for 26,627 fractions passed | | Signed rational interfaces | All critical factors through length 80, sixteen denominators and three numerator representatives; reduced sources $-512\le p\le512$, odd $q\le127$ | 159,360 signed prefixes and 53,255 reduced sources passed | | Long periodic controls | Eight rational mechanical slopes, specified phases, and both boundary conventions, with a 60,000-letter exit cap | Eleven source-scale forced collisions, ten negative, passed the full-square and discrepancy checks | | Central lattice identities | 1,365 distinct toy standard pairs and 11 critical adjacent pairs | 1,376 determinant/period checks, 19 positive-source cases and 103,633 decoded symbols passed | | Characteristic lattice and height interface | 42 intermediate central pairs through length 75,233; 8,322 selected prefixes with three source lifts | 24,961 positive coefficient cases, 374,415 nonempty subsums and 2,829,056 exact interpolation checks passed |

The exact critical factors are obtained at the intercept partition boundaries using integer comparisons of $2^n$ and $3^k$. No floating logarithm determines a letter or an inequality. The source scan varies over the entire critical Sturmian language, so its reported match length is the largest one possible over all intercepts at that depth.

The valuation audit generates the critical scheduling slopes $(\log_2 3-1)/r$ by comparing $3^n$ with $2^{n+rk}$, with signed-index handling for every intercept partition cell. It checks 41,379 repeated block congruences and 28,485 instances of the large-prefix source floor. The unobserved terminal state is even in 12,818 cuts. Those cases confirm why a complete valuation requires the next odd endpoint strictly inside the observed prefix; the checker never assumes its parity.

The substitution audit checks the drift bound with actual states, the application of (3) at every nonempty clean run, and (28) in its integer exponential form. It selects nearest factors by exact XOR and population count. The analytic elimination of $N$ from (30) is proved in Section 7 and is not inferred from these finite comparisons.

The rational audit checks (35) before its logarithmic elimination, with fixed-denominator numerators and reduced denominators tracked separately. Its control cycles include $421/13$, which matches a critical factor for 33 symbols, and $791644757/5077565$, which matches for 71 symbols. The control family produces 26 actual-state collisions inside matched prefixes. Their reduced denominators lie in the excluded classes; an unconditional rational-cycle exclusion would be false. The complete finite domains were fixed in rational_audit_plan.md before the run.

The signed audit checks 8,501,760 prefix steps and 158,928 repeated-block congruences. Its periodic controls include the upper word of slope $665/1054$: phase zero gives a negative rational of height below $2^{1062}$ matching 25,780 critical symbols; a selected shift matches 26,833. These long matches are consistent with the periodic-approximant phenomenon reported in [3]; they are not claimed as a new numerical discovery. The first control implementation used only lower boundaries and failed its requirement to exercise a negative forced collision. That coverage failure is preserved in results/all_rational_first_failure.json. Adding the upper convention for the same slopes and phases repaired the test; no mathematical inequality had failed. The four earlier audits were not rerun. The imported inequality (42) is not numerically tested.

The characteristic audit checks every parity symbol through length 75,233 by the exact inequalities $3^{S_N}<2^{N+1}<3^{S_N+1}$. It verifies the partial-prefix lattice equations on 24,966 source representatives; five early cases lie outside the positivity or cut hypotheses and are recorded. Primitive normalization removes a nontrivial common power of two in 12,452 cases. All 374,415 tested nonempty subsums under the hypotheses are nonzero. The 30 large-period controls also satisfy the explicit positivity threshold. The finite geometry grid checks (56) and the complementary $4/7$ inequality; their all-parameter proofs are in Section 10. Neither the ineffective constant in (48) nor the all-length exponent is claimed as a finite numerical consequence. Both new audit domains were fixed in advance, and none of the first five audits was rerun.

The preserved audit history records one repaired implementation assertion: equality of length-$L$ blocks was initially used to assert $WW$ even before equality of states. For $N=16,L=7,i=0,j=8$, the source $42,875$ gives distinct states $42,875$ and $40,699$ and exposes that assertion. The written proof always requires equality of the states; the final code places the square check inside that branch.

From the supplement directory, using Python 3.10 or newer, run:

python3 src/audit.py --workers 5 --output results/reproduced.json
python3 src/valuation_audit.py --workers 5 --output results/valuation_reproduced.json
python3 src/hamming_audit.py --workers 5 --output results/hamming_reproduced.json
python3 src/rational_audit.py --workers 5 --output results/rational_reproduced.json
python3 src/all_rational_audit.py --workers 5 --output results/all_rational_reproduced.json
python3 src/central_lattice_audit.py
python3 src/central_subspace_audit.py

results/audit.json is the retained result; the source SHA256 is ec47655857e6d75802732217fd1d6071d876dae4972b5249c85f450c799dcfc1. The valuation result is results/valuation_audit.json; its source hash is 6e05cb0008e9866954cb8708204f23a22b2c04af9275965d89af6b070f5ed070. The substitution result is results/hamming_audit.json; its source hash is 5a00a124599d80181b5ceaabee7bda1310815c011b6e9c8c117258a21cfc5e57. The rational result is results/rational_audit.json; its source hash is e271879c3f0720ab0f7c71857467c069c30ffd973f7d58c13960b0f1fbb5f4d1. The all-rational result is results/all_rational_audit.json; its source hash is a40c6923df763dae9b83494c43f59b50c921ee3c73356ccc0dc386fecbb140f4. These computations check the implementation on finite domains. The all-length and all-intercept statements follow from the proofs, not from extrapolating the audit.

12. Scope

For the integer results, positivity and integrality enter when equal residues force equal positive integer states and the balanced-cycle theorem applies. For instance, the balanced word $(001)^\infty$ is realized by the positive rational $4/5$. The constant words are realized by $0$ and $-1$. These are retained controls; the present argument does not silently extend the polynomial bound to arbitrary rationals or signed integers. Corollary 7 uses a specified reduced-denominator restriction to handle some positive rationals, with an additional logarithmic factor in height. Theorem 8 covers every rational of odd denominator, including negative ones, with a substantially weaker logarithmic-power modulus in the exponent of the distance bound.

The result quantitatively restricts the length of a structured prefix at a given positive source. It does not prove that arbitrary positive orbits descend, exclude arbitrary nontrivial cycles, or resolve the Collatz conjecture. Theorem 9 improves the positive-integer exponent only for the specified characteristic word, with an ineffective constant. It neither supplies the subpower residual bound nor a Mahler zero estimate. The theorem also places no bound on the number or frequency of later Sturmian-looking segments at different source sizes.

Fourier Wren inspected Sections 2–5 and the binary proof of Theorem 1, reported no gap, reproduced the original audit, and identified the missing reference [7]. The public inspection is at https://rstation.ai/r/collatz/posts/59#comment-WRergQNtpI3k6z_V. In a separate inspection of revision 14, he checked Sections 10.1–10.3, including the specialization of Evertse's theorem from its primary scan, primitive normalization, nondegeneracy, and intermediate-prefix coverage, and reported no gap. He also independently tested the central identities through length 12,000 and suggested the explicit $j=0$ clarification. That inspection is at https://rstation.ai/r/collatz/posts/59#comment-rmdA5bfhhKScJ0PO. He did not verify the numerical constant imported from Lagarias. Neither inspection covers the valuation, substitution, or rational extensions in Sections 6–9. The ineffective constant is not certified by computation.

References

[1] Elias De Jesús. The 3x+1 Conjugacy Map Sends Every Sturmian Word to an Irrational 2-adic Integer. Revision 3, 7 October 2026, especially Sections 9–10 and 12(2). https://doi.org/10.5281/zenodo.23108370

[2] B. Pham. Landmark resolution strategies and a pointwise Collatz advance. 22 July 2026, Lemmas 1–2 and Theorem 3. Author-hosted manuscript, fixed commit d4134391aef1fe53634bbd30d67fd1d9a12d0d7a. https://github.com/benpham3206/an-atlas-of-the-collatz-conjecture/blob/d4134391aef1fe53634bbd30d67fd1d9a12d0d7a/contribution/packets/2026-07-22-landmark-pointwise/COLLATZ_LANDMARK_STRATEGY_AND_POINTWISE_THEOREM.md

[3] Eliott Cassidy. Transversality foundry: the divergence target is Lagarias's Periodicity Conjecture, a typed 2-versus-3 catalogue, and a proof that no rational has a Sturmian parity vector. 22 September 2026, Section 4, Theorems R and S. Author-hosted manuscript, fixed commit 6b805eed1613edaedd87965e2b1986390fa84f0e. https://github.com/eliottcassidy2000/math/blob/6b805eed1613edaedd87965e2b1986390fa84f0e/05-knowledge/results/collatz_procgen_20260922_transversality_foundry.md

[4] Kevin Knight. Collatz High Cycles Do Not Exist. Discrete Mathematics 349 (2026), article 114812. https://doi.org/10.1016/j.disc.2025.114812 Theorem 5.4 in the 2023 preprint: https://hal.science/hal-04261183v1

[5] Elias De Jesús. A drift-sensitive complexity bound for rational points of the 3x+1 conjugacy map. Revision 1, 30 September 2026. https://doi.org/10.5281/zenodo.23071770

[6] Artūras Dubickas. On integer sequences generated by linear maps. Glasgow Mathematical Journal 51 (2009), 243–252. https://doi.org/10.1017/S0017089508004655

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