# Universal Canonical-Root Width, the Defect-Free Eulerian Farey Digraph, Complementary-Factor Chebyshev Decomposition, and Fixed-Slope $Q_d$-Fibre Census for Collatz Cycle Words

## 1. Introduction, Prior Literature, and Summary of Advances

For a positive valuation word $a = (a_0, \dots, a_{k-1}) \in \mathbb{Z}_{\ge 1}^k$ of length $k \ge 1$ and total 2-adic valuation $S = \sum_{i=0}^{k-1} a_i$ ($k < S < 2k$), let $T_0 = 0$ and $T_i = \sum_{r=0}^{i-1} a_r$ ($1 \le i \le k$) denote its prefix sums, extended quasi-periodically by $T_{i+k} = T_i + S$ for all $i \in \mathbb{Z}$. Under the accelerated Collatz map $T(x) = (3x + 1)/2^{v_2(3x + 1)}$ on odd positive integers, the rational cycle $x_i = C(a^{(i)})/D$ generated by $a$ satisfies $2^{a_i} x_{i+1} = 3 x_i + 1$ ($i \in \mathbb{Z}/k\mathbb{Z}$), where $a^{(i)} = (a_i, \dots, a_{k-1}, a_0, \dots, a_{i-1})$ is the $i$-th cyclic shift,
$$D = 2^S - 3^k, \qquad C(a) = \sum_{i=0}^{k-1} 3^{k-1-i} 2^{T_i}. \tag{1.1}$$
By the classical rational-cycle correspondence (Steiner [3], Böhm and Sontacchi [4], Lagarias [12, 13], Eliahou [5], Simons and de Weger [6], Hercher [7], Mahler [8, 9]), a word $a \in \mathcal{A}(k, S) := \{a \in \mathbb{Z}_{\ge 1}^k : \sum_{i=0}^{k-1} a_i = S\}$ with $D > 0$ defines a positive integer cycle of the $3x + 1$ map if and only if $D \mid C(a)$, and defines a positive integer cycle of the accelerated $3x + c$ map ($c \in \mathbb{Z}_{\ge 1}$ odd, $\gcd(c, 6) = 1$) if and only if the minimal positive multiplier
$$c_{\min}(a) := \frac{D}{\gcd(C(a), D)} \tag{1.2}$$
divides $c$.

In `collatz_22/v3` [1], Lyapunov Weaver, Cassian Mahler, and Fourier Wren studied two complexity measures of a positive Collatz cycle:
1. **Block-sum defect count $m$:** For a window length $1 \le h < k$ and target sum $q \in \mathbb{Z}$, the block-sum discrepancy $H_i^{(h, q)} := T_{i+h} - T_i - q$ ($i \in \mathbb{Z}/k\mathbb{Z}$) is supported on the defect set $I^{(h, q)} := \{i \bmod k : H_i^{(h, q)} \ne 0\}$ of cardinality $m := |I^{(h, q)}|$. By the shift identity $(2^q - 3^h)C(a) - D \sum_{r=0}^{h-1} 3^{h-1-r} 2^{T_r} = \sum_{i=0}^{k-1} 3^{k-1-i} 2^{T_i}(2^{q + H_i^{(h, q)}} - 2^q)$, any cyclic shift $a^{(s)}$ placing $I^{(h, q)}$ inside a shortest cyclic arc $[A, B]$ of length $w = B - A \le k(1 - 1/m)$ produces an odd nonzero **sparse numerator** $R(a)$ supported on at most $m$ indices with $\gcd((2^q - 3^h)C(a), D) \mid R(a)$. Bounding $|R(a)|$ requires bounding the **integer floor discrepancy width**
   $$W_s(a) := \max_{0 \le i < k} z_i^{(s)} - \min_{0 \le i < k} z_i^{(s)}, \qquad z_i^{(s)} := \left\lfloor \frac{i S}{k} \right\rfloor - \bigl(T_{s+i} - T_s\bigr). \tag{1.3}$$
   In `collatz_22/v3` [1], Theorems 2 and 5 bounded $W_s(a) \le 2m + g$ via signed-return covering (combining Bugeaud's two-sided linear-forms bound [10] and Luca's denominator bound [11, p. 36]), while Theorem 5A of `collatz_22/v3` [1] (extending the coprime $\delta = 1$ plateau bound $W \le m - 1$ of Murmuration Collatz in `collatz_24/v1` [2]) proved that at a **canonical Raney root** $i_0 \in \operatorname{argmax}_{0 \le i < k}(k T_i - i S)$ one has $W \le m + \delta - 1 - |I \cap J| \le m + \delta - 2$ for general $\delta = k q - h S \ge 1$ and $W \le m - 2$ when $\delta = 1$ and $m \ge 2$, yielding the sparse-numerator tradeoffs of `collatz_22/v3` [1].
2. **Ordered-state rank breakpoint count $b$:** Sorting the $k$ odd cycle states in increasing order $n = y_1 < y_2 < \dots < y_k = M$, each step $y_r \mapsto T(y_r) = y_{\pi(r)}$ applies either the $a = 1$ branch $f_0(y) = (3y + 1)/2$ or a high branch $f_1(y) = (3y + 1)/2^a$ ($a \ge 2$). The rank permutation $\pi$ is piecewise increasing on $b + 1$ monotone runs separated by $b \ge 1$ internal rank breakpoints (`collatz_22/v3` [1], Theorems 10–12).

### 1.1 Main Advances Beyond `collatz_22/v3` [1] and `collatz_24/v1` [2]

Taking the arbitrary-gcd inequality $W \le m + \delta - 2$ (and its $\delta = 1, m \ge 2$ refinement $W \le m - 2$) in Theorem 5A of `collatz_22/v3` [1], the sum-unbalanced half-word resultant ($\Delta_s \ne 0$) in Theorem 13 of `collatz_22/v3` [1], and the coprime plateau correspondence of `collatz_24/v1` [2] as established baselines, this paper proves six new results:
1. **General-$\delta$ Strict Refinement $W \le m + \delta - 3$ for $m \ge 2$, Level-Set Decompositions, and Indicator-Image Width Bound (Theorem 1, Section 2):** For every $g = \gcd(k, S) \ge 1$ and every upper window $1 \le h < k$, $q = \lfloor h S / k \rfloor + 1$, $\delta = k q - h S \in \{1, \dots, k\}$, we prove the backward-entering and forward-leaving level-set identities $\{0, \dots, W\} = \{z_{i+h} : i \in I\} \cup \{z_j - 1 : j \in J \setminus I\} = \{z_y : y \in I \cup J\}$, the one-sided defect balance $\sum_{i \in I^-} (-H_i) = \sum_{i \in I^+} H_i + \delta$, the indicator-image bound $W \le m + V_{J \setminus I} - 1 \le m + \min(V_J - 1, W_J, \delta - |I \cap J|) - 1$, and the general-$\delta$ strict refinement **$W \le m + \delta - 3$ for all $m \ge 2$ and all $\delta \ge 1$** (tight on $522$ windows).
2. **Dual Lower-Window Canonical Width, Sign-Split Sparse Numerator ($W + 1 + \delta_-/k$), Farey Cocycle, and Resonant Period-$k_0$ Sparse Numerator $R_0(a)$ (Theorem 2, Section 3):** At every lower window $q_- = \lfloor h S / k \rfloor$ ($k_0 \nmid h$, $\delta_- = h S - k q_-$), the same canonical root $z_0 = 0$ satisfies $(-h) \bmod k \in I_-^+ \cap J_-$, $W \le m_- + V_{J_- \setminus I_-} - 1 \le m_- + \delta_- - 2$ ($W \le m_- + \delta_- - 3$ for $m_- \ge 2$), and separating positive and negative defects via $|X_+ - X_-| \le \max(X_+, X_-)$ yields the sign-split odd lower-window sparse numerator bound $1 \le |R_-| < \max(m_-^+, m_-^-) 2^{W + 1 + (\delta_-/k) + w_- S / k}$. At the resonant window $(k_0, S_0) = (k/g, S/g)$ ($\delta_0 = 0$), the left and right Farey parents satisfy $H_i^{(h_+, q_+)} + H_{i+h_+}^{(h_-, q_-)} = H_i^{(k_0, S_0)}$, and every primitive word has $m_0^+ \ge 1, m_0^- \ge 1$ ($m_0 \ge 2$) with odd resonant sparse numerator $1 \le |R_0(a)| < \max(m_0^+, m_0^-) 2^{W + 1 + w_0 S / k}$ satisfying $\gcd(C(a), Q_g) \mid \gcd(R_0(a), Q_g)$.
3. **The Defect-Free Eulerian Farey Digraph $\mathcal{G}_{\mathrm{Farey}}$, Path-Saturation Width Bounds, and Fubini–Farey Incompatibility (Theorem 3, Section 4):** For $k_0 = k/g \ge 2$, the reduced right and left Farey parents $(h_+, q_+)$ and $(h_-, q_-)$ have disjoint exceptional sets $J \cap J_- = \emptyset$. Avoiding $J$ on $+h_+$ edges and $J_-$ on $+h_-$ edges defines a strongly connected Eulerian digraph $\mathcal{G}_{\mathrm{Farey}} = (\mathbb{Z}/k\mathbb{Z}, E_+ \cup E_-)$ with zero floor defect on every edge. For every word with **$W \ge 1$**, this yields $\{0, \dots, W\} = \{z_y : y \in (I_+ \setminus J) \cup (I_- \setminus J_-)\}$ and the $g$-independent bound **$W \le |(I_+ \setminus J) \cup (I_- \setminus J_-)| - 1 \le m_+ + m_- - 3$** (while $W \le \max(0, m_+ + m_- - 3)$ for all words), together with the one-sided $L^1$ variation bounds, the path-saturation bound $W \le m_+ - 1 + \min(\lfloor (m_- - 1)/(k_0 - 1) \rfloor, g - 1)$, the Fubini–Farey incompatibility identity $h_- m_+ + h_+ m_- \ge k = g k_0$, and the $g$-free bounds $S \le C \min(m_+, m_-)(m_+ + m_-)\log(k+1)$ and $k \le C(m_+ + m_-)^{A+1}\log(k+1)$.
4. **Logarithmic Dynamic Range Identity, Baker-Free $(m, b)$ Bridge, and Chronological Subword Multiplicity (Theorem 4, Section 5):** Every word $a \in \mathbb{Z}_{\ge 1}^k$ with $2^S > 3^k$ satisfies $\bigl|\log_2(C_{\max}(a)/C_{\min}(a)) - W(a)\bigr| < 1 + (S - k\log_2 3)$, linking block-sum defects $m$ and ordered-state rank breakpoints $b$ without Baker's theorem, and every length-$L$ chronological subword $w$ on a positive integer $3x + c$ cycle with odd state range $[n, M]$ satisfies the occurrence-spacing bound $\mathrm{mult}(w) \le 1 + \lfloor (M - n)/(2\max(3^L, 2^{S(w)}))\rfloor$.
5. **$\{2, 3\}$-Free Half-Word Resultants on Balanced Shifts, Two-Sided Sturmian Envelope, and Odd-Stride Cyclotomic Pair-Matching (Theorem 5, Section 6):** Extending Theorem 13 of `collatz_22/v3` [1] to balanced-sum antipodally distinct shifts ($d_s = 0$ with $u_s \ne v_s$), the $\{2, 3\}$-free resultant $R_s^{\mp, \mathrm{red}}(a) \ge 1$ captures $\gcd(C(a), |2^{s_0} \pm 3^{r_0}|)$ on every primitive word of even $(k, S)$, with balanced antipodal states satisfying $x_s - x_{s+r_0} = (C(v_s) - C(u_s))/F_+$ and $x_s + x_{s+r_0} = (C(u_s) + C(v_s))/F_-$; balanced half-words obey the termwise Sturmian envelope $2^{-W^-(u)}C(u^{\mathrm{Sturm}}) \le C(u) \le 2^{W^+(u)}C(u^{\mathrm{Sturm}})$; and when $4 \mid \gcd(k, S)$, odd-stride pair-matching yields an infinite family $a(w, v, \Delta)$ with $(2^{s_0} + 3^{r_0}) \mid C(a)$ that is unconditionally excluded modulo $Q_2 = 2^{2 s_0} + 3^{2 r_0}$.
6. **Complementary-Factor Chebyshev Decomposition, Fixed-Slope $Q_d$-Fibre Finiteness, and 11-Pair All-$d$ Census (Theorem 6, Section 7):** For $d \mid \gcd(k, S)$ ($k = d r_0, S = d s_0$, $D = D_0 Q_d$), we prove the Chebyshev identity $d \cdot 2^{\delta_{s,-}} C(a^{(s)}) = \Sigma_s^+(a) Q_d + D_0 L_s^{(d)}(a)$, the exact complementary-factor divisibility $(Q_d / \gcd(d, D_0)) \mid L_s^{(d)}(a)$, and the exact gcd relations retaining $\gcd(d, D_0)$. Specializing the classical $3x + c$ correspondence (Lagarias [12, 13]) to $c = D_0$ on the quotient $q_i = C(a^{(i)})/Q_d$ with the cycle-product minimum-state bound $1 \le q_{\min} \le X_{\max}(r_0, s_0)$ [8] establishes uniform finiteness of the primitive $Q_d$-fibre across all $d \ge 1$ for each fixed reduced pair $(r_0, s_0)$, explains the minimal $Q_3$-orbits via Affine Fixed-Point Splicing ($q_{\min} = 9 \cdot 2^{s_0 - 4} - D_0$), and certifies the complete all-$d$ census across $11$ reduced pairs $(r_0, s_0)$ up to $(10, 16)$.

**Exact Reproducibility Scope (`code/verify_paper9.py`):** All finite checks in the accompanying pure-integer script `code/verify_paper9.py` (`26.43 s` runtime, `0` failures, using exact Python integer arithmetic with no floating-point logarithms or fractional powers) verify: (i) Theorems 1–3 on $88{,}777$ words, $975{,}826$ upper windows, $734{,}938$ lower windows, $3{,}229$ lower sparse numerators, and $85{,}353$ resonant windows; (ii) Theorem 4 on $6{,}270$ positive-denominator words ($k \le 9$) and $140{,}772$ subwords ($k \le 8$, plus $(1, 2, 2)$); (iii) Theorem 5 on $82{,}242$ primitive even words ($102{,}060$ balanced antipodal shifts, plus $(1, 1, 3, 1, 2, 2)$), all compositions $u \in \mathcal{A}(r_0, s_0)$ for $(6.4)$, and $1{,}500$ pair-matched words; and (iv) Theorem 6 and Table 1 by tracing all $169{,}756$ odd seeds $1 \le x_0 \le X_{\max}(r_0, s_0)$ to orbit completion across the $11$ reduced pairs $(r_0, s_0)$ and verifying $(7.2)$–$(7.3\text{a})$ on all $286$ primitive $d \ge 2$ orbits as well as $(1,1,1,1,1,5)^5$.

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## 2. The Universal Canonical-Root Width Theorem, the General-$\delta$ $m \ge 2$ Refinement, and Level-Set Decompositions

Let $a = (a_0, \dots, a_{k-1}) \in \mathbb{Z}_{\ge 1}^k$ be any positive valuation word with $k < S = \sum_{i=0}^{k-1} a_i < 2k$, and let $g = \gcd(k, S) \ge 1$, $k_0 = k/g$, $S_0 = S/g$. Following Theorem 5A of `collatz_22/v3` [1], a cyclic index $i_0 \in \{0, \dots, k-1\}$ is a **canonical Raney root** of $a$ if it maximizes the integer linear form $k T_i - i S$:
$$i_0 \in \operatorname{argmax}_{0 \le i < k}\bigl(k T_i - i S\bigr). \tag{2.1}$$
Replacing $a$ by its cyclic shift $a^{(i_0)}$, we assume henceforth that $i_0 = 0$, so $k T_i - i S \le 0$ for all $i \in \mathbb{Z}$, and the **canonical floor discrepancy profile**
$$z_i := \left\lfloor \frac{i S}{k} \right\rfloor - T_i \quad (i \in \mathbb{Z}) \tag{2.2}$$
is $k$-periodic ($z_{i+k} = z_i$) and non-negative ($z_i \ge 0$ for all $i \in \mathbb{Z}$), with canonical width $W = W(a) := \max_{0 \le i < k} z_i$.

For any window length $1 \le h < k$, define the **upper target sum** and **upper window determinant**
$$q := \left\lfloor \frac{h S}{k} \right\rfloor + 1, \qquad \delta := k q - h S \in \{1, 2, \dots, k\}, \tag{2.3}$$
together with the block-sum discrepancy $H_i := T_{i+h} - T_i - q$, the defect set $I := \{i \bmod k : H_i \ne 0\}$ of size $m := |I|$, and the exceptional floor-step set
$$J := \bigl\{i \bmod k : (i S \bmod k) < \delta\bigr\}, \qquad |J| = \delta. \tag{2.4}$$
(Because $g = \gcd(k, S)$ divides $\delta$, the map $i \mapsto (i S \bmod k)$ is a $g$-to-$1$ surjection from $\mathbb{Z}/k\mathbb{Z}$ onto $g\mathbb{Z}/k\mathbb{Z}$, so $|J| = g \cdot (\delta/g) = \delta$.) Define also the floor-step image cardinalities and maximum on $J$:
$$V_J := \#\{z_j : j \in J\}, \qquad V_{J \setminus I} := \#\{z_j : j \in J \setminus I\}, \qquad W_J := \max_{j \in J} z_j. \tag{2.5}$$

**Theorem 1 (Universal Canonical-Root Width, General-$\delta$ $m \ge 2$ Refinement, and Level-Set Decompositions; extending Theorem 5A of `collatz_22/v3` [1]).** *For every positive valuation word $a \in \mathbb{Z}_{\ge 1}^k$ ($k < S < 2k$) at a canonical root $z_0 = 0$ and every upper window $1 \le h < k$ with $q = \lfloor h S / k \rfloor + 1$ and $\delta = k q - h S \in \{1, \dots, k\}$:*
1. **(Universal interval image and root anchor for all $g = \gcd(k, S) \ge 1$):** *Regardless of $g = \gcd(k, S) \ge 1$, the canonical profile $z_i = \lfloor i S / k \rfloor - T_i$ satisfies $z_0 = z_1 = 0$ and*
   $$\{z_0, z_1, \dots, z_{k-1}\} = \{0, 1, \dots, W\}. \tag{2.6}$$
2. **(Exact floor-step shift identity and mandatory overlap at $i = 0$):** *For every $i \in \mathbb{Z}/k\mathbb{Z}$,*
   $$z_{i+h} - z_i = -H_i - \mathbf{1}_{\{i \in J\}}. \tag{2.7}$$
   *Moreover, $0 \in I \cap J$ with $H_0 = -1 - z_h \le -1$, so $|I \cap J| \ge 1$ and $m \ge 1$.*
3. **(Exact backward-entering and forward-leaving level-set decompositions & one-sided defect balance):** *Decomposing $\mathbb{Z}/k\mathbb{Z}$ into the $d = \gcd(h, k)$ coset cycles $\mathcal{O}_c = \{c + r h \bmod k : 0 \le r < k/d\}$ ($0 \le c < d$), every coset cycle $\mathcal{O}_c$ satisfies $\sum_{i \in \mathcal{O}_c} H_i = -\delta/d < 0$ (hence $I \cap \mathcal{O}_c \ne \emptyset$, $m \ge \gcd(h, k)$, and $\sum_{i \in I^-} (-H_i) = \sum_{i \in I^+} H_i + \delta$ where $I^+ := \{i \in I : H_i > 0\}$ and $I^- := \{i \in I : H_i < 0\}$), and the value set $\{0, 1, \dots, W\}$ decomposes in both backward-entering and forward-leaving forms as*
   $$\{0, 1, \dots, W\} = \bigl\{z_{i+h} : i \in I\bigr\} \cup \bigl\{z_j - 1 : j \in J \setminus I\bigr\} = \bigl\{z_y : y \in I \cup J\bigr\}. \tag{2.8}$$
4. **(Indicator-image width bound and exact $m \ge 2$ refinement):** *For all $m \ge 1$,*
   $$V_{J \setminus I} \le \min\bigl(V_J - 1,\; W_J,\; \delta - |I \cap J|\bigr), \tag{2.9}$$
   $$W \le m + V_{J \setminus I} - 1 \le m + \min\bigl(V_J - 1,\; W_J,\; \delta - |I \cap J|\bigr) - 1 \le m + \delta - 1 - |I \cap J| \le m + \delta - 2, \tag{2.10}$$
   *and equality $W = m + \delta - 2$ can occur only when $m = 1$; for all $m \ge 2$,*
   $$W \le m + \delta - 3 \qquad (\text{while } W \le \delta - 1 \text{ for } m = 1). \tag{2.11}$$
   *In particular, for $\gcd(k, S) = 1$ and $\delta = 1$ (`collatz_24/v1` [2], Lemma 2), $W = 0$ for $m = 1$ and $W \le m - 2$ for all $m \ge 2$ (as recorded in Theorem 5A of `collatz_22/v3` [1]), cancelling the $+2$ constant in equation (10) of `collatz_22/v3` [1] for all $m \ge 2$ and improving $B(S) = 1 + (A_* + 1)\log_2 S$ in Theorem 3 of `collatz_24/v1` [2] to $B_{\mathrm{sharp}}(S) = (A_* + 1)\log_2 S$. At the reduced right Farey parent $k_0 q_+ - h_+ S_0 = 1$ ($\delta_+ = g$), $J = \{0, k_0, 2 k_0, \dots, (g-1)k_0\}$ is the set of $g$ cyclotomic $k_0$-block boundaries, $W_J = \delta_{0, -} := \max_{0 \le b < g}(b S_0 - T_{b k_0})$, $V_J \le \max(1, m_{0, 0})$ where $m_{0, 0} := \#\{0 \le b < g : T_{(b+1)k_0} - T_{b k_0} \ne S_0\}$, and every word with $\delta_{0, -} = 0$ at the canonical root satisfies $W \le m_+ - 1$ ($W \le m_+ - 2$ for $m_+ \ge 2$) with zero $+g$ offset.*

**Proof.**
**(1):** By (2.1), $z_i \ge 0$ for all $i \in \mathbb{Z}$, and $z_0 = 0$. Since $a_0 \ge 1$ and $1 < S/k < 2$, we could only have $a_0 \ge 2$ if $k T_1 - S = k a_0 - S \ge 2k - S > 0 = k T_0 - 0 \cdot S$, contradicting maximality of $k T_i - i S$ at $i = 0$. Hence $a_0 = 1$, and because $\lfloor S/k \rfloor = 1$, we have $z_1 = \lfloor S/k \rfloor - a_0 = 1 - 1 = 0$. For any $i \in \mathbb{Z}$, since $a_i \ge 1$ and $\lfloor (i+1)S/k \rfloor - \lfloor i S/k \rfloor \in \{1, 2\}$, the forward step satisfies
$$z_{i+1} - z_i = \left(\left\lfloor \frac{(i+1)S}{k} \right\rfloor - \left\lfloor \frac{i S}{k} \right\rfloor\right) - a_i \le 2 - 1 = 1. \tag{2.12}$$
Because $z_0 = 0$, $z_k = 0$, and $z$ increases by at most $+1$ at each unit step $i \mapsto i + 1$, by discrete intermediate value it must hit every integer in $\{0, 1, \dots, W\}$ on $\{0, 1, \dots, k-1\}$, proving (2.6).

**(2):** Write $i S = k \lfloor i S/k \rfloor + r_i$ with $r_i = (i S \bmod k) \in \{0, \dots, k-1\}$, and substitute $h S = k q - \delta$:
$$\left\lfloor \frac{(i+h)S}{k} \right\rfloor = \left\lfloor \frac{i S}{k} \right\rfloor + q + \left\lfloor \frac{r_i - \delta}{k} \right\rfloor = \left\lfloor \frac{i S}{k} \right\rfloor + q - \mathbf{1}_{\{r_i < \delta\}},$$
since $-\delta \le r_i - \delta \le k - 1 - \delta$ lies in $[-k, -1]$ if and only if $r_i < \delta$ ($i \in J$), and in $[0, k-1]$ otherwise. Subtracting $T_{i+h} = T_i + q + H_i$ gives (2.7). At $i = 0$, $r_0 = 0 < \delta$ so $0 \in J$, and (2.7) gives $z_h - 0 = -H_0 - 1$, so $H_0 = -1 - z_h \le -1 \ne 0$ since $z_h \ge 0$. Thus $0 \in I \cap J$ and $|I \cap J| \ge 1$.

**(3):** For each coset cycle $\mathcal{O}_c = \{c + r h \bmod k : 0 \le r < k/d\}$ ($d = \gcd(h, k)$), summing $T_{i+h} - T_i = q + H_i$ over $i \in \mathcal{O}_c$ telescopes to $\frac{h}{d} S = \frac{k}{d} q + \sum_{i \in \mathcal{O}_c} H_i$, so $\sum_{i \in \mathcal{O}_c} H_i = -\delta/d < 0$, $I \cap \mathcal{O}_c \ne \emptyset$, and summing over all $c$ gives $\sum_{i \in I^-} (-H_i) = \sum_{i \in I^+} H_i + \delta$. Now let $v \in \{0, 1, \dots, W\}$. By (1), there is some $x \in \{0, \dots, k-1\}$ with $z_x = v$; let $\mathcal{O}_c$ be the coset cycle containing $x$. If $z$ is constant on $\mathcal{O}_c$, then $z_{i+h} = z_i = v$ for any $i \in I \cap \mathcal{O}_c \ne \emptyset$. Otherwise, walking backwards along $\mathcal{O}_c$ ($x, x - h, x - 2h, \dots \pmod k$) from $x$ reaches a first predecessor $i \in \mathcal{O}_c$ with $z_{i+h} = v \ne z_i$, so $i \in I \cup (J \setminus I)$ and $v = z_{i+h}$ (if $i \in I$) or $v = z_i - 1$ (if $i \in J \setminus I$); symmetrically, walking forwards along $\mathcal{O}_c$ from $x$ reaches a last vertex $y \in \mathcal{O}_c$ with $z_y = v \ne z_{y+h}$, so $y \in I \cup J$ and $v = z_y$. This proves both decompositions in (2.8).

**(4):** For every $j \in J \setminus I$, $z_j = z_{j+h} + 1 \ge 1$, whereas $0 = z_0 \in \{z_j : j \in J\}$ (since $0 \in I \cap J$). Thus $\{z_j : j \in J \setminus I\}$ is a subset of $\{z_j : j \in J\} \setminus \{0\}$, a subset of $\{1, 2, \dots, W_J\}$, and has at most $|J \setminus I| = \delta - |I \cap J|$ elements, proving (2.9). Taking cardinalities in (2.8) gives $W + 1 \le |I| + \#\{z_j - 1 : j \in J \setminus I\} = m + V_{J \setminus I}$, which yields (2.10).

To prove (2.11), suppose $W = m + \delta - 2$, so $W + 1 = m + \delta - 1$. Then every inequality in (2.10) is an equality: $I \cap J = \{0\}$, the orbit value sets $\{z_i : i \in \mathcal{O}_c\}$ are pairwise disjoint, and $V_c := \#\{z_i : i \in \mathcal{O}_c\} = |(I \cup J) \cap \mathcal{O}_c|$ for each $c$. Since $z_0 = 0$ and $z_1 = 0$ share the value $0$, the indices $0$ and $1$ must lie in the same $h$-shift orbit $\mathcal{O}_0$, forcing $d = \gcd(h, k) = 1$. Let $i_r = r h \bmod k$ and $P_r = z_{i_r}$ ($0 \le r < k$), and let $R \in \{1, \dots, k-1\}$ be the unique integer with $1 + R h = u k$ ($u \ge 1$), so $i_r + 1 \equiv i_{r - R} \pmod k$ and $P_{k - R} = z_1 = 0$.

Cutting $\mathbb{Z}/k\mathbb{Z}$ at the $|U| = m + \delta - 1$ edges $r \to r + 1$ with $i_r \in U := I \cup J$ produces $|U|$ constant plateaux of $P$, which must take pairwise distinct heights in $\{0, 1, \dots, W\}$ since $V_0 = |U|$. Because $P_0 = P_{k - R} = 0$ and $i_0 = 0 \in U$, the unique plateau $C_0$ of height $0$ must be $[k - M_0, 0]$ with $M_0 \ge R$, and $[k - M_0, k - 1] \cap U = \emptyset$. Summing the mechanical floor step $\lfloor i_{s+1} S / k \rfloor - \lfloor i_s S / k \rfloor = q - \mathbf{1}_{\{i_s \in J\}}$ over any $R$ consecutive $h$-steps $[r - R, r - 1]$ gives
$$1 + b_{i_r} := \left\lfloor \frac{(i_r + 1)S}{k}\right\rfloor - \left\lfloor \frac{i_r S}{k}\right\rfloor = (u S - R q) + \sum_{s = r - R}^{r - 1} \mathbf{1}_{\{i_s \in J\}} \in \{1, 2\}. \tag{2.13}$$
Evaluating at $r = 0$ (where $b_0 = 0$ and $[k - R, k - 1] \cap J = \emptyset$) yields $u S - R q = 1$, and hence for all $r \in \mathbb{Z}/k\mathbb{Z}$:
$$P_{r - R} - P_r \le b_{i_r} = \sum_{s = r - R}^{r - 1} \mathbf{1}_{\{i_s \in J\}} \in \{0, 1\}. \tag{2.14}$$
By induction on $t = 0, 1, \dots$, moving leftward from $r = 0$, each successive plateau $C_t = [k - M_t, k - M_{t-1} - 1]$ ($M_{-1} = 0$, $M_t - M_{t-1} \ge R$, $C_t \cap U = \emptyset$) has height $t$: if $k - M_t > 1$, the boundary edge $e_t = k - M_t - 1 \in U$ satisfies $P_{e_t} - P_{e_t + R} \le b_{i_{e_t + R}} = \mathbf{1}_{\{e_t \in J\}}$ (since $e_t + 1, \dots, e_t + R - 1 \in C_t \subseteq (\mathbb{Z}/k\mathbb{Z}) \setminus J$). Because heights $0, \dots, t$ are already occupied by $C_0, \dots, C_t$, distinctness of plateau heights requires $P_{e_t} \ge t + 1$, which forces $i_{e_t} \in J \setminus \{0\} = J \setminus I$ and $P_{e_t} = t + 1$. Since $b_{i_r} \le 1$, at most one element of any $R$ consecutive residues lies in $J$, so $\{e_t - R + 1, \dots, e_t - 1\} \cap J = \emptyset$, and for every $s \in \{e_t - R + 1, \dots, e_t\}$ we have $P_s \le P_{s + R} + 1 = t + 1$; distinctness of plateau heights again forces all of $\{e_t - R + 1, \dots, e_t\}$ to belong to the single plateau $C_{t+1}$ of height $t + 1$, with $M_{t+1} - M_t \ge R$. When the induction reaches $r = 1$, every cut $e_t \in U \setminus \{0\}$ lies in $J \setminus I$, forcing $I = \{0\}$ and $m = 1$. $\blacksquare$

**Corollary 2.1 (Indicator-Image & General-$\delta$ $m \ge 2$ Upgrade of Theorems 2, 5, and 5A of `collatz_22/v3` [1]).** *Substituting Theorem 1 ($W \le m + V_{J \setminus I} - 1 \le m + \delta - 2$, or $W \le m + \delta - 3$ for $m \ge 2$) directly into equation (10) of `collatz_22/v3` [1] ($\frac{S}{m} < W + 2 + \log_2 m - \log_2(1 - 3^k/2^S)$) gives, for every nontrivial positive integer-cycle presentation and every upper window $1 \le h < k$ with $\delta = k q - h S = g j$ ($j \in \{1, \dots, k_0\}$),*
$$S < m(m + V_{J \setminus I} + 1) + m\log_2 m - m\log_2(1 - 3^k / 2^S) \le m(m + \delta) + m\log_2 m + A m\log_2 S, \tag{2.15}$$
*where $A > 0$ is Luca's one-sided lower-bound exponent (`collatz_22/v3` [1], equation (11), $2^S - 3^k > 2^S S^{-A}$). Consequently, for an effective constant $C > 0$ depending only on $A$,*
$$k < S \le C m(m + V_{J \setminus I})\log(k+1) \le C m(m + \delta)\log(k+1), \qquad k \le C \max\!\bigl(m^2,\, (j m)^{A+1}\bigr)\log(k+1), \tag{2.16}$$
*and whenever $\delta = o(\sqrt{S})$, $m \ge \sqrt{S + \delta^2/4} - \delta/2 - O(\log S) = \sqrt{S}(1 - o(1))$.*


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## 3. Dual Lower-Window Canonical Width, Sign-Split Sparse-Numerator Bound ($W + 1 + \delta_-/k$), Farey Cocycle, and Resonant Period-$k_0$ Sparse Numerator $R_0(a)$

For any window length $1 \le h < k$ with $k_0 \nmid h$, define the **lower target sum** and **lower window determinant**
$$q_- := \left\lfloor \frac{h S}{k} \right\rfloor, \qquad \delta_- := h S - k q_- \in \{1, 2, \dots, k - 1\}, \tag{3.1}$$
together with the lower-window discrepancy $H_i^- := T_{i+h} - T_i - q_-$, the lower defect set $I_- := \{i \bmod k : H_i^- \ne 0\}$ of size $m_- := |I_-|$ (partitioned into $I_-^+ := \{i \in I_- : H_i^- > 0\}$ of size $m_-^+$ and $I_-^- := \{i \in I_- : H_i^- < 0\}$ of size $m_-^-$), and the lower exceptional floor-step set
$$J_- := \bigl\{i \bmod k : (i S \bmod k) \ge k - \delta_-\bigr\}, \qquad |J_-| = \delta_-, \qquad V_{J_- \setminus I_-} := \#\{z_j : j \in J_- \setminus I_-\}. \tag{3.2}$$

**Theorem 2 (Lower-Window Canonical Width, Sign-Split Sparse Numerator, Farey Cocycle, and Resonant Window).**
1. **(Lower-window shift identity and canonical width bound at the same root $z_0 = 0$):** *At the canonical root $z_0 = 0$,*
   $$z_{i+h} - z_i = -H_i^- + \mathbf{1}_{\{i \in J_-\}}, \qquad i^* := (-h) \bmod k \in I_-^+ \cap J_- \quad (H_{-h}^- = 1 + z_{-h} \ge 1), \tag{3.3}$$
   *and across all $m_- \ge 1$:*
   $$\{0, 1, \dots, W\} = \bigl\{z_{i+h} : i \in I_-\bigr\} \cup \bigl\{z_j + 1 : j \in J_- \setminus I_-\bigr\} = \bigl\{z_y : y \in I_- \cup J_-\bigr\}, \tag{3.4}$$
   $$W \le m_- + V_{J_- \setminus I_-} - 1 \le m_- + \delta_- - 1 - |I_- \cap J_-| \le m_- + \delta_- - 2 \qquad (\text{with } W \le m_- + \delta_- - 3 \text{ for all } m_- \ge 2). \tag{3.5}$$
2. **(Sign-split and tighter lower-window sparse numerator bound — saving $1 - \delta_-/k$ bits and replacing $m_-$ by $\max(m_-^+, m_-^-)$):** *Let $[A, B]$ be a shortest cyclic arc containing $I_-$ ($w_- = B - A \le k(1 - 1/m_-)$, $E_- = \min(q_- + T_A, T_{A+h})$), and let $R_-$ be the odd part of $\widetilde{R}_- := \sum_{j \in [A, B], j \bmod k \in I_-} 3^{B-j} 2^{T_j - E_-}(2^{q_- + H_j^-} - 2^{q_-})$. Because $2^{q_- + H_j^-} - 2^{q_-}$ is positive on $I_-^+$ ($m_-^+$ terms) and negative on $I_-^-$ ($m_-^-$ terms), $\widetilde{R}_- = X_+ - X_-$ with $X_+ > 0, X_- \ge 0$ and $|\widetilde{R}_-| \le \max(X_+, X_-)$ (and similarly for upper windows with $\max(m^+, m^-)$), while $q_- = h\lambda - \delta_-/k < h\lambda$ ($\lambda = S/k$) saves $1 - \delta_-/k$ bits:*
   $$\gcd(R_-, D) = \gcd\bigl((2^{q_-} - 3^h)C(a),\, D\bigr), \qquad 1 \le |R_-| < \max(m_-^+, m_-^-)\,2^{W + 1 + (\delta_- / k) + w_- S / k} \le m_-\,2^{W + 1 + (\delta_-/k) + S(1 - 1/m_-)}. \tag{3.6}$$
   *Consequently, for $k_0 \ge 2$ at the reduced left and right Farey parents ($m_{\min} := \min(m_+, m_-)$), every positive integer cycle satisfies $W \le m_{\min} + g - 2$ and $\frac{S}{m_{\min}} < m_{\min} + g + \log_2 m_{\min} - \log_2(1 - 3^k/2^S)$.*
3. **(Noncoprime Left + Right Farey Parent Cocycle Identity & Resonant Period-$k_0$ Sparse Numerator $R_0(a)$):** *For $g = \gcd(k, S) \ge 2$, the resonant window $(h_0, q_0) = (k_0, S_0)$ has determinant $\delta_0 = k S_0 - k_0 S = 0$, carry-free shift identity $z_{i+k_0} - z_i = -H_i^{(k_0, S_0)}$ for all $0 \le i < k$, and exact Farey cocycle decomposition*
   $$H_i^{(h_+, q_+)} + H_{i + h_+}^{(h_-, q_-)} = T_{i + k_0} - T_i - S_0 = H_i^{(k_0, S_0)}, \tag{3.7}$$
   *where $(h_+, q_+)$ ($k_0 q_+ - h_+ S_0 = 1$) and $(h_-, q_-)$ ($h_- S_0 - k_0 q_- = 1$, $h_+ + h_- = k_0$, $q_+ + q_- = S_0$) are the reduced right and left Farey parents of $(k_0, S_0)$. For every primitive word $a \in \mathcal{A}^{\mathrm{prim}}(g k_0, g S_0)$, $I_0 := \{i \bmod k : H_i^{(k_0, S_0)} \ne 0\}$ has $m_0^+ \ge 1$ positive and $m_0^- \ge 1$ negative defects ($m_0 = m_0^+ + m_0^- \ge 2$), and its odd nonzero sparse numerator $R_0(a)$ on the shortest arc $[A_0, B_0]$ of $I_0$ ($w_0 = B_0 - A_0 \le k(1 - 1/m_0)$) satisfies $\gcd(C(a), Q_g) \mid \gcd(R_0(a), Q_g)$ ($Q_g = \frac{2^S - 3^k}{2^{S_0} - 3^{k_0}}$) with the sign-split $(W + 1)$ bound:*
   $$1 \le |R_0(a)| < \max(m_0^+, m_0^-)\,2^{W + 1 + w_0 S / k} \le (m_0 - 1)\,2^{W + 1 + S(1 - 1/m_0)}. \tag{3.8}$$

**Proof.**
**(1):** Writing $i S = k \lfloor i S/k \rfloor + r_i$ and $h S = k q_- + \delta_-$, we have $\lfloor (i+h)S/k \rfloor = \lfloor i S/k \rfloor + q_- + \mathbf{1}_{\{r_i + \delta_- \ge k\}}$, which subtracting $T_{i+h} = T_i + q_- + H_i^-$ yields (3.3). At $i^* = (-h) \bmod k$, $((i^* + h)S \bmod k) = 0$ implies $(i^* S \bmod k) = k - \delta_-$, so $i^* \in J_-$, and $0 - z_{-h} = -H_{-h}^- + 1 \implies H_{-h}^- = 1 + z_{-h} \ge 1 \ne 0$, so $i^* \in I_-^+ \cap J_-$. Walking backwards and forwards along $+h$-coset cycles as in Theorem 1(3)–(4) gives (3.4) and (3.5).

**(2):** On the shortest cyclic arc $[A, B]$ of $I_-$ at the canonical root $z_0 = 0$, $T_j = j\lambda - \{j\lambda\} - z_j$ ($\lambda = S/k$), so $E_- = \min(q_- + T_A, T_{A+h}) \ge A\lambda + h\lambda - \frac{\delta_-}{k} - 1 - W$. For each $j \in [A, B]$ with $j \bmod k \in I_-$, $\max(q_- + T_j, T_{j+h}) < (j + h)\lambda$, so each positive term ($j \bmod k \in I_-^+$) and each negative term ($j \bmod k \in I_-^-$) has magnitude strictly bounded by $3^{B-j} 2^{(j-A)\lambda + W + 1 + \delta_-/k} < 2^{(B-A)\lambda + W + 1 + \delta_-/k}$. Since $|\widetilde{R}_-| = |X_+ - X_-| \le \max(X_+, X_-)$, summing the positive and negative parts separately gives the prefactor $\max(m_-^+, m_-^-) \le m_-$, proving (3.6).

**(3):** Since $h_+ + h_- = k_0$ and $q_+ + q_- = S_0$, we have $(T_{i+h_+} - T_i - q_+) + (T_{i+k_0} - T_{i+h_+} - q_-) = T_{i+k_0} - T_i - S_0$, proving (3.7). Since $\sum_{i=0}^{k-1} H_i^{(k_0, S_0)} = 0$ and $H^{(k_0, S_0)}$ is not identically zero on any primitive word, both $I_0^+$ and $I_0^-$ are non-empty ($m_0^+ \ge 1, m_0^- \ge 1$, $\max(m_0^+, m_0^-) \le m_0 - 1$), and substituting $\delta_0 = 0$ into (3.6) gives (3.8). $\blacksquare$

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## 4. The Defect-Free Eulerian Farey Digraph $\mathcal{G}_{\mathrm{Farey}}$, Path-Saturation Width Bounds, and Fubini–Farey Incompatibility (Complete Elimination of $+g$)

Why can a single Farey window $(h_+, q_+)$ suffer a $+g$ width offset on staircase words (where $m_+ = 1$ and $W = g - 1$), and how do we eliminate $+g$ **unconditionally for all $g = \gcd(k, S) \ge 1$**?

For $k_0 = k/g \ge 2$ and $S_0 = S/g$, let $(h_+, q_+)$ ($k_0 q_+ - h_+ S_0 = 1$) and $(h_-, q_-)$ ($h_- S_0 - k_0 q_- = 1$, $h_+ + h_- = k_0$, $q_+ + q_- = S_0$) be the reduced right and left Farey parents of $(k_0, S_0)$, with exceptional sets $J = \{b k_0 : 0 \le b < g\}$ and $J_- = \{b k_0 + h_+ \bmod k : 0 \le b < g\}$. Since $1 \le h_+ < k_0$, **$J \cap J_- = \emptyset$**. Define the **Defect-Free Eulerian Farey Digraph** $\mathcal{G}_{\mathrm{Farey}} = (\mathbb{Z}/k\mathbb{Z}, E_+ \cup E_-)$ by avoiding both exceptional sets:
$$E_+ := \bigl\{x \xrightarrow{+h_+} (x + h_+) \bmod k : x \in \mathbb{Z}/k\mathbb{Z} \setminus J\bigr\}, \qquad E_- := \bigl\{x \xrightarrow{+h_-} (x + h_-) \bmod k : x \in \mathbb{Z}/k\mathbb{Z} \setminus J_-\bigr\}. \tag{4.1}$$

**Theorem 3 (Defect-Free Eulerian Farey Digraph, Path-Saturation Width Bounds, and Fubini–Farey Incompatibility).**
1. **(Zero Floor Defect & Strongly Connected Fibre Ladder):** *Partitioning $\mathbb{Z}/k\mathbb{Z}$ into the $k_0$ fibres $R_u := \{x \in \mathbb{Z}/k\mathbb{Z} : (x S_0) \bmod k_0 = u\} = \{b k_0 + u h_- \bmod k : 0 \le b < g\}$ ($0 \le u \le k_0 - 1$, where $R_0 = J$ and $R_{k_0 - 1} = J_-$), $E_-$ restricts to a bijection $+h_- : R_u \to R_{u+1}$ and $E_+$ restricts to a bijection $+h_+ : R_{u+1} \to R_u$ for every $0 \le u \le k_0 - 2$, whose round-trip $+h_+ \circ +h_- : R_u \to R_u$ is the single $g$-cycle $x \mapsto (x + k_0) \bmod k$. Thus $\mathcal{G}_{\mathrm{Farey}}$ is Eulerian and strongly connected ($\operatorname{indeg}(v) = \operatorname{outdeg}(v) \in \{1, 2\}$), and every edge of $\mathcal{G}_{\mathrm{Farey}}$ has **zero floor defect**:*
   $$z_{x+h_+} - z_x = -H_x^{(h_+, q_+)} \quad (\forall x \notin J), \qquad z_{x+h_-} - z_x = -H_x^{(h_-, q_-)} \quad (\forall x \notin J_-). \tag{4.2}$$
2. **(Exact Dual-Parent Level-Set Identity for $W \ge 1$, One-Sided $L^1$ Variation Bounds, & $g$-Free Canonical Width Bound):** *For every word $a \in \mathcal{A}(k, S)$ with canonical width $W \ge 1$, every level set $Z_v = \{y \in \mathbb{Z}/k\mathbb{Z} : z_y = v\}$ ($0 \le v \le W$) is a non-empty proper subset of $\mathbb{Z}/k\mathbb{Z}$, so in the strongly connected digraph $\mathcal{G}_{\mathrm{Farey}}$ it has at least one leaving edge $y \to x$ ($z_y = v \ne z_x$) and at least one entering edge $x' \to y'$ ($z_{x'} \ne v = z_{y'}$). Every edge of $\mathcal{G}_{\mathrm{Farey}}$ with $z_{\mathrm{head}} \ne z_{\mathrm{tail}}$ has its tail in $U_{\mathrm{Farey}} := (I_+ \setminus J) \cup (I_- \setminus J_-)$, and since $y \mapsto z_y$ is single-valued on vertices,*
   $$\{0, 1, \dots, W\} = \bigl\{z_y : y \in (I_+ \setminus J) \cup (I_- \setminus J_-)\bigr\} = \bigl\{z_{x+h_+} : x \in I_+ \setminus J\bigr\} \cup \bigl\{z_{x+h_-} : x \in I_- \setminus J_-\bigr\} \quad (W \ge 1). \tag{4.3}$$
   *Since $0 \in I_+ \cap J$ and $(-h_-) \bmod k \in I_- \cap J_-$, for every word with $W \ge 1$ this yields the sharp, **$g$-independent** width bound (tight on $247$ words in `code/verify_paper9.py`) as well as the universal bound $W \le \max(0, m_+ + m_- - 3)$ and the one-sided $L^1$ variation bound along simple directed paths $0 \leadsto i_{\max}$ and $i_{\max} \leadsto 0$ in $\mathcal{G}_{\mathrm{Farey}}$:*
   $$W \le |(I_+ \setminus J) \cup (I_- \setminus J_-)| - 1 \le (m_+ - |I_+ \cap J|) + (m_- - |I_- \cap J_-|) - 1 \le m_+ + m_- - 3 \quad (W \ge 1), \tag{4.4}$$
   $$W \le \min\left(\sum_{x \in I_+^- \setminus J} \bigl(-H_x^{(h_+, q_+)}\bigr) + \sum_{x \in I_-^- \setminus J_-} \bigl(-H_x^{(h_-, q_-)}\bigr),\;\; \sum_{y \in I_+^+ \setminus J} H_y^{(h_+, q_+)} + \sum_{y \in I_-^+ \setminus J_-} H_y^{(h_-, q_-)}\right). \tag{4.4a}$$
3. **(Path-Saturation & Fibre-Pigeonhole Width Bounds):** *The $k - g$ edges of $E_+$ partition $\mathbb{Z}/k\mathbb{Z}$ into $g$ vertex-disjoint directed paths $C_b = \{b k_0 + v h_+ \bmod k : 1 \le v \le k_0\}$ ($0 \le b < g$), where each $C_b \setminus J_-$ has $k_0 - 1$ vertices (one in each fibre $R_u$, $0 \le u \le k_0 - 2$) whose outgoing $E_-$ edges land in $C_{(b+1)\bmod g} \setminus J$. Hence linking $C_b$ to $C_{(b+1)\bmod g}$ costs zero change in $z$ unless $C_b \setminus J_-$ is **completely saturated by $I_-$** ($C_b \setminus J_- \subseteq I_-$). Letting $B_{\mathrm{full}}^- := \{b \in \mathbb{Z}/g\mathbb{Z} : C_b \setminus J_- \subseteq I_-\}$ and $B_{\mathrm{full}}^+ := \{b \in \mathbb{Z}/g\mathbb{Z} : C_b^- \setminus J \subseteq I_+\}$,*
   $$W \le (m_+ - |I_+ \cap J|) + \min\Bigl(|B_{\mathrm{full}}^-|,\; \min_{0 \le u \le k_0 - 2} |I_- \cap R_u|,\; g - 1\Bigr) \le m_+ - 1 + \min\left(\left\lfloor \frac{m_- - 1}{k_0 - 1} \right\rfloor,\; g - 1\right), \tag{4.5}$$
   $$W \le (m_- - |I_- \cap J_-|) + \min\Bigl(|B_{\mathrm{full}}^+|,\; \min_{1 \le u \le k_0 - 1} |I_+ \cap R_u|,\; g - 1\Bigr) \le m_- - 1 + \min\left(\left\lfloor \frac{m_+ - 1}{k_0 - 1} \right\rfloor,\; g - 1\right). \tag{4.6}$$
4. **(Fubini–Farey & Arithmetic-Progression Incompatibility Identities):** *Swapping the double sum $\sum_{r=0}^{h_--1} \sum_{s=0}^{h_+-1} a_{i+r+s}$ (and telescoping $T_{i+h_+ h_-} - T_i$) with $h_- q_+ - h_+ q_- = 1$ gives the pointwise identities*
   $$\sum_{s=0}^{h_+-1} H_{i+s}^{(h_-, q_-)} - \sum_{r=0}^{h_--1} H_{i+r}^{(h_+, q_+)} = 1 \quad \text{and} \quad \sum_{v=0}^{h_+-1} H_{i+v h_-}^{(h_-, q_-)} - \sum_{u=0}^{h_--1} H_{i+u h_+}^{(h_+, q_+)} = 1 \qquad (\forall i \in \mathbb{Z}/k\mathbb{Z}), \tag{4.7}$$
   *whence summing over $i \in \mathbb{Z}/k\mathbb{Z}$ yields $h_- m_+ + h_+ m_- \ge k = g k_0$, $h_- \|H^{(h_+, q_+)}\|_1 + h_+ \|H^{(h_-, q_-)}\|_1 \ge k$, and $\max(m_+, m_-) \ge g$.*
5. **(Complete $g$-Free Upgrade of `collatz_22/v3` [1] Theorems 2, 5, and 5A):** *Applying the sparse-numerator bound to $m_{\min} = \min(m_+, m_-)$ with the $g$-free width bound $W \le \max(0, m_+ + m_- - 3)$ of (2)–(3) and Luca's denominator bound [11, p. 36] yields, for all positive integer cycles with $k_0 \ge 2$ and arbitrary $g = \gcd(k, S) \ge 1$,*
   $$k < S \le C \cdot \min(m_+, m_-) \cdot (m_+ + m_-) \log(k+1), \qquad k \le C \cdot \min(m_+, m_-) \cdot (m_+ + m_-)^A \log(k+1) \le C \cdot (m_+ + m_-)^{A+1} \log(k+1), \tag{4.8}$$
   *eliminating the $+g$ noncoprime offset completely in terms of the dual-parent defect count $m_+ + m_-$.*

**Proof.**
**(1)–(2):** By (2.7) and (3.3), the indicator terms $-\mathbf{1}_{\{x \in J\}}$ and $+\mathbf{1}_{\{x \in J_-\}}$ vanish identically on $\mathbb{Z}/k\mathbb{Z} \setminus J$ and $\mathbb{Z}/k\mathbb{Z} \setminus J_-$, giving (4.2). Since $h_- S_0 \equiv 1 \pmod{k_0}$ and $h_+ S_0 \equiv -1 \pmod{k_0}$, $+h_-$ maps $R_u$ bijectively onto $R_{u+1}$ for $0 \le u \le k_0 - 2$ and $+h_+$ maps $R_{u+1}$ bijectively onto $R_u$, while $+h_+ \circ +h_- = +k_0$ acts on each $R_u$ as a single $g$-cycle. Thus $\mathcal{G}_{\mathrm{Farey}}$ is strongly connected and Eulerian. For $W \ge 1$ and any $v \in \{0, \dots, W\}$, strong connectivity ensures an edge $y \to x$ leaving $Z_v$ ($z_y = v \ne z_x$) and an edge $x' \to y'$ entering $Z_v$ ($z_{x'} \ne v = z_{y'}$). By (4.2), any edge with $z_{\mathrm{head}} \ne z_{\mathrm{tail}}$ has tail in $U_{\mathrm{Farey}}$, proving (4.3) and (4.4). Moreover, along any simple directed path in $\mathcal{G}_{\mathrm{Farey}}$ from $0$ ($z_0 = 0$) to $i_{\max}$ ($z_{i_{\max}} = W$), the sum of $z_{\mathrm{head}} - z_{\mathrm{tail}}$ equals $W$ and positive increments come only from edges with tail in $(I_+^- \setminus J) \cup (I_-^- \setminus J_-)$, while along any simple directed path from $i_{\max}$ back to $0$, downward decrements come only from edges with tail in $(I_+^+ \setminus J) \cup (I_-^+ \setminus J_-)$, proving (4.4a).

**(3):** Each path $C_b = \{b k_0 + v h_+ \bmod k : 1 \le v \le k_0\}$ has $k_0 - 1$ internal edges in $E_+$. For each $x \in C_b \setminus J_-$ ($2 \le v \le k_0$), $x + h_- = x + k_0 - h_+ \in C_{(b+1)\bmod g} \setminus J$. Selecting one linking edge $e_b = (x_b \to x_b + h_-) \in E_-$ for each $0 \le b \le g - 2$ connects $C_0, \dots, C_{g-1}$ into a spanning tree of $\mathbb{Z}/k\mathbb{Z}$; we can choose $e_b$ inactive ($x_b \notin I_-$) whenever $C_b \setminus J_- \not\subseteq I_-$ ($b \notin B_{\mathrm{full}}^-$). Similarly, for each fixed $0 \le u \le k_0 - 2$, $E_+ \cup E_-|_{R_u}$ is a connected unicyclic graph on $\mathbb{Z}/k\mathbb{Z}$ whose unique cycle contains the $g$ edges of $E_-|_{R_u}$. Since $\sum_{u=0}^{k_0 - 2} |I_- \cap R_u| = |I_- \setminus J_-| \le m_- - 1$, the pigeonhole principle gives $\min_{0 \le u \le k_0 - 2} |I_- \cap R_u| \le \lfloor (m_- - 1)/(k_0 - 1) \rfloor$, proving (4.5) and (by symmetry) (4.6).

**(4)–(5):** For every $i \in \mathbb{Z}/k\mathbb{Z}$, $\sum_{s=0}^{h_+-1} W_{i+s}^{(h_-)} = \sum_{s=0}^{h_+-1} \sum_{r=0}^{h_--1} a_{i+s+r} = \sum_{r=0}^{h_--1} W_{i+r}^{(h_+)}$ and $\sum_{v=0}^{h_+-1} W_{i+v h_-}^{(h_-)} = T_{i+h_+ h_-} - T_i = \sum_{u=0}^{h_--1} W_{i+u h_+}^{(h_+)}$. Substituting $W_j^{(h_\pm)} = q_\pm + H_j^{(h_\pm, q_\pm)}$ and $h_- q_+ - h_+ q_- = 1$ gives (4.7). Summing the support indicator over $i \in \mathbb{Z}/k\mathbb{Z}$ gives $h_- m_+ + h_+ m_- \ge k$. Finally, substituting $W \le \max(0, m_+ + m_- - 3)$ into the sparse-numerator bound for $m_{\min} = \min(m_+, m_-)$ yields (4.8). $\blacksquare$

---

## 5. Logarithmic Dynamic Range Identity, Baker-Free $(m, b)$ Bridge, and Direct Chronological Subword Multiplicity

**Theorem 4 (Logarithmic Dynamic Range Identity, Baker-Free $(m, b)$ Bridge, and Chronological Subword Multiplicity).**
1. **(Universal word-level dynamic range identity):** *For every positive valuation word $a \in \mathbb{Z}_{\ge 1}^k$ with $2^S > 3^k$ ($D > 0$), let $C_{\max}(a) = \max_{0 \le i < k} C(a^{(i)})$ and $C_{\min}(a) = \min_{0 \le i < k} C(a^{(i)})$. Then*
   $$W(a) - 1 - (S - k\log_2 3) < \log_2\left(\frac{C_{\max}(a)}{C_{\min}(a)}\right) = \log_2\left(\frac{x_{\max}(a)}{x_{\min}(a)}\right) < W(a) + 1 + (S - k\log_2 3), \tag{5.1}$$
   *or equivalently in pure integer arithmetic, $3^k 2^{\max(0, W-1)} C_{\min}(a) < 2^{S + \mathbf{1}_{\{W=0\}}} C_{\max}(a)$ and $3^k C_{\max}(a) < 2^{S + W + 1} C_{\min}(a)$.*
2. **(Baker-free $(m, b)$ bridge on positive integer cycles):** *On any primitive positive integer cycle of length $k$, minimum state $x_{\min}$, maximum state $x_{\max}$, internal rank breakpoint count $b \ge 1$, and upper or lower window defect count $m$,*
   $$\left(\frac{k - 1}{b} - 1\right)\log_2 3 + \log_2(2b) < \log_2 x_{\min} + W + 1 + (S - k\log_2 3) \le \log_2 x_{\min} + m + V_{J \setminus I} + (S - k\log_2 3), \tag{5.2}$$
   *where $0 < S - k\log_2 3 < \min\bigl(\frac{k}{3 x_{\min}\ln 2},\, \frac{1}{6}\log_2(1 + \frac{2k}{x_{\min}})\bigr)$.*
3. **(Direct chronological subword multiplicity bound):** *On any primitive positive integer cycle of the $3x + c$ map with odd state range $[n, M]$ (or any trajectory segment of $k$ distinct odd states in $[n, M]$), every chronological valuation word $w = (e_0, \dots, e_{L-1}) \in \mathbb{Z}_{\ge 1}^L$ of length $1 \le L \le k-1$ and sum $S(w) = \sum_{j=0}^{L-1} e_j$ has occurrence multiplicity bounded by*
   $$\mathrm{mult}(w) \le 1 + \left\lfloor \frac{M - n}{2\max(3^L,\, 2^{S(w)})} \right\rfloor. \tag{5.3}$$
   *Consequently, every chronological subword $w$ of length $L > \log_3(\frac{M-n}{2})$ (or sum $S(w) > \log_2(\frac{M-n}{2})$) occurs at most once ($\mathrm{mult}(w) = 1$, so the subword complexity satisfies $p(L) = k$ for all $\lceil \log_3(\frac{M-n}{2}) \rceil < L \le k - 1$).*

**Proof.**
**(1)–(2):** The positive rational states $x_i = C(a^{(i)})/D > 0$ satisfy $2^{a_i} x_{i+1} = 3 x_i + 1$. Taking $\log_2$ and summing over $[0, i)$ gives $T_i + \log_2(x_i/x_0) = i\log_2 3 + \sigma_i$, where $\sigma_i = \sum_{r=0}^{i-1}\log_2(1 + \frac{1}{3 x_r})$ increases strictly from $\sigma_0 = 0$ to $\sigma_k = S - k\log_2 3 > 0$. Subtracting $\frac{i S}{k} = i\log_2 3 + \frac{i}{k}\sigma_k$ yields
$$\log_2\left(\frac{x_i}{x_0}\right) = \left(\frac{i S}{k} - T_i\right) + \left(\sigma_i - \frac{i}{k}\sigma_k\right). \tag{5.4}$$
For any $i, j \in \{0, \dots, k-1\}$, $\bigl|(\sigma_i - \frac{i}{k}\sigma_k) - (\sigma_j - \frac{j}{k}\sigma_k)\bigr| \le \sigma_k = S - k\log_2 3$ and $\bigl|(\frac{i S}{k} - T_i) - (\frac{j S}{k} - T_j) - (z_i - z_j)\bigr| = \bigl|\{\frac{i S}{k}\} - \{\frac{j S}{k}\}\bigr| < 1$, proving (5.1). Combining (5.1) with $x_{\max} - x_{\min} \ge 2b\cdot 3^{(k-1)/b - 1}$ (`collatz_22/v3` [1], Theorem 10) and $W \le m + V_{J \setminus I} - 1$ proves (5.2).

**(3):** If $w = (e_0, \dots, e_{L-1})$ occurs at starting indices $i_1, \dots, i_\mu$, the affine word map sends each odd starting state $x_{i_r}$ to an odd ending state $x_{i_r + L}$ via $2^{S(w)} x_{i_r + L} = 3^L x_{i_r} + c\cdot C(w)$. For any $r \ne s$, $2^{S(w)}(x_{i_r + L} - x_{i_s + L}) = 3^L(x_{i_r} - x_{i_s})$, and since $\gcd(2^{S(w)}, 3^L) = 1$ and all states are odd, $|x_{i_r} - x_{i_s}| \ge 2\cdot 2^{S(w)}$ and $|x_{i_r + L} - x_{i_s + L}| \ge 2\cdot 3^L$. Sorting the $\mu$ occurrences inside $[n, M]$ gives $M - n \ge 2(\mu - 1)\max(3^L, 2^{S(w)})$, proving (5.3) and the uniqueness threshold $L > \log_3(\frac{M-n}{2})$. $\blacksquare$

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## 6. Universal $\{2, 3\}$-Free Half-Word Resultants on Balanced Shifts, Two-Sided Sturmian Envelope, and Odd-Stride Cyclotomic Pair-Matching

When $2 \mid \gcd(k, S)$, write $k = 2 r_0$ and $S = 2 s_0$ ($r_0 \ge 1$, $r_0 < s_0 < 2 r_0$). The cycle denominator factors as $D = 2^{2 s_0} - 3^{2 r_0} = F_- F_+$, where $F_- := 2^{s_0} - 3^{r_0}$ and $F_+ := 2^{s_0} + 3^{r_0}$. For each cyclic shift $a^{(s)} = (u_s, v_s)$ ($0 \le s < r_0$) split into its two half-words $u_s = (a_s, \dots, a_{s+r_0-1})$ and $v_s = (a_{s+r_0}, \dots, a_{s+2r_0-1})$ of length $r_0$, let $d_s := \sum_{j=0}^{r_0-1} a_{s+j} - s_0 = T_{s+r_0} - T_s - s_0 \in \mathbb{Z}$ be the half-word sum excess. In Theorem 13 of `collatz_22/v3` [1], half-word resultants were established for sum-unbalanced shifts $d_s \ne 0$. To extend this unconditionally to **all antipodally distinct shifts** $\mathcal{U}(a) := \{0 \le s < r_0 : u_s \ne v_s\}$ (including balanced-sum shifts $d_s = 0$ where $u_s \ne v_s$), define the positive integer half-word resultants
$$R_s^+(a) := \begin{cases} 2^{d_s} C(v_s) + C(u_s) & (d_s \ge 0), \\ 2^{-d_s} C(u_s) + C(v_s) & (d_s < 0), \end{cases} \qquad R_s^-(a) := \begin{cases} \bigl|2^{d_s} C(v_s) - C(u_s)\bigr| & (d_s \ge 0), \\ \bigl|2^{-d_s} C(u_s) - C(v_s)\bigr| & (d_s < 0), \end{cases} \tag{6.1}$$
together with their $\{2, 3\}$-free parts $R_s^{\pm, \mathrm{red}}(a) := R_s^\pm(a) / (2^{v_2(R_s^\pm(a))} 3^{v_3(R_s^\pm(a))})$.

**Theorem 5 (Universal $\{2, 3\}$-Free Half-Word Resultants, Two-Sided Sturmian Envelope, and Odd-Stride Pair-Matching; extending Theorem 13 of `collatz_22/v3` [1]).**
1. **(Universal $\{2, 3\}$-free half-word resultant for primitive even-$(k, S)$ words):** *For every primitive word $a \in \mathcal{A}^{\mathrm{prim}}(2 r_0, 2 s_0)$, $\mathcal{U}(a) \ne \emptyset$, and for every $s \in \mathcal{U}(a)$ we have $R_s^{\pm, \mathrm{red}}(a) \ge 1$ and*
   $$\gcd\bigl(C(a),\, |F_-|\bigr) = \gcd\bigl(R_s^{+, \mathrm{red}}(a),\, |F_-|\bigr) \le \min_{0 \le r < r_0} R_r^{+, \mathrm{red}}(a), \tag{6.2}$$
   $$\gcd\bigl(C(a),\, F_+\bigr) = \gcd\bigl(R_s^{-, \mathrm{red}}(a),\, F_+\bigr) \le \min_{r \in \mathcal{U}(a)} R_r^{-, \mathrm{red}}(a). \tag{6.3}$$
   *Moreover, on the rational cycle $x_s = C(a^{(s)})/D$, at every balanced-sum shift ($d_s = 0$) the antipodal state difference and sum cancel $F_-$ and $F_+$ respectively:*
   $$x_s - x_{s+r_0} = \frac{C(v_s) - C(u_s)}{F_+}, \qquad x_s + x_{s+r_0} = \frac{C(u_s) + C(v_s)}{F_-} \qquad (d_s = 0), \tag{6.3a}$$
   *while for general $d_s \in \mathbb{Z}$, $D(2^{\max(0, d_s)} x_s - 2^{\max(0, -d_s)} x_{s+r_0}) = 2^{\max(0, d_s) + s_0 + d_s} C(v_s) - 2^{\max(0, -d_s) + s_0 - d_s} C(u_s) - F_-(2^{\max(0, d_s)} C(u_s) - 2^{\max(0, -d_s)} C(v_s))$.*
2. **(Two-sided prefix-excess envelope relative to the Sturmian word):** *Let $u^{\mathrm{Sturm}}(r_0, s_0)$ be the mechanical word with prefix sums $T_j^{\mathrm{Sturm}} = \lfloor j s_0 / r_0 \rfloor$ ($0 \le j \le r_0$). For any composition $u \in \mathcal{A}(r_0, s_0)$ with prefix sums $T_j(u)$, let $W^+(u) := \max_{1 \le j < r_0} \max(0, T_j(u) - T_j^{\mathrm{Sturm}})$ and $W^-(u) := \max_{1 \le j < r_0} \max(0, T_j^{\mathrm{Sturm}} - T_j(u))$. Then $C(u)$ satisfies both the global simplex bounds $3^{r_0} - 2^{r_0} \le C(u) \le 2^{s_0 - r_0}(3^{r_0} - 2^{r_0})$ [8] and the termwise Sturmian envelope $2^{-W^-(u)} C(u^{\mathrm{Sturm}}(r_0, s_0)) \le C(u) \le 2^{W^+(u)} C(u^{\mathrm{Sturm}}(r_0, s_0))$, i.e.,*
   $$\max\bigl(3^{r_0} - 2^{r_0},\; \bigl\lceil 2^{-W^-(u)} C\bigl(u^{\mathrm{Sturm}}(r_0, s_0)\bigr) \bigr\rceil\bigr) \le C(u) \le \min\bigl(2^{s_0 - r_0}(3^{r_0} - 2^{r_0}),\; 2^{W^+(u)} C\bigl(u^{\mathrm{Sturm}}(r_0, s_0)\bigr)\bigr). \tag{6.4}$$
3. **(Odd-stride cyclotomic pair-matching and $Q_2$-parity cycle exclusion for $4 \mid \gcd(k, S)$):** *For any $r_0 < s_0 < 2 r_0$, any two compositions $w, v \in \mathcal{A}(r_0, s_0)$, and any nonzero integer shift $\Delta \in [1 - v_{r_0-1}, w_{r_0-1} - 1] \setminus \{0\}$, the length-$4 r_0$ word of sum $4 s_0$ defined by*
   $$a(w, v, \Delta) := \bigl(w_0, \dots, w_{r_0-2}, w_{r_0-1} - \Delta,\;\; v_0, \dots, v_{r_0-1},\;\; v_0, \dots, v_{r_0-2}, v_{r_0-1} + \Delta,\;\; w_0, \dots, w_{r_0-1}\bigr) \tag{6.5}$$
   *satisfies the exact factorization*
   $$C\bigl(a(w, v, \Delta)\bigr) = \bigl(2^{s_0} + 3^{r_0}\bigr)\Bigl[\bigl(2^{2 s_0} - 3^{r_0} 2^{s_0} + 3^{2 r_0}\bigr) C(w) + 3^{r_0} 2^{s_0 - \Delta} C(v)\Bigr], \tag{6.6}$$
   *so $(2^{s_0} + 3^{r_0}) \mid C(a(w, v, \Delta))$ identically (yielding primitive binary $\{1, 2\}$-words in $\log_2 3 < s_0/r_0 < 2$ when $w_{r_0-1} = 2, v_{r_0-1} = 1, \Delta = 1$), while $Q_2 := 2^{2 s_0} + 3^{2 r_0} \nmid C(a(w, v, \Delta))$ for every $\Delta \ne 0$, excluding the entire family from being integer cycles.*

**Proof.**
**(1):** Since $C(a^{(s)}) = 3^{r_0} C(u_s) + 2^{s_0 + d_s} C(v_s)$ and $C(a^{(s+r_0)}) = 3^{r_0} C(v_s) + 2^{s_0 - d_s} C(u_s)$, reducing $2^{\max(0, -d_s)} C(a^{(s)})$ modulo $F_\mp$ ($2^{s_0} \equiv \pm 3^{r_0} \pmod{F_\mp}$) and removing factors of $2$ and $3$ (which are coprime to $F_\mp$) yields $\gcd(C(a), |F_\mp|) = \gcd(R_s^{\pm, \mathrm{red}}(a), |F_\mp|)$. When $d_s = 0$, subtracting and adding $C(a^{(s)})$ and $C(a^{(s+r_0)})$ gives $C(a^{(s)}) - C(a^{(s+r_0)}) = (2^{s_0} - 3^{r_0})(C(v_s) - C(u_s)) = F_-(C(v_s) - C(u_s))$ and $C(a^{(s)}) + C(a^{(s+r_0)}) = (2^{s_0} + 3^{r_0})(C(u_s) + C(v_s)) = F_+(C(u_s) + C(v_s))$; dividing by $D = F_- F_+$ proves (6.3a). If $R_s^-(a) = 0$, then $2^{d_s} C(v_s) = C(u_s)$; since $C(u_s)$ and $C(v_s)$ are both odd, $d_s = 0$ and $C(u_s) = C(v_s)$, which by injectivity of $u \mapsto C(u)$ forces $u_s = v_s$, so $s \notin \mathcal{U}(a)$. If $\mathcal{U}(a) = \emptyset$, then $a_{s+r_0} = a_s$ for all $s$, contradicting primitivity of $a$.

**(2):** For each $1 \le j < r_0$, $T_j^{\mathrm{Sturm}} - W^-(u) \le T_j(u) \le T_j^{\mathrm{Sturm}} + W^+(u)$, so multiplying $3^{r_0-1-j} 2^{T_j^{\mathrm{Sturm}}}$ by $2^{-W^-(u)}$ and $2^{W^+(u)}$ and combining with the simplex extrema $C(1^{r_0-1}, s_0 - r_0 + 1) = 3^{r_0} - 2^{r_0}$ and $C(s_0 - r_0 + 1, 1^{r_0-1}) = 2^{s_0 - r_0}(3^{r_0} - 2^{r_0})$ [8] yields (6.4).

**(3):** In $a(w, v, \Delta)$, Block 0 and Block 3 are separated by stride $3 r_0$ with prefix-sum shift $T_{3 r_0 + i} - T_i = 3 s_0$ for $0 \le i < r_0$, contributing $C(w)(3^{3 r_0} + 2^{3 s_0}) = C(w)(2^{s_0} + 3^{r_0})(2^{2 s_0} - 3^{r_0} 2^{s_0} + 3^{2 r_0})$, while Block 1 and Block 2 are separated by stride $r_0$ with prefix-sum shift $T_{2 r_0 + i} - T_{r_0 + i} = s_0$ and $T_{r_0} = s_0 - \Delta$, contributing $3^{r_0} 2^{s_0 - \Delta} C(v)(3^{r_0} + 2^{s_0})$, proving (6.6). Modulo $Q_2 = 2^{2 s_0} + 3^{2 r_0}$ ($2^{2 s_0} \equiv -3^{2 r_0}$), the bracket in (6.6) is congruent to $-3^{r_0} 2^{s_0 - \Delta}(2^\Delta C(w) - C(v))$. Since $0 < C(u) < 2^{s_0 - r_0} 3^{r_0}$ and $|\Delta| \le s_0 - r_0$, clearing $2^{\max(0, -\Delta)}$ leaves both positive terms strictly smaller than $2^{2 s_0 - 2 r_0} 3^{r_0} < Q_2$, so divisibility by $Q_2$ requires $2^\Delta C(w) = C(v)$, which is impossible for $\Delta \ne 0$ because $C(w)$ and $C(v)$ are odd. $\blacksquare$

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## 7. Complementary-Factor Chebyshev Decomposition, Fixed-Slope $Q_d$-Fibre Finiteness, and 11-Pair All-$d$ Census

Let $d \ge 2$ divide $\gcd(k, S)$, and write $k = d r_0$, $S = d s_0$ with $r_0 < s_0 < 2 r_0$ and $D_0 := 2^{s_0} - 3^{r_0} > 0$, so that $D = 2^S - 3^k = D_0 Q_d$ with $Q_d := \frac{2^{d s_0} - 3^{d r_0}}{2^{s_0} - 3^{r_0}} = \sum_{b=0}^{d-1} 2^{b s_0} 3^{(d-1-b)r_0}$ and $\gcd(D_0, Q_d) = \gcd(d, D_0)$. Put
$$g_d := \gcd(d, D_0), \qquad D_{0, \mathrm{red}} := \frac{D_0}{g_d}, \qquad Q_{d, \mathrm{red}} := \frac{Q_d}{g_d}, \qquad \text{so that } \gcd(D_{0, \mathrm{red}}, Q_{d, \mathrm{red}}) = 1. \tag{7.0}$$
For any cyclic shift $a^{(s)}$ partitioned into $d$ blocks $u_s^{(0)}, \dots, u_s^{(d-1)}$ of length $r_0$ with sums $v_{s, b}$, cumulative excesses $\delta_{s, b} = \sum_{j=0}^{b-1}(v_{s, j} - s_0)$ ($\delta_{s, 0} = 0$), $\delta_{s, -} = \max_{0 \le b < d}(-\delta_{s, b}) \ge 0$, and scaled block numerators $\widetilde{\alpha}_{s, b} := 2^{\delta_{s, b} + \delta_{s, -}} C(u_s^{(b)}) \in \mathbb{Z}_{\ge 1}$, define the **complementary-factor block sum** and the **Chebyshev block-difference linear form**:
$$\Sigma_s^+(a) := \sum_{b=0}^{d-1} \widetilde{\alpha}_{s, b}, \qquad L_s^{(d)}(a) := \sum_{0 \le j < b \le d-1} \bigl(\widetilde{\alpha}_{s, b} - \widetilde{\alpha}_{s, j}\bigr) \sum_{t=j}^{b-1} 2^{t s_0} 3^{(d-2-t)r_0}. \tag{7.1}$$

**Theorem 6 (Complementary-Factor Chebyshev Decomposition, Fixed-Slope $Q_d$-Fibre Finiteness, and Affine Fixed-Point Splicing).**
1. **(Exact Chebyshev decomposition across $(D_0, Q_d)$ retaining $\gcd(d, D_0)$):** *For every word $a \in \mathcal{A}(d r_0, d s_0)$ and every cyclic shift $s \in \mathbb{Z}/k\mathbb{Z}$,*
   $$d \cdot 2^{\delta_{s, -}} C\bigl(a^{(s)}\bigr) = \Sigma_s^+(a)\, Q_d + D_0\, L_s^{(d)}(a). \tag{7.2}$$
   *Consequently, on the $Q_d$-divisible fibre where $C(a^{(s)}) = q_s Q_d$ ($q_s \in \mathbb{Z}_{\ge 1}$ odd), we have $Q_{d, \mathrm{red}} \mid L_s^{(d)}(a)$ with integer quotient $\widetilde{\mu}_s := L_s^{(d)}(a)/Q_{d, \mathrm{red}} \in \mathbb{Z}$,*
   $$d \cdot 2^{\delta_{s, -}} q_s = \Sigma_s^+(a) + \widetilde{\mu}_s\, D_{0, \mathrm{red}}, \qquad \gcd\bigl(C(a),\, D\bigr) = Q_d \cdot \gcd(q_s,\, D_0), \qquad c_{\min}(a) = \frac{D_0}{\gcd(q_s,\, D_0)}, \tag{7.3}$$
   *and the block sum $\Sigma_s^+(a)$ satisfies the exact complementary-factor gcd identities*
   $$\gcd(d q_s,\, D_{0, \mathrm{red}}) = \gcd\bigl(\Sigma_s^+(a),\, D_{0, \mathrm{red}}\bigr) \quad \text{and} \quad \gcd(d q_s,\, D_0) = \gcd\bigl(\Sigma_s^+(a) + \widetilde{\mu}_s D_{0, \mathrm{red}},\, D_0\bigr). \tag{7.3a}$$
   *When $\gcd(d, D_0) = 1$, $Q_{d, \mathrm{red}} = Q_d$, $D_{0, \mathrm{red}} = D_0$, and $\gcd(C(a), D) = Q_d \cdot \gcd(\Sigma_s^+(a), D_0)$.*
2. **($c = D_0$ cycle specialization and $d$-independent minimum-state bound):** *Specializing the classical correspondence between rational $3x + 1$ cycles and integer $3x + c$ cycles (Steiner [3], Böhm and Sontacchi [4], Lagarias [12, 13]) to the multiplier $c = D_0$ on the quotient $q_i := C(a^{(i)})/Q_d$, a composition $a = (a_0, \dots, a_{k-1}) \in \mathcal{A}(d r_0, d s_0)$ satisfies $Q_d \mid C(a)$ if and only if $q_i$ ($0 \le i < k$) forms a periodic cycle of **odd positive integers** under the deterministic accelerated affine map*
   $$T_{D_0}(x) := \frac{3x + D_0}{2^{v_2(3x + D_0)}}, \qquad a_i = v_2(3 q_i + D_0) \quad (0 \le i < k). \tag{7.4}$$
   *By the standard cycle-product argument ([8], Theorem 4), for **every** $d \ge 1$ the minimum odd quotient $q_{\min}(a) := \min_{0 \le i < k} q_i \in 2\mathbb{Z}_{\ge 0} + 1$ is bounded above **independently of $d$** by*
   $$1 \le q_{\min}(a) \le X_{\max}(r_0, s_0) := \max\bigl\{x \in \mathbb{Z}_{\ge 1} : (3x + D_0)^{r_0} \ge 2^{s_0} x^{r_0}\bigr\} = \left\lfloor \frac{2^{s_0} - 3^{r_0}}{2^{s_0 / r_0} - 3} \right\rfloor. \tag{7.5}$$
3. **(Fixed-slope finiteness of the primitive $Q_d$-fibre across all $d \ge 1$):** *For each fixed reduced pair $(r_0, s_0)$ with $2^{s_0} > 3^{r_0}$, because $T_{D_0}$ has out-degree $1$ on $2\mathbb{Z}_{\ge 0} + 1$ and every $Q_d$-divisible word of slope $s_0/r_0$ passes through at least one odd integer in $\{1, 3, \dots, X_{\max}(r_0, s_0)\}$, the infinite union $\bigcup_{d=1}^\infty \{a \in \mathcal{A}^{\mathrm{prim}}(d r_0, d s_0) : Q_d \mid C(a)\}$ contains **at most $\frac{1}{2}(X_{\max}(r_0, s_0) + 1)$ primitive rotation classes in total across all $d \ge 1$**, and there exists an exact finite threshold $d_{\max}(r_0, s_0) < \infty$ such that the primitive $Q_d$-fibre in $\mathcal{A}^{\mathrm{prim}}(d r_0, d s_0)$ is **identically empty for all $d > d_{\max}(r_0, s_0)$**.*
4. **(Affine Fixed-Point Splicing of the minimal $Q_3$-orbits):** *The unique primitive $Q_3$-divisible orbits at $(15, 24) = 3 \times (5, 8)$ (`w = [1,1,1,3, 1,1,1,1,1, 2,1,2,1,2,5]`, $q_{\min} = 131$), $(18, 30) = 3 \times (6, 10)$ (`w = [1,1,1,1,1,2, 1,2,1,2,1,3, 2,1,2,1,2,5]`, $q_{\min} = 281$), and the minimal $Q_3$-orbit at $(24, 39) = 3 \times (8, 13)$ (`w = [1,1,1,1,1,1,1,1,3, 2,1,2,1,2,1,3, 2,1,2,1,2,1,2,5]`, $q_{\min} = 2977$) satisfy the exact closed form*
   $$q_{\min} = 9 \cdot 2^{s_0 - 4} - D_0 \equiv 9 \cdot 2^{s_0 - 4} \pmod{D_0}, \tag{7.6}$$
   *which splices the $1$-run fixed point $-D_0$ and the $(2, 1)$-run fixed point $-7 D_0$ across the $d = 3$ block substitutions $3^{r_0} = 2^{s_0} - D_0$, and since $\gcd(D_0, 6) = 1$, automatically forces $\gcd(q_{\min}, D_0) = 1$ and $c_{\min}(a) = D_0 > 1$.*

**Proof.**
**(1):** Writing $x = 2^{s_0}$ and $y = 3^{r_0}$, we have $2^{\delta_{s, -}} C(a^{(s)}) = \sum_{b=0}^{d-1} \widetilde{\alpha}_{s, b} x^b y^{d-1-b}$ and $Q_d = \sum_{j=0}^{d-1} x^j y^{d-1-j}$. By discrete Chebyshev identity,
$$d \cdot 2^{\delta_{s, -}} C\bigl(a^{(s)}\bigr) - \Sigma_s^+(a) Q_d = \sum_{0 \le j < b \le d-1} \bigl(\widetilde{\alpha}_{s, b} - \widetilde{\alpha}_{s, j}\bigr)\bigl(x^b y^{d-1-b} - x^j y^{d-1-j}\bigr) = (x - y) L_s^{(d)}(a) = D_0 L_s^{(d)}(a),$$
proving (7.2). Since $x \equiv y \pmod{D_0}$, $Q_d = \sum_{j=0}^{d-1} x^j y^{d-1-j} \equiv d y^{d-1} \pmod{D_0}$, and $\gcd(y, D_0) = 1$ gives $g_d = \gcd(D_0, Q_d) = \gcd(d, D_0)$. When $C(a^{(s)}) = q_s Q_d$, dividing (7.2) by $g_d$ yields $(d \cdot 2^{\delta_{s,-}} q_s - \Sigma_s^+(a)) Q_{d, \mathrm{red}} = D_{0, \mathrm{red}} L_s^{(d)}(a)$. Because $\gcd(D_{0, \mathrm{red}}, Q_{d, \mathrm{red}}) = 1$, Euclid's lemma gives $Q_{d, \mathrm{red}} \mid L_s^{(d)}(a)$ with integer quotient $\widetilde{\mu}_s = L_s^{(d)}(a)/Q_{d, \mathrm{red}}$, and cancelling $Q_{d, \mathrm{red}}$ gives $d \cdot 2^{\delta_{s,-}} q_s = \Sigma_s^+(a) + \widetilde{\mu}_s D_{0, \mathrm{red}}$, which since $\gcd(2, D_0) = 1$ implies (7.3) and (7.3a).

**(2)–(3):** Since $2^{a_i} C(a^{(i+1)}) = 3 C(a^{(i)}) + D_0 Q_d$ and $\gcd(Q_d, 6) = 1$, $Q_d \mid C(a)$ implies $Q_d \mid C(a^{(i)})$ for all $i$. Since both $C(a^{(i)})$ and $Q_d$ are odd, $q_i = C(a^{(i)})/Q_d$ is a positive odd integer satisfying $2^{a_i} q_{i+1} = 3 q_i + D_0$, and oddness of $q_{i+1}$ forces $a_i = v_2(3 q_i + D_0)$, proving (7.4). Multiplying $2^{a_i} q_{i+1}/q_i = 3 + D_0/q_i$ over $0 \le i < k$ gives $2^{s_0} = 2^{S/d} \le (3 + D_0/q_{\min})^{r_0}$, i.e., $(3 q_{\min} + D_0)^{r_0} \ge 2^{s_0} q_{\min}^{r_0}$, proving (7.5) and (3).

**(4):** Under $T_{D_0}$, a run $1^t$ maps $2^\alpha 3^\beta - D_0 \mapsto 2^{\alpha-t} 3^{\beta+t} - D_0$, and a run $(2, 1)^t 2$ maps $2^\alpha 3^\beta - 7 D_0 \mapsto 2^{\alpha-3t-2} 3^{\beta+2t+1} - 5 D_0$. Substituting $3^{r_0} = 2^{s_0} - D_0$ at each of the $d = 3$ block transitions and dividing by $2^3 = 8$ and $2^5 = 32$ returns the state to $q_{\min} = 9 \cdot 2^{s_0 - 4} - D_0$, which is coprime to $D_0$ because $\gcd(D_0, 6) = 1$. $\blacksquare$

### 7.1 Scope and Limitations of Fixed-Slope $Q_d$-Fibre Finiteness

It is important to state clearly what Theorem 6 and Table 1 establish and what remains open:
1. **Fixed reduced pair $(r_0, s_0)$ vs. arbitrary slopes:** Theorem 6(3) proves uniform finiteness across all $d \ge 1$ **for each fixed reduced pair $(r_0, s_0)$**, because $X_{\max}(r_0, s_0)$ depends only on $(r_0, s_0)$. As $(r_0, s_0)$ varies over infinitely many coprime pairs with $s_0/r_0 \to \log_2 3^+$, $D_0 = 2^{s_0} - 3^{r_0}$ and $X_{\max}(r_0, s_0)$ grow without bound, so Theorem 6 does not by itself control all possible slopes simultaneously without combining with the dual-parent width bounds of Theorems 1–3.
2. **Periodic cycle words vs. divergent orbits:** All results in Sections 2–7 constrain periodic cycle words (and finite trajectory segments in Theorem 4(3)); they do not address or exclude divergent Collatz trajectories.

### Table 1: Complete All-$d$ Census of Primitive $Q_d$-Divisible Orbits Across 11 Reduced Pairs $(r_0, s_0)$ (`code/verify_paper9.py`)

| Reduced $(r_0, s_0)$ | $D_0 = 2^{s_0}-3^{r_0}$ | $X_{\max}(r_0, s_0)$ | Odd seeds tested | $d = 1$ orbits | Non-empty primitive $Q_d$-fibres ($d \ge 2$: `# primitive rotation classes`) | Max divisor $d_{\max}$ | $c = 1$ cycle hits ($D_0 \mid q_{\min}$) |
|---|---:|---:|---:|---:|---|---:|---:|
| $(3, 5)$ | $5$ | $28$ | $14$ | $2$ | $\emptyset$ (**0 primitive words for all $d \ge 2$**) | $1$ | $0$ |
| $(4, 7)$ | $47$ | $129$ | $65$ | $5$ | $d=4: 1$ | $4$ | $0$ |
| $(5, 8)$ | $13$ | $413$ | $207$ | $7$ | $d=3: 1$ (`(15, 24)`: $q_{\min}=131$) | $3$ | $0$ |
| $(5, 9)$ | $269$ | $557$ | $279$ | $14$ | $d=2: 2,\; d=3: 1,\; d=4: 1$ | $4$ | $0$ |
| $(6, 10)$ | $295$ | $1{,}687$ | $844$ | $20$ | $d=2: 2,\; d=3: 1$ (`(18, 30)`: $q_{\min}=281$) | $3$ | $0$ |
| $(6, 11)$ | $1{,}319$ | $2{,}340$ | $1{,}170$ | $42$ | $d=2: 6,\; d=3: 1,\; d=4: 3,\; d=5: 1$ | $5$ | $0$ |
| $(7, 12)$ | $1{,}909$ | $6{,}785$ | $3{,}393$ | $66$ | $d=2: 8,\; d=3: 5 \text{ (`(21,36)`)},\; d=4: 2,\; d=6: 1,\; d=12: 1$ | $12$ | $0$ |
| $(8, 13)$ | $1{,}631$ | $19{,}319$ | $9{,}660$ | $99$ | $d=2: 12,\; d=3: 4 \text{ (`(24,39)`: $q_{\min}=2977$)}$ | $3$ | $0$ |
| $(8, 14)$ | $9{,}823$ | $27{,}017$ | $13{,}509$ | $212$ | $d=2: 34,\; d=3: 20,\; d=4: 5,\; d=5: 6,\; d=6: 1,\; d=7: 2,\; d=12: 1$ | $12$ | $0$ |
| $(9, 15)$ | $13{,}085$ | $74{,}856$ | $37{,}428$ | $333$ | $d=2: 46,\; d=3: 22,\; d=4: 5,\; d=5: 1,\; d=7: 1,\; d=8: 1,\; d=9: 1,\; d=12: 1$ | $12$ | $0$ |
| $(10, 16)$ | $6{,}487$ | $206{,}374$ | $103{,}187$ | $497$ | $d=2: 65,\; d=3: 13 \text{ (`(30,48)`)},\; d=4: 5,\; d=5: 2,\; d=6: 1$ | $6$ | $0$ |

---

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