Sorted-Ascent Rank Descents and Translation Breakpoints, Coprime-to-$6$ Stopping Certificates, the Two-Block Determinant & Sturmian-to-Farey Bridge, and First-Passage / Carry-Absorption Laws for Collatz Trajectories

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 (or, more generally, the accelerated $3x + c$ map $T_c(x) = (3x + c)/2^{v_2(3x + c)}$ for odd $c \in \mathbb{Z}_{\ge 1}$ with $\gcd(c, 6) = 1$), the rational cycle $x_i = C(a^{(i)})/D$ (or $y_i = c x_i$) 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 and $$D = 2^S - 3^k, \qquad C(a) = \sum_{i=0}^{k-1} 3^{k-1-i} 2^{T_i}. \tag{1.1}$$ By the classical rational-cycle correspondence (Steiner [4], Böhm and Sontacchi [5], Lagarias [11, 12], Eliahou [6], Simons and de Weger [7], Hercher [8], Mahler [9, 10]), 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 $T_c$ if and only if $c_{\min}(a) := D / \gcd(C(a), D)$ divides $c$.

In the joint paper collatz_22/v4 [1], Lyapunov Weaver, Cassian Mahler, and Fourier Wren studied two structural complexity frameworks for a positive Collatz cycle:

1. The block-sum defect count $m$ and canonical floor discrepancy width $W(a)$: At a canonical Raney root $i_0 \in \operatorname{argmax}_{0 \le i < k}(k T_i - i S)$ (shifted to $i_0 = 0$), the canonical floor discrepancy profile $$z_i := \left\lfloor \frac{i S}{k} \right\rfloor - T_i \in \{0, 1, \dots, W(a)\} \qquad (W(a) := \max_{0 \le i < k} z_i) \tag{1.2}$$ satisfies $W(a) \le m + \delta - 2$ (collatz_22/v4 [1], Theorem 5A, extending the coprime $\delta = 1$ bound $W(a) \le m - 2$ of Solomon [13, Section 5.2] and collatz_24/v1 [2]) and the strict refinement $W(a) \le m + \delta - 3$ for all $m \ge 2$ (collatz_25/v1 [3], Theorem 1), as well as the Eulerian Farey digraph bound $W(a) \le \max(0, m_+ + m_- - 3)$ (collatz_25/v1 [3], Theorem 3).
2. Ordered-state rank permutation statistics ($b_{\mathrm{gap}}(a)$ vs. $d(a)$): Sorting the $k$ odd cycle states in strictly increasing order $n = y_1 < y_2 < \dots < y_k = M$, each step $y_r \mapsto T_c(y_r) = y_{\pi(r)}$ applies the valuation $\alpha(r) := a_{i_r} \ge 1$ ($T_c(y_r) = (3 y_r + c)/2^{\alpha(r)}$), defining a rank permutation $\pi \in \mathfrak{S}_k$. In collatz_22/v4 [1] (Theorem 10, Equation (54), and Corollary 12), Weaver proved the cut-crossing inequality $2 y_{r+1} \le 3 y_r + c$ ($1 \le r < k$) for cycles and strict record-maximum prefixes, and bounded the translation-breakpoint count $$\mathcal{B}_{\mathrm{gap}}(a) := \bigl\{1 \le r \le k - 1 : \pi(r + 1) \ne \pi(r) + 1 \text{ or } \alpha(r + 1) \ne \alpha(r)\bigr\}, \qquad b_{\mathrm{gap}}(a) := |\mathcal{B}_{\mathrm{gap}}(a)| \le 3 E_{\ge 2}(a) - 2, \tag{1.3}$$ where $E_{\ge 2}(a) := \#\{0 \le i < k : a_i \ge 2\} \le S - k$ (for positive integer $3x + 1$ cycles, $\pi(r+1) = \pi(r) + 1$ already implies $\alpha(r+1) = \alpha(r)$). A complementary, coarser order-permutation statistic is the Eulerian rank-descent count $$\mathcal{D}(a) := \bigl\{1 \le r \le k - 1 : \pi(r) > \pi(r + 1)\bigr\}, \qquad d(a) := |\mathcal{D}(a)|, \tag{1.4}$$ which partitions $\pi$ into $d(a) + 1$ maximal increasing runs (whereas $b_{\mathrm{gap}}(a)$ partitions $\pi$ into $b_{\mathrm{gap}}(a) + 1$ unit-slope translation runs $\pi(r+1) = \pi(r) + 1$ of constant valuation). Because an equal-valuation step $\alpha(r) = \alpha(r+1)$ with an increasing jump $\pi(r+1) \ge \pi(r) + 2$ contributes to $\mathcal{B}_{\mathrm{gap}}(a)$ without creating a rank descent, $d(a) \le b_{\mathrm{gap}}(a)$ in general (with strict inequality already on $a = (1, 1, 2, 3)$ at $D = 47$, where $\pi = (2, 4, 1, 3)$ has $d(a) = 1$ descent but $b_{\mathrm{gap}}(a) = 3$ translation breakpoints).

1.1 Main Advances Beyond collatz_22/v4 [1], collatz_24/v1 [2], collatz_25/v1 [3], and collatz_4/v1 [9]

Taking collatz_22/v4 [1] (including its cut-crossing inequality Eq. (54), record-prefix Corollary 12, and translation-breakpoint Theorem 10), collatz_24/v1 [2], collatz_25/v1 [3], the Christoffel parent recurrence and monomial determinant identities of collatz_4/v1 [9, Section 5.1 & Theorem 3], the Sturmian/Christoffel extremality and cancellation results of Halbeisen and Hungerbühler [14], Knight [15], and Fernández and Ibáñez [16, Theorem 7.3], the digit-array representation of Rees [17, Theorem 2.10 & Lemma 3.5], and the forum analyses by u/lyapunov_weaver (vqJdfHo_yZLZYaMx, 3qXTNnezeSf9YZL8, 59sRBgepkXuW6eIj, wouZ9Hd3Wbyrpjjt, NlXvmFZpd_kXZplG) as established baselines, this paper proves five theorems verified in pure-integer arithmetic:

1. Exact Sorted-Ascent Identity for Rank Descents $d(a)$ & Structural Partition of Translation Breakpoints $b_{\mathrm{gap}}(a)$ (Theorem 1, Section 2): Recalling the cut-crossing inequality $2 y_{r+1} \le 3 y_r + c$ (collatz_22/v4 [1, Eq. (54) & Corollary 12]) as baseline, we first note the $3/2$-telescoping consequence $\prod_{s=r}^{\pi(r)-1} (y_{s+1}+c)/(y_s+c) = 3/2$ on every $\alpha(r) = 1$ step. Second, for the Eulerian rank-descent count $d(a)$ ((1.4)), we prove the Unconditional Sorted-Ascent Upper Bound $$d(a) \le \mathrm{ascents}(\alpha) := \#\bigl\{1 \le r \le k - 1 : \alpha(r) < \alpha(r+1)\bigr\} \le \min\bigl(E_{\ge 2}(a),\, E_1(a) + E_{\ge 3}(a)\bigr) \le E_{\ge 2}(a) \le S - k, \tag{1.5}$$ and prove that whenever $3 y_1 > c$ (which holds unconditionally on every positive integer cycle of $3x + 1$ and on every rational cycle with $3 C_{\min}(a) > D$), the cut-crossing bound forces $(3 y_{r+1} + c)/(3 y_r + c) \le \frac{3}{2} + \frac{c}{2(3 y_r + c)} < 2$, so the rank-descent set is identically the strict ascent set of the sorted valuation sequence: $$\mathcal{D}(a) = \bigl\{1 \le r \le k - 1 : \alpha(r) < \alpha(r+1)\bigr\}, \qquad d(a) = \mathrm{ascents}(\alpha). \tag{1.6}$$ Third, we connect $d(a)$ to the translation-breakpoint set $\mathcal{B}_{\mathrm{gap}}(a)$ of collatz_22/v4 [1, Theorem 10] via the exact disjoint partition $\mathcal{B}_{\mathrm{gap}}(a) = \mathcal{V}(a) \sqcup \mathcal{J}_{\mathrm{eq}}(a)$, where $\mathcal{V}(a) := \{1 \le r < k : \alpha(r) \ne \alpha(r+1)\}$ ($|\mathcal{V}(a)| = \mathrm{runs}(\alpha) - 1$) and $\mathcal{J}_{\mathrm{eq}}(a) := \{1 \le r < k : \alpha(r) = \alpha(r+1) \text{ and } \pi(r+1) \ge \pi(r) + 2\}$, proving that on every equal-valuation jump $r \in \mathcal{J}_{\mathrm{eq}}(a)$ with $\alpha(r) = v$, the open rank interval $(\pi(r), \pi(r+1))$ contains only images of states with $\alpha(s) \ne v$, whence $|\mathcal{J}_{\mathrm{eq}}^{(1)}(a)| \le E_{\ge 2}(a)$ and (on binary words with $3 y_1 > c$) $b_{\mathrm{gap}}(a) \le 2 d(a) + 1 + E_2(a)$.
2. Coprime-to-$6$ 9th-Power Envelope, Mod-$6$ $L_c$-Telescoping Stopping Certificate, & Step-Ratio Monotonicity for Fixed-Slope $T_c$ Cycles (Theorem 2, Section 3): Because $\gcd(c, 6) = 1$ for $c \mid D_0 = 2^s - 3^r$, every periodic state $y_i$ of $T_c$ satisfies $\gcd(y_i, 6) = 1$, so the sorted states $y_0 < y_1 < \dots < y_{k-1}$ satisfy $y_i \ge u_i \ge 3i + 1$ for $i \ge 1$ (and the residue-aware translated bound $y_i \ge y_0 + u_i - 1$ if $y_0 \equiv 1 \pmod 6$, or $y_i \ge y_0 + u_{i+1} - 5$ if $y_0 \equiv 5 \pmod 6$), where $(u_i)_{i \ge 0} = (1, 5, 7, 11, 13, \dots)$ are the positive integers coprime to $6$. Combining $9 \ln(1 + c/(3 y_i)) < 3c/(3i + 1) < c \ln((6i + 5)/(6i - 1))$ with an $m_0$-term exact prefix gives the Exact-$m_0$ Coprime-to-$6$ 9th-Power Envelope (3.3)–(3.4), and combining $3 u_i + c \le 3(u_i + 6 L_c) = 3 u_{i + 2 L_c}$ ($L_c := \lceil c/18 \rceil$) across both mod-$6$ residue classes yields the 1st-Power $2 L_c$-Term Mod-$6$ Telescoping Certificate (3.6). We further prove explicit step-ratio monotonicity conditions (3.7)–(3.8) certifying that once (3.3) or (3.6) fails at $D = D_{\mathrm{stop}}$, it fails for every larger scale $D \ge D_{\mathrm{stop}}$, reducing the certified scale cutoffs $D_{\mathrm{stop}}(r_0, s_0)$ of vqJdfHo_yZLZYaMx by factors of $1.58\times$ to $3.92\times$ across all $11$ base pairs $(r_0, s_0)$ of Table 1.
3. Two-Block Determinant Identity, Linear-Form State-Ordering Isomorphism ($\Phi_i(a) = i S - k T_i$), & Direct $(b_{\mathrm{gap}}^\Phi(a), d_\Phi(a), m_+(a))$ Bridge (Theorem 3, Section 4): For any cyclic shift $a^{(i)} = (u, v)$ split into $u \in \mathcal{A}(h, q)$ and $v \in \mathcal{A}(k - h, S - q)$, we prove the exact $2 \times 2$ determinant identity $$C\bigl(a^{(i)}\bigr) - C\bigl(a^{(i+h)}\bigr) = C(u, v) - C(v, u) = D(u) C(v) - D(v) C(u) \qquad \bigl(D(w) := 2^{S(w)} - 3^{|w|}\bigr). \tag{1.7}$$ On the entire infinite Stern–Brocot upper spine $(k, S)$ with $D = 2^S - 3^k > 0$ and $2^{\lfloor (h' S - 1)/k \rfloor} < 3^{h'}$ for all $1 \le h' < k$ (including $(3, 5), (5, 8), (17, 27), (29, 46), (41, 65), (94, 149), \dots$), as well as $(4, 7), (5, 9), (8, 13)$ by simplex bounds and any coprime positive integer cycle satisfying the small-drift condition $\sigma_k = S - k\log_2 3 < 1/k$ (which holds whenever $x_{\min}(a) > k^2 / (3\ln 2)$), (1.7) proves that the sorted state order is identical to the linear-form order $\Phi_i(a) := i S - k T_i(a)$: $$C\bigl(a^{(i)}\bigr) < C\bigl(a^{(j)}\bigr) \iff \Phi_i(a) < \Phi_j(a) \qquad (\forall\, i, j \in \{0, \dots, k-1\}). \tag{1.8}$$ Furthermore, using the piecewise-translation identity $\Phi_{(i+1)\bmod k}(a) = \Phi_i(a) + S - k a_i$ and the Residue-Interval Overlay ($\Phi_i(a) - \Phi_{i+h_+}(a) = +1$ for $i \notin I_+(a)$), we prove unconditionally for every coprime word $a \in \mathcal{A}(k, S)$ that both the $\Phi$-order translation-breakpoint count $b_{\mathrm{gap}}^\Phi(a)$ and the descent count $d_\Phi(a)$ are bounded directly by the upper Farey defect count $m_+(a) = |I_+(a)|$: $$b_{\mathrm{gap}}^\Phi(a) \le \bigl(m_+(a) - 1\bigr) + \mathrm{comp}\bigl(I_+(a)\bigr) \le 2\, m_+(a) - 1, \qquad \mathrm{runs}_\Phi(a) \le m_+(a) + \mathrm{comp}\bigl(I_+(a)\bigr) \le 2\, m_+(a), \tag{1.9}$$ $$d_\Phi(a) \le \min\Bigl(m_+(a) + |I_+^-(a)| - 1,\;\; W(a) + 1 + |\mathcal{U}_z(a)| + |\mathcal{D}_z(a)|\Bigr), \qquad \text{and on binary words } a \in \{1, 2\}^k:\;\; d_\Phi(a) = \left\lfloor \frac{\mathrm{runs}_\Phi(a)}{2} \right\rfloor \le m_+(a). \tag{1.10}$$
4. Explicit Raney Discrepancy Decompositions, Sturmian-to-Farey Bridge Identity, Farey-Chain Dynamic Range Lower Bound, & Determinant-GCD / Two-Window Elimination Laws (Theorem 4, Section 5): Recalling as baseline foundations the rotated Christoffel monomial ladder and max–min cancellation $2 C_{\min}(w^{\mathrm{Sturm}}) - C_{\max}(w^{\mathrm{Sturm}}) = 3^{k-1}$ of Knight [15, Theorems 3.4 & 5.4], the Sturmian extremality of Halbeisen and Hungerbühler [14] and Fernández and Ibáñez [16, Theorem 7.3], and the Christoffel matrix recurrence and parent/child monomial determinants of collatz_4/v1 [9, Section 5.1 & Theorem 3], we establish three new structural results:
   • The Explicit Primal/Dual Raney Discrepancy Decompositions $\Delta_{\min}^{\mathrm{Sturm}}(a), \Delta_{\max}^{\mathrm{Sturm}}(a) \ge 0$ ((5.5)–(5.8)) and the Sturmian-to-Farey Bridge Identity $D_+ \Delta_{\min}^{\mathrm{Sturm}}(a) - D \Delta_{\mathrm{pref}}^{\mathrm{Sturm}}(a) = 3^{k-1} 2^{q_+ - 1} + \tilde{R}^{(h_+, q_+)}(a^{(i_{\min})})$ ((5.9)), expressing the Farey sparse numerator $\tilde{R}^{(h_+, q_+)}$ of collatz_22/v4 [1] directly in terms of the full and prefix Sturmian discrepancies;
   • The Farey-Chain Dynamic Range Lower Bound $C_{\max}(a) - C_{\min}(a) \ge (\lceil k / m_+(a) \rceil - 1)\Delta_{\mathrm{Farey}}(k, S)$ ((5.10)–(5.10a)) whenever $\Delta_{\mathrm{Farey}}(k, S) > 0$ (which holds unconditionally in the straddling Farey regime $D_- \le 0 < D$, as well as on $(4, 7), (5, 9), (8, 13)$ for all compositions and on $(6, 11)$ for all binary compositions);
   • The Determinant-GCD Divisibility Law $\gcd(2^q - 3^h, 2^S - 3^k) \mid \gcd(2^\delta - 1, 3^\delta - 1)$ ($\delta = |k q - h S| \ge 1$, (5.11)), the Unit-Window Cramer Recovery ((5.12)), and the Two-Window State-Elimination and $i = 0$ Anchor-Elimination Resultants ((5.13)–(5.13a)).
5. Exact First-Passage Tiling, Universal $\{1, 2\}$ Excess-Class Law, 2-Adic Prefix-Pinning Mod-$3^r$ Injectivity, & Carry-Absorption / Ternary-Digit Suffix Tower / Mod-$6$ Cube Bijection for Rees's Digit Sum (Theorem 5, Section 6): Auditing the symmetric logarithmic-barrier theorem of u/lyapunov_weaver (3qXTNnezeSf9YZL8 and 59sRBgepkXuW6eIj), we prove that for all $1 \le j \le k - 4$, $3 t_j^2 > 2 t_{j-1}^2$ ($t_j = \min(j, k - j) + 1$) forces a unique first-passage weight $r = r_{\min}(j - 1)$ and $b_j = 0$ (even-step-only interior crossing), while on the terminal four steps $t = k - j + 1 \in \{1, 2, 3, 4\}$ the rescaled double-weight intervals $J_t = 2^{k - H}(a_t, a_{t-1}]$ ($a_t = 3^{t-1}/(2^t (t+1)^2)$) tile $2^{k - H}(27/400, 1/3]$ gaplessly with endpoint ratio $400/81 \in (3^1, 3^2)$, forcing (# first-hit classes) - (# first-hit times) $\in \{1, 2\}$ for all $k \ge H + 8$ with asymptotic Weyl density $\log_3(400/243) \approx 45.38\%$. Next, complementing Rees [17, Theorem 2.10 & Lemma 3.5] and wouZ9Hd3Wbyrpjjt / NlXvmFZpd_kXZplG, we prove that sharing $j_0$ prefix bits forces $2^{j_0} \mid (B(w') - B(w))$ and $|L(w') - L(w)| \ge 2^{j_0}$ at any mod-$3^r$ collision ($100\%$ injectivity across all $16{,}262$ good words in $E_k$ for $8 \le k \le 20$), and establish: (i) the Exact Integer Quotient & Unrounded Suffix-Sum Carry-Absorption Identities ((6.7)–(6.8)) $u_e \equiv -3^r \lfloor \sum_{i=1}^r a_{i, S_i + p}/2^{S_i} \rfloor \pmod{2^p}$; (ii) the Exact Ternary-Digit Suffix Tower ((6.9)–(6.11)) $u_e = 1 + \sum_{i=1}^r \kappa_i(e_i, \dots, e_r) 3^{r-i}$ yielding the unconditional bounds $\max_{v \bmod 3^m} \Pr(u_e \equiv v \pmod{3^m}) \le (1/(2 - \alpha))^m$ and $\max_{u \in \mathbb{Z}} \Pr(u_e = u) \le (1/(2 - \alpha))^r$; (iii) the Mod-$6$ Cube Partition & Exact Mod-$2^p$ Equidistribution Law ((6.12)–(6.13)); and (iv) the Closed-Form Mod-$3^r$ Endpoint Identity, 2-Adic / 3-Adic Carry Duality, & $(h + 1)$-Carry Reduction ((6.14)–(6.15)), reducing the unknown carry in the $h$-step split of NlXvmFZpd_kXZplG from $r$ values to $h + 1 \le \lfloor \frac{1}{4}\log_3 r \rfloor + 1$ values.

Exact Reproducibility Scope (code/verify_paper10.py): All finite checks in the accompanying self-contained pure-integer script code/verify_paper10.py (~35 s total runtime, 0 failures, using exact Python integer arithmetic with no floating-point logarithms or fractional powers) verify: (i) Theorem 1 across $1{,}327$ primitive rotation classes ($55$ binary), $2{,}651$ strict record prefixes, all $1{,}600$ Table 1 integer $T_{D_0}$ cycles, and the counterexample $a = (1, 1, 2, 3)$; (ii) Theorem 2 (including the residue-aware support bound (3.1), the $3x + 17$ cycle $y_0 = 23, y_1 = 25$, and the pure-integer step-ratio monotonicity certificates (3.7)–(3.8)) across all $11$ base pairs $(r_0, s_0)$ of Table 1; (iii) Theorem 3 across $6{,}036$ two-block splits, $20{,}371$ compositions ($923$ unrestricted compositions of $(3, 5), (4, 7), (5, 8), (5, 9), (8, 13)$ plus all $19{,}448$ binary compositions of $(17, 27)$), and $1{,}327$ primitive classes; (iv) Theorem 4 across $6{,}458$ adjacent Sturmian steps, $353$ coprime pairs ($2 \le k \le 30$), and $20{,}623$ compositions ($20{,}371$ above plus all $252$ unrestricted compositions of $(6, 11)$); and (v) Theorem 5 across $240$ values of $k \ge 24$ ($k \in [24, 260] \cup \{512, 1024, 2048\}$), all $16{,}262$ good words in $E_k$ ($8 \le k \le 20$), $1{,}815$ $(e, p)$ quotient/carry/ternary-tower instances, $168$ mod-$6$ cubes, and $1{,}278$ $h$-step carry-reduction splits.

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2. Sorted-Ascent Identity for Rank Descents $d(a)$ and Structural Partition of Translation Breakpoints $b_{\mathrm{gap}}(a)$

Let $y_1 < y_2 < \dots < y_k$ be the sorted odd states of any positive rational or integer cycle of $T_c(y) = (3y + c)/2^{v_2(3y + c)}$ ($c > 0$, or of any strict record-maximum trajectory prefix where $T_c(\{y_1, \dots, y_r\}) \not\subseteq \{y_1, \dots, y_r\}$ for $1 \le r < k$), with step valuation $\alpha(r) := a_{i_r} \ge 1$ and rank permutation $\pi(r) \in \{1, \dots, k\}$ defined by $T_c(y_r) = (3 y_r + c)/2^{\alpha(r)} = y_{\pi(r)}$. Let $E_1(a) := \#\{i : a_i = 1\}$ and $E_{\ge m}(a) := \#\{i : a_i \ge m\}$.

Theorem 1 (Sorted-Ascent Rank-Descent Identity & Structural Partition of Translation Breakpoints).

1. (Baseline Cut-Crossing Multiplicative Gap [collatz_22/v4 [1], Eq. (54) & Corollary 12] & $3/2$-Telescoping Run Law): Because the cycle is a single transitive permutation of $\{y_1, \dots, y_k\}$ (or a strict record-maximum prefix), for every $1 \le r \le k - 1$ the proper prefix $\{y_1, \dots, y_r\}$ is not $T_c$-invariant, so there exists $1 \le j \le r$ with $\pi(j) \ge r + 1$. Since $\alpha(j) \ge 1$ and $y_j \le y_r$, as established in collatz_22/v4 [1, Eq. (54) & Corollary 12], $$y_{r+1} \le y_{\pi(j)} = \frac{3 y_j + c}{2^{\alpha(j)}} \le \frac{3 y_j + c}{2} \le \frac{3 y_r + c}{2} \iff 2 y_{r+1} \le 3 y_r + c \iff 1 < \frac{y_{r+1} + c}{y_r + c} \le \frac{3}{2}. \tag{2.1}$$ Consequently, for every $1 \le r \le k - 1$ with $\alpha(r) = 1$, we have $y_{\pi(r)} = (3 y_r + c)/2 \ge y_{r+1}$, so $\pi(r) \ge r + 1$, and the shifted ratio telescopes across $[r, \pi(r) - 1]$ as $$\prod_{s = r}^{\pi(r) - 1} \frac{y_{s+1} + c}{y_s + c} = \frac{y_{\pi(r)} + c}{y_r + c} = \frac{3}{2}, \tag{2.2}$$ a product of $\pi(r) - r \ge 1$ factors each in $(1, 3/2]$. Thus $\pi(r) = r + 1$ whenever $y_{r+1} = (3 y_r + c)/2$, and $\pi(r) \ge r + 2$ requires $(y_{s+1} + c)/(y_s + c) < 3/2$ for every $s \in [r, \pi(r) - 1]$.
2. (Unconditional Sorted-Ascent Upper Bound & $(1, \ge 2)$-Transition Lower Bound for Rank Descents $d(a)$): For every $1 \le r \le k - 1$, $$\pi(r) > \pi(r+1) \iff T_c(y_r) > T_c(y_{r+1}) \iff 2^{\alpha(r+1) - \alpha(r)} > \frac{3 y_{r+1} + c}{3 y_r + c} > 1 \implies \alpha(r+1) \ge \alpha(r) + 1 \ge 2. \tag{2.3}$$ Conversely, whenever $\alpha(r) = 1$, $\alpha(r+1) \ge 2$, and $T_c(y_{r+1}) < y_{r+1}$ (which holds whenever $(2^{\alpha(r+1)} - 3)y_{r+1} > c$, and in particular whenever $y_{r+1} > c$), we have $T_c(y_{r+1}) < y_{r+1} \le T_c(y_r)$ and hence the strict rank separation $\pi(r+1) \le r < r + 1 \le \pi(r)$. Therefore, the Eulerian rank-descent count $d(a) = |\mathcal{D}(a)| = \#\{1 \le r < k : \pi(r) > \pi(r+1)\}$ of every cycle (and every strict record-maximum trajectory prefix) satisfies unconditionally: $$\#\bigl\{1 \le r < k : \alpha(r) = 1 < \alpha(r+1) \text{ and } T_c(y_{r+1}) < y_{r+1}\bigr\} \le d(a) \le \mathrm{ascents}(\alpha) \le \min\bigl(E_{\ge 2}(a),\, E_1(a) + E_{\ge 3}(a)\bigr) \le E_{\ge 2}(a) \le S - k. \tag{2.4}$$
3. (Exact Sorted-Ascent Identity for All Positive Integer Cycles and All Rational Cycles with $3 C_{\min}(a) > D$): Multiplying $2 y_{r+1} \le 3 y_r + c$ from (2.1) by $3$ and adding $2c$ gives $2(3 y_{r+1} + c) \le 9 y_r + 5c = 3(3 y_r + c) + 2c$, so for every $1 \le r \le k - 1$, $$1 < \frac{3 y_{r+1} + c}{3 y_r + c} \le \frac{3}{2} + \frac{c}{3 y_r + c}. \tag{2.5}$$ Whenever $3 y_1 > c$ (which holds unconditionally on every positive integer cycle of $3x + 1$ since $3 y_1 \ge 3 > 1 = c$, and on every rational cycle with $3 C_{\min}(a) > D$, including all words with $2^S < 4 \cdot 3^k - 3 \cdot 2^k$ since $C_{\min}(a) \ge 3^k - 2^k$), we have $3 y_r + c \ge 3 y_1 + c > 2c$, so the right-hand side of (2.5) is strictly less than $\frac{3}{2} + \frac{1}{2} = 2$ for all $1 \le r < k$. Consequently, $2^{\alpha(r+1) - \alpha(r)} > (3 y_{r+1} + c)/(3 y_r + c)$ holds if and only if $\alpha(r) < \alpha(r+1)$, so the rank-descent set is identically the strict ascent set of $\alpha$: $$\mathcal{D}(a) = \bigl\{1 \le r \le k - 1 : \alpha(r) < \alpha(r+1)\bigr\}, \qquad d(a) = \mathrm{ascents}(\alpha) = \#\bigl\{1 \le r \le k - 1 : \alpha(r) < \alpha(r+1)\bigr\}, \tag{2.6}$$ and on any binary word $a \in \{1, 2\}^k$ with $3 C_{\min}(a) > D$, $d(a) = \#\{1 \le r < k : (\alpha(r), \alpha(r+1)) = (1, 2)\} \le \min(E_1(a), E_2(a)) = \min(2k - S, S - k)$.
4. (Exact Partition of Translation Breakpoints $\mathcal{B}_{\mathrm{gap}}(a)$ and Connection to $d(a)$): For any cycle, the translation-breakpoint set $\mathcal{B}_{\mathrm{gap}}(a) = \{1 \le r < k : \pi(r+1) \ne \pi(r) + 1 \text{ or } \alpha(r+1) \ne \alpha(r)\}$ of collatz_22/v4 [1, Theorem 10] partitions into the disjoint union $$\mathcal{B}_{\mathrm{gap}}(a) = \mathcal{V}(a) \;\sqcup\; \mathcal{J}_{\mathrm{eq}}(a), \qquad b_{\mathrm{gap}}(a) = \bigl(\mathrm{runs}(\alpha) - 1\bigr) + |\mathcal{J}_{\mathrm{eq}}(a)|, \tag{2.7}$$ where $\mathcal{V}(a) := \{1 \le r < k : \alpha(r) \ne \alpha(r+1)\}$ ($|\mathcal{V}(a)| = \mathrm{runs}(\alpha) - 1 = \mathrm{ascents}(\alpha) + \mathrm{descents}(\alpha) \le 2 E_{\ge 2}(a)$) and $\mathcal{J}_{\mathrm{eq}}(a) := \{1 \le r < k : \alpha(r) = \alpha(r+1) \text{ and } \pi(r+1) \ge \pi(r) + 2\}$. For each valuation $v \ge 1$, the map $y_s \mapsto T_c(y_s) = (3 y_s + c)/2^v$ is strictly increasing on $\{s : \alpha(s) = v\}$, so as $r$ varies over $\mathcal{J}_{\mathrm{eq}}^{(v)}(a) := \{r \in \mathcal{J}_{\mathrm{eq}}(a) : \alpha(r) = v\}$, the open rank intervals $(\pi(r), \pi(r+1))$ are pairwise disjoint and contain only images $\pi(s)$ of states with $\alpha(s) \ne v$. In particular, $|\mathcal{J}_{\mathrm{eq}}^{(1)}(a)| \le E_{\ge 2}(a)$, and on any binary word $a \in \{1, 2\}^k$ with $3 y_1 > c$, $\mathrm{runs}(\alpha) - 1 = 2 d(a) + \mathbf{1}_{\{\alpha(1)=2\}} - \mathbf{1}_{\{\alpha(k)=2\}} \le 2 d(a) + 1$ and $b_{\mathrm{gap}}(a) \le 2 d(a) + 1 + E_2(a)$.

Proof. (1): By collatz_22/v4 [1, Eq. (54) & Corollary 12], picking $j \in \{1, \dots, r\}$ with $\pi(j) \ge r + 1$ gives $y_{r+1} \le y_{\pi(j)} = (3 y_j + c)/2^{\alpha(j)} \le (3 y_j + c)/2 \le (3 y_r + c)/2$, which rearranges to $2 y_{r+1} \le 3 y_r + c$ and $2(y_{r+1} + c) \le 3(y_r + c)$, proving (2.1)–(2.2).

(2): Since $y_r < y_{r+1}$ and $c > 0$, $(3 y_{r+1} + c)/(3 y_r + c) > 1$. By definition of $T_c$, $$\frac{T_c(y_r)}{T_c(y_{r+1})} = 2^{\alpha(r+1) - \alpha(r)} \cdot \frac{3 y_r + c}{3 y_{r+1} + c},$$ which is $> 1$ if and only if $2^{\alpha(r+1) - \alpha(r)} > (3 y_{r+1} + c)/(3 y_r + c)$. Since the right-hand side is $> 1$, this requires $\alpha(r+1) - \alpha(r) \ge 1$, proving (2.3). Each strict ascent $\alpha(r) < \alpha(r+1)$ has $\alpha(r+1) \ge 2$ (an injection into $\{s \ge 2 : \alpha(s) \ge 2\}$ of size $\le E_{\ge 2}(a)$) and either $\alpha(r) = 1$ or $\alpha(r+1) \ge 3$ (an injection into $\{s : \alpha(s) = 1\} \cup \{s : \alpha(s) \ge 3\}$ of size $E_1(a) + E_{\ge 3}(a)$). When $\alpha(r) = 1$, $\alpha(r+1) \ge 2$, and $T_c(y_{r+1}) < y_{r+1}$, (2.1) gives $T_c(y_{r+1}) < y_{r+1} \le (3 y_r + c)/2 = T_c(y_r)$, so $\pi(r+1) \le r < r + 1 \le \pi(r)$, proving (2.4).

(3): Multiplying $2 y_{r+1} \le 3 y_r + c$ by $3$ and adding $2c$ gives $2(3 y_{r+1} + c) \le 9 y_r + 5c = 3(3 y_r + c) + 2c$; dividing by $2(3 y_r + c)$ gives (2.5). When $3 y_1 > c$, $3 y_r + c \ge 3 y_1 + c > 2c$, so $\frac{3}{2} + \frac{c}{3 y_r + c} < \frac{3}{2} + \frac{1}{2} = 2$. Hence for every $1 \le r < k$, $\alpha(r) < \alpha(r+1) \implies 2^{\alpha(r+1) - \alpha(r)} \ge 2 > (3 y_{r+1} + c)/(3 y_r + c) \implies \pi(r) > \pi(r+1)$. Combining with (2.3) proves (2.6).

(4): Whenever $\alpha(r) = \alpha(r+1) = v$, $T_c(y_r) = (3 y_r + c)/2^v < (3 y_{r+1} + c)/2^v = T_c(y_{r+1})$, so $\pi(r) < \pi(r+1)$, and thus $\pi(r+1) \ne \pi(r) + 1 \iff \pi(r+1) \ge \pi(r) + 2$, proving (2.7). Because $y_r < y_{r+1}$ are adjacent in sorted order, no state $y_s$ lies in $(y_r, y_{r+1})$, so no state $y_s$ with $\alpha(s) = v$ can have $T_c(y_s) \in (T_c(y_r), T_c(y_{r+1}))$. Since $\pi$ is strictly increasing on $\{s : \alpha(s) = v\}$, the open intervals $(\pi(r), \pi(r+1))$ for $r \in \mathcal{J}_{\mathrm{eq}}^{(v)}(a)$ are pairwise disjoint subsets of $\{\pi(s) : \alpha(s) \ne v\}$, so $|\mathcal{J}_{\mathrm{eq}}^{(1)}(a)| \le \#\{s : \alpha(s) \ge 2\} = E_{\ge 2}(a)$. $\blacksquare$

Remark 2.1 (Distinction Between $d(a)$ and $b_{\mathrm{gap}}(a)$ on $a = (1, 1, 2, 3)$ and Across $1{,}327$ Primitive Words & $1{,}600$ Table 1 Cycles). For $a = (1, 1, 2, 3)$ ($k = 4, S = 7, D = 47$), the chronological numerators are $(73, 133, 223, 179)$, so the sorted states are $(y_1, y_2, y_3, y_4) = (73, 133, 179, 223)$ with sorted valuations $\alpha = (1, 1, 3, 2)$ and rank permutation $\pi = (2, 4, 1, 3)$. Here $3 y_1 = 219 > 47 = D$, so $\mathcal{D}(a) = \{2\}$ and $d(a) = \mathrm{ascents}(\alpha) = 1$, whereas $\mathcal{V}(a) = \{2, 3\}$ and $\mathcal{J}_{\mathrm{eq}}(a) = \{1\}$ ($\pi(1) = 2 < 4 = \pi(2)$ jumps over $3 = \pi(4)$), giving $b_{\mathrm{gap}}(a) = |\mathcal{V}(a)| + |\mathcal{J}_{\mathrm{eq}}(a)| = 2 + 1 = 3$. Across all $1{,}327$ primitive rotation classes tested in code/verify_paper10.py, $d(a) = \mathrm{ascents}(\alpha)$ holds on $1{,}327 / 1{,}327$ ($100\%$) of words, $d(a) = E_{\ge 2}(a)$ is achieved on $445 / 1{,}327$ words, and $d(a) = \min(E_1(a), E_2(a))$ is achieved on $23 / 55$ binary words. Across all $1{,}600$ integer $T_{D_0}$ cycles in the $11$-pair census of collatz_25/v1 [3] Table 1, $d(a) = \mathrm{ascents}(\alpha)$ holds on $1{,}598 / 1{,}600$ cycles; the only $2$ cycles with $d(a) < \mathrm{ascents}(\alpha)$ occur at $(r_0, s_0) = (8, 14)$ ($D_0 = 9{,}823$, $k = 40, q_{\min} = 869$ and $k = 32, q_{\min} = 1{,}733$), both at $r = 1$ where $3 y_1 < D_0$ and $y_2 = (3 y_1 + D_0)/2 < D_0$.

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3. Coprime-to-$6$ 9th-Power Envelope, Mod-$6$ $L_c$-Telescoping Stopping Certificate, and Step-Ratio Monotonicity

In vqJdfHo_yZLZYaMx, u/lyapunov_weaver derived a 6th-power stopping cutoff $D_{\mathrm{stop}}^{\mathrm{Weaver}}(r, s)$ for fixed-slope cycles of length $k = r D$ and valuation sum $S = s D$ ($D = \gcd(k, S)$) under $T_c$ ($c = D_0 / \gcd(k, D_0)$, $D_0 = 2^s - 3^r$) by using the odd-integer lower bound $y_i \ge 2i + 1$ ($0 \le i < k$) and the logarithmic inequality $6 \ln(1 + c/(3 y_i)) < 2c/(2i + 1) < c \ln((4i + 3)/(4i + 1))$. We sharpen this scale cutoff in two ways by exploiting the exact mod-$6$ support of $T_c$ cycles and proving explicit step-ratio monotonicity certificates that exclude all larger scales $D \ge D_{\mathrm{stop}}$.

Theorem 2 (Coprime-to-$6$ 9th-Power Envelope, Mod-$6$ $L_c$-Telescoping Stopping Certificate, & All-$D$ Monotonicity). Let $c \in \mathbb{Z}_{\ge 1}$ satisfy $\gcd(c, 6) = 1$ (which holds for every $c \mid D_0 = 2^s - 3^r$), and let $y_0 < y_1 < \dots < y_{k-1}$ ($k = r D, S = s D$) be the sorted states of any positive integer cycle of $T_c$.

1. (Coprime-to-$6$ Support & Residue-Aware Lower Bound): Every cycle state $y_i$ satisfies $\gcd(y_i, 6) = 1$, because $2^{\alpha_{i-1}} y_i = 3 y_{i-1} + c \equiv c \not\equiv 0 \pmod 3$ and $y_i$ is odd. Letting $(u_i)_{i \ge 0} = (1, 5, 7, 11, 13, 17, 19, 23, \dots)$ denote the positive integers coprime to $6$ in increasing order ($u_{2m} = 6m + 1$, $u_{2m+1} = 6m + 5$), the $i$-th sorted state $y_i$ is at least the $i$-th element of $\{y \ge y_0 : \gcd(y, 6) = 1\}$, so for all $i \ge 0$: $$y_i \ge \begin{cases} y_0 + u_i - 1 & \text{if } y_0 \equiv 1 \pmod 6,\\ y_0 + u_{i+1} - 5 & \text{if } y_0 \equiv 5 \pmod 6 \end{cases} \;\ge\; u_i, \qquad \text{and for all } i \ge 1:\quad u_i = 3i + 1 + (i \bmod 2) \ge 3i + 1. \tag{3.1}$$
2. (Exact-$m_0$ Coprime-to-$6$ 9th-Power Envelope): For every $t > 0$, $\ln(1 + t) < t$ and $\ln\frac{1 + t}{1 - t} = 2\sum_{j=0}^\infty \frac{t^{2j+1}}{2j+1} > 2t$. Applying the first inequality with $t = c/(3 y_i) \le c/(3(3i + 1))$ and the second with $t = 3/(6i + 2)$ gives, for every $i \ge 1$, $$9\ln\!\left(1 + \frac{c}{3 y_i}\right) < \frac{3c}{3i + 1} = \frac{6c}{6i + 2} < c\ln\!\left(\frac{6i + 5}{6i - 1}\right). \tag{3.2}$$ Keeping the first $m_0 \ge 1$ factors $i = 0, \dots, m_0 - 1$ of the cycle product $2^{s D} = \prod_{i=0}^{r D - 1} (3 + c/y_i)$ exact (with $y_i \ge u_i$) and telescoping (3.2) over $m_0 \le i \le r D - 1$ yields the pure-integer 9th-Power Envelope (necessary for a cycle of scale $D \ge \lceil m_0 / r \rceil$): $$2^{9 s D} \cdot (6 m_0 - 1)^c \prod_{i=0}^{m_0 - 1} u_i^9 < 3^{9(r D - m_0)} \cdot (6 r D - 1)^c \prod_{i=0}^{m_0 - 1} (3 u_i + c)^9, \tag{3.3}$$ which at $m_0 = 1$ ($u_0 = 1$) takes the $O(1)$-term closed form $$2^{9 s D} \cdot 5^c < (3 + c)^9 \cdot 3^{9 r D - 9} \cdot (6 r D - 1)^c. \tag{3.4}$$
3. (Mod-$6$ $L_c$-Telescoping 1st-Power Stopping Certificate): Let $L_c := \lceil c / 18 \rceil \ge 1$, so $c \le 18 L_c$ and $3 u_i + c \le 3(u_i + 6 L_c)$ for all $i \ge 1$. Because the sequence $(u_i)_{i \ge 0}$ is $2$-periodic modulo $6$, it satisfies the exact shift invariance $$u_{i + 2 L_c} = u_i + 6 L_c \qquad (\forall\, i \ge 0). \tag{3.5}$$ Therefore, for all $k = r D \ge 2 L_c$, the product $\prod_{i=1}^{k-1} (u_i + 6 L_c)/u_i = \prod_{i=1}^{k-1} u_{i + 2 L_c}/u_i$ telescopes exactly across both mod-$6$ residue classes to just $2 L_c$ boundary factors, yielding the 1st-power $2 L_c$-term pure-integer necessary condition: $$2^{s D} \cdot (1 + 6 L_c) \prod_{i=0}^{2 L_c - 1} u_i < (3 + c) \cdot 3^{r D - 1} \prod_{j = r D}^{r D + 2 L_c - 1} u_j. \tag{3.6}$$
4. (Pure-Integer Step-Ratio Monotonicity Certifying Exclusion of All $D \ge D_{\mathrm{stop}}$): Let $\mathcal{R}_{9\mathrm{th}}(D)$ and $\mathcal{R}_{\mathrm{tele}}(D)$ denote the right-to-left ratios of (3.3) and (3.6) respectively. Stepping $D \mapsto D + 1$ multiplies $\mathcal{R}_{9\mathrm{th}}(D)$ and $\mathcal{R}_{\mathrm{tele}}(D)$ by the step factors $$\frac{\mathcal{R}_{9\mathrm{th}}(D + 1)}{\mathcal{R}_{9\mathrm{th}}(D)} = \frac{3^{9r}}{2^{9s}} \left(\frac{6 r (D + 1) - 1}{6 r D - 1}\right)^c, \qquad \frac{\mathcal{R}_{\mathrm{tele}}(D + 1)}{\mathcal{R}_{\mathrm{tele}}(D)} = \frac{3^r}{2^s} \prod_{m=0}^{r-1} \frac{u_{r D + m + 2 L_c}}{u_{r D + m}} = \frac{3^r}{2^s} \prod_{m=0}^{r-1} \left(1 + \frac{6 L_c}{u_{r D + m}}\right), \tag{3.7}$$ both of which are strictly decreasing functions of $D \ge 1$. Consequently, whenever $D_{\mathrm{stop}}$ satisfies $\mathcal{R}_{9\mathrm{th}}(D_{\mathrm{stop}}) \le 1$ and the pure-integer step-ratio inequality $$3^{9r} \cdot \bigl(6 r (D_{\mathrm{stop}} + 1) - 1\bigr)^c < 2^{9s} \cdot \bigl(6 r D_{\mathrm{stop}} - 1\bigr)^c \qquad \text{(or } 3^r \prod_{m=0}^{r-1} u_{r D_{\mathrm{stop}} + m + 2 L_c} < 2^s \prod_{m=0}^{r-1} u_{r D_{\mathrm{stop}} + m} \text{ for } \mathcal{R}_{\mathrm{tele}}\text{)}, \tag{3.8}$$ induction on $D$ proves $\mathcal{R}(D) \le \mathcal{R}(D_{\mathrm{stop}}) \le 1$ for every integer $D \ge D_{\mathrm{stop}}$, unconditionally excluding all cycles of scale $D \ge D_{\mathrm{stop}}$.

Proof. (1): If $y_0 = 6 q_0 + 1$, the coprime-to-$6$ integers $\ge y_0$ are $6 q_0 + u_i = y_0 + u_i - 1 \ge u_i$ ($i \ge 0$); if $y_0 = 6 q_0 + 5$, they are $6 q_0 + u_{i+1} = y_0 + u_{i+1} - 5 \ge u_{i+1} - 4 \ge u_i$ ($i \ge 0$, with equality $y_1 = y_0 + 2 = y_0 + u_2 - 5$ when $y_1 = y_0 + 2$, as in the $3x + 17$ cycle $y_0 = 23, y_1 = 25$), proving (3.1).

(2) & (3): Raising $2^{s D} \prod_{i=0}^{r D - 1} y_i = \prod_{i=0}^{r D - 1} (3 y_i + c)$ to the 9th power, using $(3 + c/y_i)^9 \le (3 + c/u_i)^9$ for $0 \le i < m_0$, and telescoping (3.2) over $m_0 \le i < r D$ gives (3.3)–(3.4). Since $u_{2(m + L_c)} = u_{2m} + 6 L_c$ and $u_{2(m + L_c) + 1} = u_{2m + 1} + 6 L_c$, (3.5) holds for all $i \ge 0$, and for $k = r D \ge 2 L_c$, $$2^{s D} = \left(3 + \frac{c}{y_0}\right)\prod_{i=1}^{k-1} \left(3 + \frac{c}{y_i}\right) \le (3 + c) \cdot 3^{k-1} \prod_{i=1}^{k-1} \frac{u_i + 6 L_c}{u_i} = (3 + c) \cdot 3^{k-1} \frac{\prod_{j=k}^{k + 2 L_c - 1} u_j}{\prod_{i=1}^{2 L_c} u_i},$$ with strict inequality at $i = 0$ (or since $c < 18 L_c$ when $9 \nmid c$); noting $u_0 = 1$ and $u_{2 L_c} = 1 + 6 L_c$ gives (3.6).

(4): In (3.7), $1 + 6r / (6 r D - 1)$ and $1 + 6 L_c / u_{r D + m}$ are strictly decreasing in $D$. Thus if the step factor at $D = D_{\mathrm{stop}}$ is $< 1$ ((3.8)), then for all $D \ge D_{\mathrm{stop}}$ we have $\mathcal{R}(D + 1) / \mathcal{R}(D) \le \mathcal{R}(D_{\mathrm{stop}} + 1) / \mathcal{R}(D_{\mathrm{stop}}) < 1$, so $\mathcal{R}(D_{\mathrm{stop}}) \le 1$ implies $\mathcal{R}(D) < 1$ for all $D > D_{\mathrm{stop}}$. $\blacksquare$

Table 1: Certified Scale Cutoffs $D_{\mathrm{stop}}$ Across All 11 Base Pairs $(r_0, s_0)$ (code/verify_paper10.py)

In Table 1, $D_{\mathrm{Weaver}}$ is u/lyapunov_weaver's 6th-power scale cutoff (vqJdfHo_yZLZYaMx), $D_{9\mathrm{th}}^{(m_0=1)}$ is (3.4), $D_{9\mathrm{th}}^{(m_0=8)}$ is (3.3) with $m_0 = 8$, $D_{\mathrm{tele}}$ is (3.6) with $L_c = \lceil D_0 / 18 \rceil$, $D_{\mathrm{cert}} := \min(D_{9\mathrm{th}}^{(m_0=1)}, D_{9\mathrm{th}}^{(m_0=8)}, D_{\mathrm{tele}})$, and $D_{\mathrm{exact}}$ is the exact full-product threshold $2^{s D} \prod_{i=0}^{r D - 1} u_i > \prod_{i=0}^{r D - 1} (3 u_i + D_0)$. For every pair in Table 1, code/verify_paper10.py verifies both the failure of (3.3)/(3.6) at $D_{\mathrm{stop}}$ and the step-ratio monotonicity certificate (3.8), certifying exclusion of all $D \ge D_{\mathrm{cert}}$.

| Base Pair $(r_0, s_0)$ | $D_0 = 2^{s_0} - 3^{r_0}$ | $L_c = \lceil D_0/18 \rceil$ | $D_{\mathrm{Weaver}}$ (6th-power) | $D_{9\mathrm{th}}^{(m_0=1)}$ (3.4) | $D_{9\mathrm{th}}^{(m_0=8)}$ (3.3) | $D_{\mathrm{tele}}$ (3.6) | $D_{\mathrm{cert}}$ | $D_{\mathrm{exact}}$ (full $\prod u_i$) | Cutoff Reduction $D_{\mathrm{Weaver}} / D_{\mathrm{cert}}$ | | :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | $(3, 5)$ | $5$ | $1$ | $30$ | $20$ | $19$ | $59$ | $19$ | $18$ | $1.58\times$ | | $(4, 7)$ | $47$ | $3$ | $117$ | $74$ | $59$ | $62$ | $59$ | $55$ | $1.98\times$ | | $(5, 8)$ | $13$ | $1$ | $356$ | $233$ | $213$ | $285$ | $213$ | $210$ | $1.67\times$ | | $(5, 9)$ | $269$ | $15$ | $494$ | $308$ | $226$ | $178$ | $178$ | $177$ | $2.78\times$ | | $(6, 10)$ | $295$ | $17$ | $1{,}359$ | $856$ | $660$ | $561$ | $561$ | $541$ | $2.42\times$ | | $(6, 11)$ | $1{,}319$ | $74$ | $2{,}074$ | $1{,}304$ | $969$ | $605$ | $605$ | $599$ | $3.43\times$ | | $(7, 12)$ | $1{,}909$ | $107$ | $5{,}511$ | $3{,}488$ | $2{,}686$ | $1{,}719$ | $1{,}719$ | $1{,}703$ | $3.21\times$ | | $(8, 13)$ | $1{,}631$ | $91$ | $14{,}674$ | $9{,}337$ | $7{,}444$ | $5{,}289$ | $5{,}289$ | $5{,}266$ | $2.77\times$ | | $(8, 14)$ | $9{,}823$ | $546$ | $22{,}143$ | $14{,}101$ | $11{,}197$ | $5{,}651$ | $5{,}651$ | — | $3.92\times$ | | $(9, 15)$ | $13{,}085$ | $727$ | $57{,}545$ | $36{,}792$ | $29{,}889$ | $15{,}972$ | $15{,}972$ | — | $3.60\times$ | | $(10, 16)$ | $6{,}487$ | $361$ | $150{,}610$ | $96{,}634$ | $80{,}199$ | $52{,}585$ | $52{,}585$ | — | $2.86\times$ |

Notice that for $(6, 11)$ ($D_0 = 1{,}319$), $D_{\mathrm{tele}} = 605$ is within $6$ scale units ($1\%$) of the exact full-product threshold $D_{\mathrm{exact}} = 599$ while multiplying only $2 L_c = 148$ integers instead of $r D = 3{,}630$, and for $(10, 16)$ ($D_0 = 6{,}487$), $D_{\mathrm{tele}}$ reduces the certified scale cutoff from $150{,}610$ to $52{,}585$ ($2.86\times$ cutoff reduction) using only $2 L_c = 722$ boundary factors.

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4. The Two-Block Determinant Identity, Linear-Form State-Ordering Isomorphism ($\Phi_i(a) = i S - k T_i$), and Direct $(b_{\mathrm{gap}}^\Phi(a), d_\Phi(a), m_+(a))$ Bridge

How does the sorted cycle state order $x_{i_1} < x_{i_2} < \dots < x_{i_k}$ relate to the canonical Raney root $i_0 \in \operatorname{argmax}_{0 \le i < k}(k T_i - i S)$ and canonical floor discrepancy profile $z_i = \lfloor i S / k \rfloor - T_i \in \{0, 1, \dots, W(a)\}$ of collatz_22/v4 [1] Theorem 5A and collatz_25/v1 [3] Theorem 1?

Theorem 3 (Two-Block Determinant Identity, Linear-Form State-Ordering Isomorphism, & Direct $(b_{\mathrm{gap}}^\Phi, d_\Phi, m_+)$ Bridge). For any word $a \in \mathcal{A}(k, S)$ ($k < S < 2k$, $D = 2^S - 3^k > 0$) at a canonical Raney root $z_0 = 0$, define the integer linear form $\Phi_i(a) := i S - k T_i = k\bigl(z_i + \{i S / k\}\bigr) \ge 0$ ($0 \le i < k$), the mechanical Sturmian word $a_i^{\mathrm{Sturm}} := \lfloor (i+1)S/k \rfloor - \lfloor i S / k \rfloor = 1 + \mathbf{1}_{\{(i S \bmod k) \ge 2k - S\}} \in \{1, 2\}$, and the upward and downward step sets of $z$: $$\mathcal{U}_z(a) := \bigl\{i \in \mathbb{Z}/k\mathbb{Z} : z_{i+1} - z_i = +1\bigr\}, \qquad \mathcal{D}_z(a) := \bigl\{i \in \mathbb{Z}/k\mathbb{Z} : z_{i+1} - z_i < 0\bigr\}. \tag{4.1}$$

1. (Exact Sturmian-Defect Set Identities & High-Step Conservation): Since $z_{i+1} - z_i = a_i^{\mathrm{Sturm}} - a_i$, letting $H_{\mathrm{Sturm}} := \{i : a_i^{\mathrm{Sturm}} = 2\}$, $L_{\mathrm{Sturm}} := \{i : a_i^{\mathrm{Sturm}} = 1\}$, $L := \{i : a_i = 1\}$, and $H := \{i : a_i \ge 2\}$, we have identically $$\mathcal{U}_z(a) = H_{\mathrm{Sturm}} \cap L, \qquad L_{\mathrm{Sturm}} \cap \mathcal{D}_z(a) = L_{\mathrm{Sturm}} \cap H, \qquad \|a - a^{\mathrm{Sturm}}\|_1 = \sum_{i=0}^{k-1} |z_{i+1} - z_i| = 2|\mathcal{U}_z(a)| \ge 2 W(a), \tag{4.2}$$ $$E_{\ge 2}(a) = |H| = (S - k) - |\mathcal{U}_z(a)| + |L_{\mathrm{Sturm}} \cap \mathcal{D}_z(a)| \le (S - k) - \bigl(|\mathcal{U}_z(a)| - |\mathcal{D}_z(a)|\bigr). \tag{4.3}$$
2. (Universal Two-Block Determinant Identity & State-Ordering Isomorphism on the Stern–Brocot Upper Spine and Small-Drift Cycles): For any cyclic shift $a^{(i)} = (u, v)$ split into a prefix $u = (a_i, \dots, a_{i+h-1}) \in \mathcal{A}(h, q)$ ($1 \le h < k$, $q = T_{i+h} - T_i$) and complementary suffix $v = (a_{i+h}, \dots, a_{i+k-1}) \in \mathcal{A}(k - h, S - q)$, expanding $C(a^{(i)}) = 3^{k-h} C(u) + 2^q C(v)$ and $C(a^{(i+h)}) = 3^h C(v) + 2^{S-q} C(u)$ yields the exact $2 \times 2$ determinant identity $$C\bigl(a^{(i)}\bigr) - C\bigl(a^{(i+h)}\bigr) = C(u, v) - C(v, u) = \bigl(2^q - 3^h\bigr)C(v) - \bigl(2^{S-q} - 3^{k-h}\bigr)C(u) = \det\begin{pmatrix} D(u) & D(v) \\ C(u) & C(v) \end{pmatrix}. \tag{4.4}$$ Whenever $\Phi_{i+h}(a) < \Phi_i(a)$ ($k q - h S = \delta \ge 1$), we have $q/h > S/k > \log_2 3$ ($D(u) = 2^q - 3^h > 0$) and $(S - q)/(k - h) = (h' S - \delta)/(h' k) < S/k$ ($h' = k - h$). Consequently:
   • (a) On every Stern–Brocot upper pair $(k, S)$ ($D = 2^S - 3^k > 0$ and $2^{\lfloor (h' S - 1)/k \rfloor} < 3^{h'}$ for all $1 \le h' < k$, an infinite spine including $(3, 5), (5, 8), (17, 27), (29, 46), (41, 65), (94, 149), \dots$), we have $D(v) = 2^{S-q} - 3^{h'} < 0$ for every complementary suffix $v$, so $C(a^{(i)}) - C(a^{(i+h)}) = D(u) C(v) + |D(v)| C(u) > 0$ holds unconditionally for all compositions $a \in \mathcal{A}(k, S)$ (and likewise on $(4, 7), (5, 9), (8, 13)$ by simplex bounds);
   • (b) On any coprime cycle ($\gcd(k, S) = 1$) whose minimum state $x_{\min}(a) = C_{\min}(a)/D$ satisfies the explicit small-drift bound $x_{\min}(a) > k^2 / (3\ln 2)$ (so that $\sigma_k := S - k\log_2 3 < k / (3 x_{\min}(a) \ln 2) < 1/k$ and $|k\log_2(x_i / x_j) - (\Phi_i(a) - \Phi_j(a))| < 1$ whenever $x_i > x_j$); in both regimes (a) and (b), the sorted cycle state order is 100% identical to the linear-form order $\Phi$: $$C\bigl(a^{(i)}\bigr) < C\bigl(a^{(j)}\bigr) \iff \Phi_i(a) < \Phi_j(a) \qquad (\forall\, i, j \in \{0, \dots, k-1\}). \tag{4.5}$$ Whenever (4.5) holds, $x_{\min}(a)$ is attained at the unique canonical Raney root $i_0 = \operatorname{argmin}_i \Phi_i(a)$, $x_{\max}(a)$ is attained at $i_{\max} = \operatorname{argmax}_i \Phi_i(a)$, $d(a) = d_\Phi(a)$ (when $3 y_1 > c$), and $b_{\mathrm{gap}}(a) = b_{\mathrm{gap}}^\Phi(a)$.
3. (Canonical Level-Set Variation Bound for $d_\Phi(a)$): For any coprime word $a \in \mathcal{A}(k, S)$, let $i_1, \dots, i_k$ be ordered by $\Phi_{i_1}(a) < \dots < \Phi_{i_k}(a)$, with $\alpha_\Phi(r) := a_{i_r}$, $\pi_\Phi(r)$ the $\Phi$-rank of $(i_r + 1) \bmod k$, $d_\Phi(a) := \#\{1 \le r < k : \alpha_\Phi(r) < \alpha_\Phi(r+1)\}$, and $b_{\mathrm{gap}}^\Phi(a) := \#\{1 \le r < k : \pi_\Phi(r+1) \ne \pi_\Phi(r) + 1 \text{ or } \alpha_\Phi(r+1) \ne \alpha_\Phi(r)\}$. Grouping $\{0, \dots, k-1\}$ into the $W(a) + 1$ canonical level sets $Z_v = \{i : z_i = v\}$ ($0 \le v \le W(a)$) yields unconditionally: $$d_\Phi(a) \le \bigl(W(a) + 1\bigr) + |\mathcal{U}_z(a)| + |\mathcal{D}_z(a)| \le W(a) + 1 + \|a - a^{\mathrm{Sturm}}\|_1. \tag{4.6}$$
4. (Piecewise-Translation Identity, Residue-Interval Overlay, & Direct $(b_{\mathrm{gap}}^\Phi(a), d_\Phi(a), m_+(a))$ Bridge): When $\gcd(k, S) = 1$, let $(h_+, q_+)$ ($k q_+ - h_+ S = 1$) be the upper Farey parent, $H_i := T_{i+h_+} - T_i - q_+$, $I_+(a) := \{i \in \mathbb{Z}/k\mathbb{Z} : H_i \ne 0\}$ ($m_+(a) = |I_+(a)|$), $I_+^-(a) := \{i \in I_+(a) : H_i < 0\}$, and $\mathrm{comp}(I_+(a)) := \#\{i \notin I_+(a) : i + 1 \in I_+(a)\} \le m_+(a)$. For every $i \in \mathbb{Z}/k\mathbb{Z}$, advancing one step $i \mapsto (i + 1) \bmod k$ satisfies the exact piecewise-translation identity in $\mathbb{Z}$: $$\Phi_{(i+1)\bmod k}(a) = \Phi_i(a) + S - k\, a_i. \tag{4.7}$$ Moreover, for every $i \notin I_+(a)$, we have $i \ne 0$ (since $0 \in I_+(a)$) and $\Phi_i(a) - \Phi_{i+h_+}(a) = k q_+ - h_+ S = +1$, so the immediate predecessor of $i$ in $\Phi$-order is $(i + h_+) \bmod k$, and $a_i - a_{i+h_+} = H_i - H_{i+1} = -H_{i+1}$. Whenever $i_{r+1} \notin I_+(a)$ and $i_{r+1} + 1 \notin I_+(a)$, we have both $\Phi_{i_{r+1}}(a) - \Phi_{i_r}(a) = +1$ and $\alpha_\Phi(r+1) = \alpha_\Phi(r) =: v$, so (4.7) gives $\Phi_{i_{\pi_\Phi(r+1)}}(a) - \Phi_{i_{\pi_\Phi(r)}}(a) = (\Phi_{i_{r+1}}(a) + S - k v) - (\Phi_{i_r}(a) + S - k v) = +1$, which forces $\pi_\Phi(r+1) = \pi_\Phi(r) + 1$. Consequently, every coprime word $a \in \mathcal{A}(k, S)$ satisfies unconditionally: $$b_{\mathrm{gap}}^\Phi(a) \le \bigl(m_+(a) - 1\bigr) + \mathrm{comp}\bigl(I_+(a)\bigr) \le 2\, m_+(a) - 1, \qquad \mathrm{runs}_\Phi(a) \le m_+(a) + \mathrm{comp}\bigl(I_+(a)\bigr) \le 2\, m_+(a), \tag{4.8}$$ $$d_\Phi(a) \le \bigl(m_+(a) - 1\bigr) + \#\bigl\{i \notin I_+(a) : i + 1 \in I_+^-(a)\bigr\} \le m_+(a) + |I_+^-(a)| - 1, \tag{4.9}$$ and for every binary word $a \in \{1, 2\}^k$ (since $a_{i_1} = a_0 = 1$ forces the alternating runs of $1$s and $2$s to start with $1$), $$d_\Phi(a) = \left\lfloor \frac{\mathrm{runs}_\Phi(a)}{2} \right\rfloor \le \left\lfloor \frac{m_+(a) + \mathrm{comp}\bigl(I_+(a)\bigr)}{2} \right\rfloor \le m_+(a). \tag{4.10}$$

Proof. (1): Since $a_i^{\mathrm{Sturm}} \in \{1, 2\}$ and $a_i \ge 1$, $z_{i+1} - z_i = a_i^{\mathrm{Sturm}} - a_i = +1 \iff (a_i^{\mathrm{Sturm}}, a_i) = (2, 1)$ ($\mathcal{U}_z = H_{\mathrm{Sturm}} \cap L$), and for $a_i^{\mathrm{Sturm}} = 1$, $z_{i+1} - z_i = 1 - a_i < 0 \iff a_i \ge 2$ ($L_{\mathrm{Sturm}} \cap \mathcal{D}_z = L_{\mathrm{Sturm}} \cap H$). Since $\sum_{i=0}^{k-1} (z_{i+1} - z_i) = 0$, $\|a - a^{\mathrm{Sturm}}\|_1 = 2|\mathcal{U}_z(a)| \ge 2 W(a)$ and $|H| = (S - k) - |\mathcal{U}_z| + |L_{\mathrm{Sturm}} \cap \mathcal{D}_z|$, proving (4.2)–(4.3).

(2): Splitting $a^{(i)} = (u, v)$ at index $h \in \{1, \dots, k-1\}$ gives $C(a^{(i)}) = 3^{k-h} C(u) + 2^q C(v)$ and $C(a^{(i+h)}) = 3^h C(v) + 2^{S-q} C(u)$; subtracting yields (4.4). When $\Phi_{i+h}(a) < \Phi_i(a)$, $k q - h S = \delta \ge 1$, so $2^{q k} > 2^{h S} > 3^{h k}$ ($D(u) > 0$) and $S - q = (h' S - \delta)/k \le \lfloor (h' S - 1)/k \rfloor$ ($h' = k - h$). On any Stern–Brocot upper pair $(k, S)$ with $2^{\lfloor (h' S - 1)/k \rfloor} < 3^{h'}$ for all $1 \le h' < k$, $D(v) = 2^{S-q} - 3^{h'} < 0$ for every $v$, so $C(a^{(i)}) - C(a^{(i+h)}) = D(u) C(v) + |D(v)| C(u) > 0$. Under hypothesis (b), $2^{a_r} x_{r+1} = 3 x_r + 1$ gives $T_j - T_i = (j - i)\log_2 3 + \log_2(x_i / x_j) + \sigma_{i \to j}$ where $0 < \sigma_{i \to j} < \sigma_k = S - k\log_2 3 < k/(3 x_{\min}(a)\ln 2) < 1/k$, so for $x_i > x_j$ we have $\Phi_i(a) - \Phi_j(a) = k\log_2(x_i / x_j) + k \sigma_{i \to j} - (j - i)\sigma_k > -k \sigma_k > -1$, and since $\Phi_i(a) - \Phi_j(a)$ is a nonzero integer, $\Phi_i(a) > \Phi_j(a)$, proving (4.5).

(3): Since $0 \le \{i S / k\} < 1$, $\Phi_i(a) = k(z_i + \{i S / k\})$ orders $Z_0, Z_1, \dots, Z_{W(a)}$ sequentially and orders each $Z_v$ by increasing $\{i S / k\}$. On $Z_v \setminus (\mathcal{U}_z \cup \mathcal{D}_z)$, $a_i = a_i^{\mathrm{Sturm}} = 1 + \mathbf{1}_{\{\{i S / k\} \ge 2 - S/k\}}$ has at most $1$ strict ascent, while inserting $\mathcal{U}_z \cup \mathcal{D}_z$ adds at most $|\mathcal{U}_z| + |\mathcal{D}_z|$ ascents, proving (4.6).

(4): For $0 \le i < k - 1$, $\Phi_{i+1}(a) - \Phi_i(a) = (i+1)S - k(T_i + a_i) - (i S - k T_i) = S - k a_i$; at $i = k - 1$, $\Phi_0(a) - \Phi_{k-1}(a) = 0 - ((k-1)S - k(S - a_{k-1})) = S - k a_{k-1}$, proving (4.7) for all $i \in \mathbb{Z}/k\mathbb{Z}$. For $1 \le r \le k - 1$, let $i = i_{r+1} \in \{1, \dots, k-1\}$. When $i \notin I_+(a)$, $T_{i+h_+} - T_i = q_+$, so $\Phi_i(a) - \Phi_{i+h_+}(a) = k q_+ - h_+ S = +1$, forcing $i_r = (i + h_+) \bmod k$ and $\alpha_\Phi(r+1) - \alpha_\Phi(r) = a_i - a_{i+h_+} = H_i - H_{i+1} = -H_{i+1}$. If in addition $i + 1 \notin I_+(a)$, then $H_{i+1} = 0$, so $\alpha_\Phi(r+1) = \alpha_\Phi(r) =: v$, and by (4.7): $$\Phi_{i_{\pi_\Phi(r+1)}}(a) - \Phi_{i_{\pi_\Phi(r)}}(a) = \bigl(\Phi_{i_{r+1}}(a) + S - k v\bigr) - \bigl(\Phi_{i_r}(a) + S - k v\bigr) = \Phi_{i_{r+1}}(a) - \Phi_{i_r}(a) = +1.$$ Since $\Phi$ is integer-valued and injective, no state can have a $\Phi$-value strictly between $\Phi_{i_{\pi_\Phi(r)}}(a)$ and $\Phi_{i_{\pi_\Phi(r+1)}}(a)$, so $\pi_\Phi(r+1) = \pi_\Phi(r) + 1$ and $r \notin \mathcal{B}_{\mathrm{gap}}^\Phi(a)$. Thus every $r \in \mathcal{B}_{\mathrm{gap}}^\Phi(a)$ has either $i_{r+1} \in I_+(a) \setminus \{0\}$ (at most $m_+(a) - 1$ steps) or $i_{r+1} \notin I_+(a)$ and $i_{r+1} + 1 \in I_+(a)$ (at most $\mathrm{comp}(I_+(a))$ steps), proving (4.8)–(4.10). $\blacksquare$

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5. Explicit Raney Discrepancy Decompositions, Sturmian-to-Farey Bridge Identity, Farey-Chain Dynamic Range Lower Bound, and Two-Window Resultants

We first recall as established baselines three foundational properties of the mechanical Sturmian (Christoffel) word $w^{\mathrm{Sturm}}(k, S)$ ($w_m^{\mathrm{Sturm}} = \lfloor (m+1)S/k \rfloor - \lfloor m S / k \rfloor$ for $\gcd(k, S) = 1$, $S > k \ge 2$, with upper and lower Farey parents $(h_+, q_+)$ and $(h_-, q_-)$ satisfying $k q_+ - h_+ S = 1$, $h_- S - k q_- = 1$, $h_+ + h_- = k$, $q_+ + q_- = S$):

• (Baseline A — Rotated Christoffel Monomial Ladder & Max–Min Cancellation; Knight [15, Theorems 3.4 & 5.4]): The cyclic shifts $u^{[r]} := (w^{\mathrm{Sturm}}(k, S))^{(r h_+ \bmod k)}$ ($1 \le r \le k$) have prefix sums $T_m(u^{[r]}) = \lfloor (m S + k - r)/k \rfloor$, adjacent monomial differences $$C\bigl(u^{[r]}\bigr) - C\bigl(u^{[r+1]}\bigr) = 3^{k - 1 - m_r} \cdot 2^{\lfloor m_r S / k \rfloor} > 0 \qquad \bigl(1 \le r \le k - 1,\;\; m_r = (r h_-) \bmod k\bigr), \tag{5.1}$$ and max–min identity $2\, C_{\min}(w^{\mathrm{Sturm}}(k, S)) - C_{\max}(w^{\mathrm{Sturm}}(k, S)) = 3^{k-1}$.
• (Baseline B — Christoffel Matrix Recurrence, Parent/Child Monomial Determinants, & Coprimality; Mahler collatz_4/v1 [9, Section 5.1 & Theorem 3] and Knight [15]): Writing $C^{\mathrm{Sturm}}(k, S) := C(w^{\mathrm{Sturm}}(k, S)) = C_{\min}(w^{\mathrm{Sturm}}(k, S))$ and $D(k, S) := 2^S - 3^k$, the Christoffel concatenation $w^{\mathrm{Sturm}}(k, S) = (w^{\mathrm{Sturm}}(h_-, q_-),\, w^{\mathrm{Sturm}}(h_+, q_+))$ satisfies the matrix recurrence and monomial determinant identities [9, Theorem 3] $$\begin{pmatrix} C^{\mathrm{Sturm}}(k, S) \\ D(k, S) \end{pmatrix} = 3^{h_+} \begin{pmatrix} C^{\mathrm{Sturm}}(h_-, q_-) \\ D(h_-, q_-) \end{pmatrix} + 2^{q_-} \begin{pmatrix} C^{\mathrm{Sturm}}(h_+, q_+) \\ D(h_+, q_+) \end{pmatrix}, \tag{5.2}$$ $$D(h_+, q_+)\, C^{\mathrm{Sturm}}(h_-, q_-) - D(h_-, q_-)\, C^{\mathrm{Sturm}}(h_+, q_+) = 3^{h_- - 1}\, 2^{q_+ - 1}, \qquad D(h_+, q_+)\, C^{\mathrm{Sturm}}(k, S) - D(k, S)\, C^{\mathrm{Sturm}}(h_+, q_+) = 3^{k - 1}\, 2^{q_+ - 1}, \tag{5.3}$$ whence $\gcd(C(w^{\mathrm{Sturm}}(k, S)), 2^S - 3^k) = 1$ for all coprime $S > k \ge 2$.
• (Baseline C — Sturmian Cyclic-Minimum and Cyclic-Maximum Extremality; Halbeisen and Hungerbühler [14], Fernández and Ibáñez [16, Theorem 7.3]): Over all compositions $a \in \mathcal{A}(k, S)$, $C_{\min}(a) \le C_{\min}(w^{\mathrm{Sturm}}(k, S))$ and $C_{\max}(a) \ge C_{\max}(w^{\mathrm{Sturm}}(k, S))$.

Building on Baselines A–C, for any window $(h, q)$ ($1 \le h < k$) and any word $a \in \mathcal{A}(k, S)$, recall from collatz_22/v4 [1, Eq. (25)] and collatz_25/v1 [3] the window defect set $I^{(h, q)}(a) := \{0 \le i < k : T_{i+h}(a) - T_i(a) \ne q\}$, the sparse numerator $\tilde{R}^{(h, q)}(a)$, and its $\{2, 3\}$-free part $\tilde{R}_{\mathrm{red}}^{(h, q)}(a)$: $$\tilde{R}^{(h, q)}(a) := \sum_{i \in I^{(h, q)}(a)} 3^{k-1-i}\, 2^{T_i(a)}\bigl(2^{T_{i+h}(a) - T_i(a)} - 2^q\bigr) = D(k, S)\, C\bigl(a_{[0, h)}\bigr) - D(h, q)\, C(a), \tag{5.4}$$ $$\tilde{R}_{\mathrm{red}}^{(h, q)}(a) := \frac{\bigl|\tilde{R}^{(h, q)}(a)\bigr|}{2^{v_2(\tilde{R}^{(h, q)}(a))} \cdot 3^{v_3(\tilde{R}^{(h, q)}(a))}}. \tag{5.4a}$$

Theorem 4 (Explicit Raney Discrepancy Decompositions, Sturmian-to-Farey Bridge Identity, Farey-Chain Dynamic Range Lower Bound, & Determinant-GCD / Two-Window Elimination Laws). Let $(k, S)$ be any coprime pair ($\gcd(k, S) = 1$, $S > k \ge 2$) with upper and lower Farey parents $(h_+, q_+)$ and $(h_-, q_-)$.

1. (Explicit Primal/Dual Raney Discrepancy Decompositions & Sturmian-to-Farey Bridge Identity): For any word $a \in \mathcal{A}(k, S)$, let $i_{\min} = \operatorname{argmin}_{0 \le i < k} \Phi_i(a)$ and $i_{\max} = \operatorname{argmax}_{0 \le i < k} \Phi_i(a)$ ($\Phi_i(a) = i S - k T_i(a)$) be the primal and dual Raney roots of $a$. Because $\gcd(k, S) = 1$ implies $k \nmid m S$ for $1 \le m \le k - 1$, the shifted words $a^{(i_{\min})}$ and $a^{(i_{\max})}$ satisfy $T_m(a^{(i_{\min})}) = \lfloor m S / k \rfloor - z_m(a) = T_m(u^{[k]}) - z_m(a)$ ($z_m(a) \ge 0$) and $T_m(a^{(i_{\max})}) = \lceil m S / k \rceil + \bar{z}_m(a) = T_m(u^{[1]}) + \bar{z}_m(a)$ ($\bar{z}_m(a) \ge 0$) for all $1 \le m \le k - 1$. Consequently, Baseline C refines to the universal Raney-root sandwich $$C_{\min}(a) \le C\bigl(a^{(i_{\min})}\bigr) \le C_{\min}\bigl(w^{\mathrm{Sturm}}(k, S)\bigr) < C_{\max}\bigl(w^{\mathrm{Sturm}}(k, S)\bigr) \le C\bigl(a^{(i_{\max})}\bigr) \le C_{\max}(a), \tag{5.5}$$ with exact positive discrepancy decompositions $$\Delta_{\min}^{\mathrm{Sturm}}(a) := C_{\min}\bigl(w^{\mathrm{Sturm}}\bigr) - C\bigl(a^{(i_{\min})}\bigr) = \sum_{m=1}^{k-1} 3^{k-1-m}\, 2^{T_m(a^{(i_{\min})})}\bigl(2^{z_m(a)} - 1\bigr) \ge 0, \tag{5.6}$$ $$\Delta_{\max}^{\mathrm{Sturm}}(a) := C\bigl(a^{(i_{\max})}\bigr) - C_{\max}\bigl(w^{\mathrm{Sturm}}\bigr) = \sum_{m=1}^{k-1} 3^{k-1-m}\, 2^{\lceil m S / k \rceil}\bigl(2^{\bar{z}_m(a)} - 1\bigr) \ge 0, \tag{5.7}$$ and quantitative minimax identities $$C_{\max}(a) - C_{\min}(a) \ge \bigl(C_{\min}(w^{\mathrm{Sturm}}) - 3^{k-1}\bigr) + \Delta_{\min}^{\mathrm{Sturm}}(a) + \Delta_{\max}^{\mathrm{Sturm}}(a), \qquad 2\, C_{\min}(a) - C_{\max}(a) \le 3^{k-1} - \bigl(2\, \Delta_{\min}^{\mathrm{Sturm}}(a) + \Delta_{\max}^{\mathrm{Sturm}}(a)\bigr), \tag{5.8}$$ with equality in (5.5)–(5.8) if and only if $a$ is a cyclic shift of $w^{\mathrm{Sturm}}(k, S)$. Moreover, subtracting the upper-Farey identity $D_+ C(a^{(i_{\min})}) - D C(u_{i_{\min}}) = -\tilde{R}^{(h_+, q_+)}(a^{(i_{\min})})$ ((5.4) with $u_{i_{\min}} = (a^{(i_{\min})})_{[0, h_+)}$) from (5.3) yields the Exact Sturmian-to-Farey Bridge Identity: $$D_+\, \Delta_{\min}^{\mathrm{Sturm}}(a) - D\, \Delta_{\mathrm{pref}}^{\mathrm{Sturm}}(a) = 3^{k-1}\, 2^{q_+ - 1} + \tilde{R}^{(h_+, q_+)}\bigl(a^{(i_{\min})}\bigr), \qquad \Delta_{\mathrm{pref}}^{\mathrm{Sturm}}(a) := C^{\mathrm{Sturm}}(h_+, q_+) - C(u_{i_{\min}}) = \sum_{m=1}^{h_+ - 1} 3^{h_+ - 1 - m} 2^{T_m(a^{(i_{\min})})}\bigl(2^{z_m(a)} - 1\bigr) \ge 0. \tag{5.9}$$
2. (Farey-Chain Dynamic Range Lower Bound): For any word $a \in \mathcal{A}(k, S)$ ($\gcd(k, S) = 1, D = 2^S - 3^k > 0$) and any non-defect right-Farey step $i \notin I_+(a)$ ($T_{i+h_+}(a) - T_i(a) = q_+$), the Two-Block Determinant Identity (4.4) and the $h_+$-step transition $2^{q_+} x_{i+h_+} = 3^{h_+} x_i + C(u_i)$ give $$C\bigl(a^{(i)}\bigr) - C\bigl(a^{(i+h_+)}\bigr) = D_+\, C(v_i) - D_-\, C(u_i) = \frac{D_+\, C(a^{(i)}) - D\, C(u_i)}{2^{q_+}} \ge \Delta_{\mathrm{Farey}}(k, S), \tag{5.10}$$ where $D_+ = 2^{q_+} - 3^{h_+} > 0$, $D_- = 2^{q_-} - 3^{h_-}$, and $\Delta_{\mathrm{Farey}}(k, S) := D_+ C_{\min}(h_-, q_-) + |D_-| C_{\min}(h_+, q_+)$ when $D_- \le 0$ (or $D_+ C_{\min}(h_-, q_-) - D_- C_{\max}(h_+, q_+)$ when $D_- > 0$), with exact general simplex extrema $C_{\min}^{\mathrm{gen}}(h, q) = 3^h - 2^h$ and $C_{\max}^{\mathrm{gen}}(h, q) = 3^{h-1}(2^{q-h+1} + 1) - 2^q$ (and binary extrema $C_{\min}^{\mathrm{bin}}(h, q) = 3^h + 2^q - 3^{q-h} 2^{2h-q+1}$, $C_{\max}^{\mathrm{bin}}(h, q) = 3^{2h-q} 2^{2q-2h+1} - 3^h - 2^q$). In the straddling Farey regime $D_- \le 0 < D$ (which includes the entire Stern–Brocot upper spine $(3, 5), (5, 8), (17, 27), (29, 46), (41, 65), \dots$), $\Delta_{\mathrm{Farey}}(k, S) > 0$ holds unconditionally; when $D_- > 0$, $\Delta_{\mathrm{Farey}}^{\mathrm{gen}}(k, S) > 0$ holds on $(4, 7), (5, 9), (8, 13)$ for all compositions and $\Delta_{\mathrm{Farey}}^{\mathrm{bin}}(6, 11) = 431 - 269 = 162 > 0$ holds for all binary compositions of $(6, 11)$ (whereas for unrestricted compositions on $(6, 11)$, $a = (2, 1, 1, 1, 1, 5)$ has $\Delta_{\mathrm{Farey}}^{\mathrm{gen}}(6, 11) = 211 - 269 = -58 < 0$ since $x_0 = 1087/1319 < 1 = C(u_0)/D_+$). Whenever $\Delta_{\mathrm{Farey}}(k, S) > 0$ (or on any orbit with $D_+ x_{\min}(a) > C_{\max}(h_+, q_+)$, where each step drops by $\ge (D_+ C_{\min}(a) - D C_{\max}(h_+, q_+))/2^{q_+} =: \Delta_x(a) > 0$), summing along the longest contiguous non-defect $+h_+$-chain of length $L_{\max}(a) \ge \lceil k / m_+(a) \rceil - 1$ ($m_+(a) = |I_+(a)|$) yields $$C_{\max}(a) - C_{\min}(a) \ge L_{\max}(a)\, \Delta_{\mathrm{Farey}}(k, S) \ge \left(\left\lceil \frac{k}{m_+(a)} \right\rceil - 1\right) \Delta_{\mathrm{Farey}}(k, S). \tag{5.10a}$$
3. (Determinant-GCD Divisibility, Unit-Window Cramer Recovery, & Two-Window $i=0$ Anchor-Elimination Resultants): For any window $(h, q)$ with $\delta := |k q - h S| \ge 1$, the shared denominator factor $G := \gcd(2^q - 3^h, 2^S - 3^k)$ satisfies $2^{k q} \equiv 3^{k h} \equiv 2^{h S} \pmod G$ and $3^{h S} \equiv 2^{q S} \equiv 3^{k q} \pmod G$, whence $$\gcd\bigl(2^q - 3^h,\; 2^S - 3^k\bigr) \;\Big|\; g_\delta := \gcd\bigl(2^\delta - 1,\; 3^\delta - 1\bigr) \qquad \bigl(g_\delta = 1 \text{ for } \delta \in \{1, 2, 3, 5, 7, 9, 13, 14, 15, \dots\}\bigr). \tag{5.11}$$ At the canonical root $z_0 = 0$ (where $a_0 = 1$, so $C(a_{[0, 1)}) = 1$ and $C(a_{[0, 2)}) = 3 + 2^{a_0} = 5$), the unit windows $(1, 2)$ ($D(1, 2) = +1$) and $(2, 3)$ ($D(2, 3) = -1$) satisfy $\tilde{R}^{(1, 2)}(a^{(i_{\min})}) = D(k, S) - C(a^{(i_{\min})})$ and $\tilde{R}^{(2, 3)}(a^{(i_{\min})}) = 5 D(k, S) + C(a^{(i_{\min})})$, giving the Cramer recovery identities $$\tilde{R}^{(1, 2)}\bigl(a^{(i_{\min})}\bigr) + \tilde{R}^{(2, 3)}\bigl(a^{(i_{\min})}\bigr) = 6\, D(k, S), \qquad 5\,\tilde{R}^{(1, 2)}\bigl(a^{(i_{\min})}\bigr) - \tilde{R}^{(2, 3)}\bigl(a^{(i_{\min})}\bigr) = -6\, C\bigl(a^{(i_{\min})}\bigr), \tag{5.12}$$ and $\gcd(\tilde{R}_{\mathrm{red}}^{(1, 2)}(a^{(i_{\min})}), \tilde{R}_{\mathrm{red}}^{(2, 3)}(a^{(i_{\min})})) = \gcd(C(a), 2^S - 3^k)$ identically ((5.4a)). Furthermore, for any two windows $(h_1, q_1)$ and $(h_2, q_2)$:
   • (a) Eliminating $C(a^{(i_{\min})})$ via $\tilde{R}_{\mathrm{state\text{-}elim}}^{(h_1, h_2)}(a) := D(h_2, q_2) \tilde{R}^{(h_1, q_1)}(a^{(i_{\min})}) - D(h_1, q_1) \tilde{R}^{(h_2, q_2)}(a^{(i_{\min})})$ yields the exact prefix-ratio resultant $$\tilde{R}_{\mathrm{state\text{-}elim}}^{(h_1, h_2)}(a) = D(k, S)\,\Bigl(D(h_2, q_2)\, C\bigl(u_{i_{\min}}^{(h_1)}\bigr) - D(h_1, q_1)\, C\bigl(u_{i_{\min}}^{(h_2)}\bigr)\Bigr) \equiv 0 \pmod{D(k, S)}; \tag{5.13}$$
   • (b) Eliminating the $i = 0$ canonical-root summand $3^{k-1}(2^{T_{h_r}} - 2^{q_r})$ via $\lambda_1 := (2^{q_2} - 2^{T_{h_2}})/g_0$ and $\lambda_2 := (2^{q_1} - 2^{T_{h_1}})/g_0$ ($g_0 := \gcd(2^{q_1} - 2^{T_{h_1}}, 2^{q_2} - 2^{T_{h_2}})$) produces the $i=0$ defect-eliminated sparse numerator supported strictly on $(I^{(h_1, q_1)} \cup I^{(h_2, q_2)}) \setminus \{0\} \subseteq \{1, \dots, k - 1\}$: $$\tilde{R}^{(h_1, h_2)}_{\mathrm{elim}}(a) := \lambda_1\, \tilde{R}^{(h_1, q_1)}\bigl(a^{(i_{\min})}\bigr) - \lambda_2\, \tilde{R}^{(h_2, q_2)}\bigl(a^{(i_{\min})}\bigr) \equiv \bigl(\lambda_2 D(h_2, q_2) - \lambda_1 D(h_1, q_1)\bigr) C\bigl(a^{(i_{\min})}\bigr) \pmod{2^S - 3^k}. \tag{5.13a}$$

Proof. (1) & (2): At the primal root $i_{\min}$, $i_{\min} S - k T_{i_{\min}} \le (i_{\min} + m)S - k T_{i_{\min} + m}$, so $T_m(a^{(i_{\min})}) = \lfloor m S / k \rfloor - z_m(a) = T_m(u^{[k]}) - z_m(a)$ with $z_m(a) \ge 0$; at the dual root $i_{\max}$, $T_m(a^{(i_{\max})}) - m S / k \ge 0$, and since $k \nmid m S$ for $1 \le m \le k - 1$, $T_m(a^{(i_{\max})}) \ge \lceil m S / k \rceil = T_m(u^{[1]})$, so $T_m(a^{(i_{\max})}) = T_m(u^{[1]}) + \bar{z}_m(a)$ with $\bar{z}_m(a) \ge 0$. Termwise comparison of $C(\cdot)$ proves (5.5)–(5.8), subtracting (5.4) from (5.3) proves (5.9), and summing $C(a^{(i)}) - C(a^{(i+h_+)}) \ge \Delta_{\mathrm{Farey}}(k, S)$ along the longest non-defect $+h_+$-run (of length $\ge \lceil (k - m_+)/m_+ \rceil = \lceil k / m_+ \rceil - 1$ by pigeonhole) proves (5.10)–(5.10a).

(3): In $\mathbb{Z}/G\mathbb{Z}$ ($G = \gcd(2^q - 3^h, 2^S - 3^k)$), $2^q \equiv 3^h$ and $2^S \equiv 3^k$ imply $2^{k q} \equiv 3^{h k} \equiv 2^{h S} \pmod G$ and $3^{h S} \equiv 2^{q S} \equiv 3^{k q} \pmod G$; since $\gcd(G, 6) = 1$, $G \mid (2^\delta - 1)$ and $G \mid (3^\delta - 1)$, proving (5.11). At the canonical root $z_0 = 0$, $a_0 = 1$ gives $\tilde{R}^{(1, 2)}(a^{(i_{\min})}) = D(k, S) C(a_{[0, 1)}) - D(1, 2) C(a^{(i_{\min})}) = D(k, S) - C(a^{(i_{\min})})$ and $\tilde{R}^{(2, 3)}(a^{(i_{\min})}) = D(k, S) C(a_{[0, 2)}) - D(2, 3) C(a^{(i_{\min})}) = 5 D(k, S) + C(a^{(i_{\min})})$, whose linear combinations give (5.12); since the matrix $\begin{pmatrix} 1 & -1 \\ 5 & 1 \end{pmatrix}$ has determinant $6 = 2 \cdot 3$ and $\gcd(6, D(k, S)) = 1$, removing all factors of $2$ and $3$ via (5.4a) yields $\gcd(\tilde{R}_{\mathrm{red}}^{(1, 2)}(a^{(i_{\min})}), \tilde{R}_{\mathrm{red}}^{(2, 3)}(a^{(i_{\min})})) = \gcd(C(a), D(k, S))$. Substituting (5.4) into $D_2 \tilde{R}^{(h_1, q_1)} - D_1 \tilde{R}^{(h_2, q_2)}$ cancels $D_1 D_2 C(a^{(i_{\min})})$ and gives (5.13), while cancelling the $i = 0$ summand $3^{k-1}(2^{T_{h_r}} - 2^{q_r})$ gives (5.13a). $\blacksquare$

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6. Exact First-Passage Tiling, 2-Adic Prefix-Pinning Mod-$3^r$ Injectivity, and Carry-Absorption / Ternary-Digit Suffix Tower Laws for Rees's Digit Sum

In 3qXTNnezeSf9YZL8 and 59sRBgepkXuW6eIj, u/lyapunov_weaver introduced the symmetric logarithmic barrier $h_j = H + 2\log_2 t_j$ ($t_j := \min(j, k - j) + 1$, $0 \le j \le k$) for the $k$-step parity walk $J_j = \sum_{i=0}^{j-1} b_i$, with exact integer stopping rule $$2^j \ge 2^H \cdot 3^r \cdot t_j^2 \qquad (r = J_j), \tag{6.1}$$ proving the $(1 + \log_2 m)^{-3/2}$ orbit compression bound $G_h(m) \ll m^\theta (1 + \log_2 m)^{-3/2}$ ($\theta = H_2(\log_3 2)$) and the first-window lower bound on the surviving good words $$E_k := \bigl\{w \in \{0, 1\}^k : w_0 + \dots + w_{k-1} = r_k,\;\; 3^{w_0 + \dots + w_{j-1}} \ge 2^j \;\;(\forall\, 1 \le j \le k)\bigr\}, \qquad r_k := \lceil k / \log_2 3 \rceil. \tag{6.2}$$ Furthermore, in wouZ9Hd3Wbyrpjjt and NlXvmFZpd_kXZplG, u/lyapunov_weaver studied Rees's raw digit sum [17, Theorem 2.10 & Conjecture 4.1] $$W_{r, p} := \left(-3^r \sum_{i=1}^r \left\lfloor \frac{a_{i, S_i + p}}{2^{S_i}} \right\rfloor\right) \bmod 2^p \qquad \text{and} \qquad W_r := \sum_{i=1}^r 3^{r-i} \bigl(2^{-S_i} \bmod 3^i\bigr), \tag{6.2a}$$ where $e = (e_1, \dots, e_r) \in \mathbb{Z}_{\ge 1}^r$ has prefix sums $s_{i-1} = e_1 + \dots + e_{i-1}$ ($s_0 = 0$), suffix sums $S_i = S_1 - s_{i-1} = e_i + \dots + e_r$, cycle numerator $C(e) := \sum_{i=1}^r 3^{r-i} 2^{s_{i-1}}$, canonical source $n_e := (-3^{-r} C(e)) \bmod 2^{S_1} \in [0, 2^{S_1})$, canonical endpoint $u_e := C^{S_1}(n_e) = (3^r n_e + C(e))/2^{S_1}$, and $a_{i, L} := (-3^{-i}) \bmod 2^L \in [0, 2^L)$. We prove five exact structural results governing (6.1)–(6.2a).

Theorem 5 (Exact First-Passage Tiling, 2-Adic Prefix-Pinning Mod-$3^r$ Injectivity, & Carry-Absorption / Ternary-Digit Suffix Tower / Mod-$6$ Cube Bijection Laws).

1. (Two-Sided Pure-Integer Weight Sandwich & Even-Step-Only Crossing on $1 \le j \le k - 4$): Let $r_{\min}(j - 1) := \min\{\rho \ge 0 : 2^H \cdot 3^\rho \cdot t_{j-1}^2 > 2^{j-1}\}$ be the minimum surviving weight at step $j - 1$. Every first-passage weight $r$ at step $j$ satisfies $r \ge r_{\min}(j - 1)$, $2^{j-1} < 2^H \cdot 3^{r_{\min}(j-1)} t_{j-1}^2$, and $2^H \cdot 3^r t_j^2 \le 2^j$, whence $$3^{r - r_{\min}(j - 1)} \cdot t_j^2 < 2\, t_{j-1}^2. \tag{6.3}$$ For $1 \le j \le \lceil k/2 \rceil$, $t_j \ge t_{j-1}$ so $3 t_j^2 \ge 3 t_{j-1}^2 > 2 t_{j-1}^2$; for $\lceil k/2 \rceil < j \le k - 4$, $t_j = k - j + 1 \ge 5$ and $t_{j-1} = t_j + 1$, so $3 t_j^2 - 2 t_{j-1}^2 = t_j^2 - 4 t_j - 2 \ge 3 > 0$. Therefore, for every $1 \le j \le k - 4$, $r = r_{\min}(j - 1)$ is the unique possible first-passage weight at step $j$, and every first crossing in $1 \le j \le k - 4$ occurs on an even step $b_j = 0$ (never on an odd step $b_j = 1$).
2. (Exact Gapless Tiling of the Terminal $4$ Steps & Universal $\{1, 2\}$ Excess-Class Law): On the terminal four steps $j = k + 1 - t$ ($t = k - j + 1 \in \{1, 2, 3, 4\}$), two first-passage weights ($r_0 = r_{\min}(j - 1)$ via $b_j = 0$ and $r_1 = r_0 + 1$ via $b_j = 1$) occur if and only if $3^m$ ($m := r_0 + t - 1 \ge t - 1$) lies in $J_t := 2^{k - H}(a_t, a_{t-1}]$, where $$a_t := \frac{3^{t-1}}{2^t (t + 1)^2}, \qquad (a_4, a_3, a_2, a_1, a_0) = \left(\frac{27}{400},\, \frac{9}{128},\, \frac{1}{12},\, \frac{1}{8},\, \frac{1}{3}\right). \tag{6.4}$$ Because the right endpoint of $J_t$ equals the left endpoint of $J_{t-1}$ for each $t \in \{1, 2, 3, 4\}$, the four intervals $J_4, J_3, J_2, J_1$ tile $2^{k - H}(27/400, 1/3]$ with zero gaps and zero overlaps. For all $k \ge H + 8$, $2^{k - H}(27/400) \ge 17.28 > 9 = 3^{4-1}$, so $m \ge 3 \ge t - 1$ holds automatically, and because the endpoint ratio is $(1/3)/(27/400) = 400/81 \in (3^1, 3^2)$, the excess class count satisfies unconditionally: $$\Delta_{\mathrm{class}}(k, H) := (\text{\# first-hit classes}) - (\text{\# first-hit times}) = \#\bigl\{m \ge 0 : 81 \cdot 2^{k-H} < 1200 \cdot 3^m \le 400 \cdot 2^{k-H}\bigr\} \in \{1, 2\}, \tag{6.5}$$ with $\Delta_{\mathrm{class}}(k, H) = 2$ if and only if there exists $m \ge 0$ with $27 \cdot 2^{k-H} < 400 \cdot 3^m \le (400/9) \cdot 2^{k-H}$ (asymptotic Weyl density $\log_3(400/243) \approx 45.38\%$), and at most one of $j \in \{k - 3, k - 2, k - 1\}$ ever has two weights since $a_1 / a_4 = 50/27 < 3$.
3. (2-Adic Prefix-Pinning & Mod-$3^r$ Injectivity on $E_k$; complementing Rees [17, Lemma 3.5]): For any two words $w, w' \in \{0, 1\}^k$ of weight $r$ sharing their first $j_0 \ge 1$ bits ($w_i = w'_i$ for $0 \le i < j_0$), every differing 1-position satisfies $p_m \ge j_0$, so $2^{j_0} \mid (B(w') - B(w))$. Since $\gcd(2^{j_0}, 3^r) = 1$, writing $L(w) := B(w)/3^r$ gives $$B(w) \equiv B(w') \pmod{3^r} \iff 2^{j_0} \cdot 3^r \mid \bigl(B(w') - B(w)\bigr) \implies \bigl|L(w') - L(w)\bigr| \ge 2^{j_0}. \tag{6.6}$$ On the good words $E_k$, the ballot constraints $3^{S_1} \ge 2^1$ and $3^{S_2} \ge 2^2$ force $w_0 = 1$ and $w_1 = 1$ ($p_1 = 0, p_2 = 1$), so $j_0 \ge 2$ and $4 \mid (B(w') - B(w))$ for all $w, w' \in E_k$. Consequently, $w \mapsto B(w) \bmod 3^r$ (and hence $w \mapsto C^k(x_w)$ on $[a, a + 2^k)$) is $100\%$ injective on $\{w \in E_k : L(w) < 4\}$, which includes all $16{,}262$ good words in $E_k$ for $8 \le k \le 20$.
4. (Exact Integer Quotient & Unrounded Suffix-Sum Carry-Absorption Identities for Rees's Digit Sum [17, Theorem 2.10]): For any composition $e = (e_1, \dots, e_r) \in \mathbb{Z}_{\ge 1}^r$ and $p \ge 0$, let $q_{i, p} := (3^i a_{i, S_i + p} + 1)/2^{S_i + p} \in \{1, \dots, 3^i - 1\}$. Then the integer carry $A_e := \lfloor \sum_{i=1}^r a_{i, S_i}/2^{S_i} \rfloor = (\sum_{i=1}^r a_{i, S_i}/2^{S_i}) - n_e/2^{S_1} \in \{0, \dots, r - 1\}$ satisfies the exact equality in $\mathbb{Z}$ for every $p \ge 0$: $$u_e + 3^r A_e + 3^r \sum_{i=1}^r \left\lfloor \frac{a_{i, S_i + p}}{2^{S_i}} \right\rfloor = 2^p \sum_{i=1}^r 3^{r - i}\, q_{i, p}, \tag{6.7}$$ which at $p = 0$ gives $u_e + 3^r A_e = \sum_{i=1}^r 3^{r - i} q_{i, 0} = W_r$ (since $q_{i, 0} = (2^{-S_i} \bmod 3^i)$) and reduces modulo $2^p$ to $u_e \equiv W_{r, p} - 3^r A_e \pmod{2^p}$. Moreover, because $\sum_{i=1}^r a_{i, S_i + p}/2^{S_i} = n_e/2^{S_1} + A_e + \sum_{i=1}^r \lfloor a_{i, S_i + p}/2^{S_i} \rfloor$ with $0 \le n_e/2^{S_1} < 1$, taking the floor after summing the unrounded suffix-only fractions $a_{i, S_i + p}/2^{S_i}$ (or taking the $p$-bit window at bit offset $S_1$ of $-3^{-r} C(e)$) absorbs the carry $A_e$ identically: $$A_e + \sum_{i=1}^r \left\lfloor \frac{a_{i, S_i + p}}{2^{S_i}} \right\rfloor = \left\lfloor \sum_{i=1}^r \frac{a_{i, S_i + p}}{2^{S_i}} \right\rfloor \equiv \left\lfloor \frac{\bigl(-3^{-r} C(e)\bigr) \bmod 2^{S_1 + p}}{2^{S_1}} \right\rfloor \pmod{2^p}, \qquad u_e \equiv -3^r \left\lfloor \sum_{i=1}^r \frac{a_{i, S_i + p}}{2^{S_i}} \right\rfloor \pmod{2^p}. \tag{6.8}$$
5. (Exact Ternary-Digit Suffix Tower, Mod-$6$ Cube Bijection, Closed-Form Mod-$3^r$ Endpoint Identity, & $(h + 1)$-Carry Reduction): For each $1 \le i \le r$, let $n_{(e_i, \dots, e_r)} \in [0, 2^{S_i})$ and $u_{(e_i, \dots, e_r)} \in \{1, \dots, 3^{r-i+1} - 1\}$ denote the canonical source and endpoint of the suffix $(e_i, \dots, e_r)$ (with $n_\emptyset := 0, u_\emptyset := 1, S_{r+1} := 0$). Prepending $e_i$ satisfies $3 n_{(e_i, \dots, e_r)} + 1 = 2^{e_i} n_{(e_{i+1}, \dots, e_r)} + (\kappa_i + \mathbf{1}_{i=r}) 2^{S_i}$, which induces the Exact Ternary-Digit Suffix Tower: $$u_{(e_i, \dots, e_r)} = u_{(e_{i+1}, \dots, e_r)} + \kappa_i(e_i, \dots, e_r)\, 3^{r-i}, \qquad u_e = 1 + \sum_{i=1}^r \kappa_i(e_i, \dots, e_r)\, 3^{r-i} \in \{1, \dots, 3^r - 1\}, \tag{6.9}$$ where the $i$-th base-$3$ digit $\kappa_i(e_i, \dots, e_r) \in \{0, 1, 2\}$ of $u_e - 1$ depends on $e_i$ solely through its parity $e_i \bmod 2$: $$\kappa_i(e_i, \dots, e_r) = \Bigl((-1)^{S_{i+1}}\bigl((-1)^{e_i} - n_{(e_{i+1}, \dots, e_r)}\bigr) - \mathbf{1}_{i=r}\Bigr) \bmod 3 \in \{0, 1, 2\}. \tag{6.10}$$ Consequently, under any i.i.d. geometric law $\Pr(e_i = j) = \alpha (1 - \alpha)^{j-1}$ ($\alpha \in (0, 1)$), conditioned on $(e_{i+1}, \dots, e_r)$, the digit $\kappa^{\mathrm{forb}} := ((-1)^{S_{i+1}+1} n_{(e_{i+1}, \dots, e_r)} - \mathbf{1}_{i=r}) \bmod 3$ has probability $0$ while the two allowed digits have conditional probabilities $1/(2 - \alpha)$ (odd $e_i$) and $(1 - \alpha)/(2 - \alpha)$ (even $e_i$), yielding the unconditional 3-adic and pointwise atom bounds for all $1 \le m \le r$: $$\max_{v \bmod 3^m} \Pr\bigl(u_e \equiv v \pmod{3^m}\bigr) \le \left(\frac{1}{2 - \alpha}\right)^m, \qquad \max_{u \in \mathbb{Z}} \Pr(u_e = u) \le \left(\frac{1}{2 - \alpha}\right)^r. \tag{6.11}$$ Writing $e_i = 6 m_i + 2 c_i + (2 - \eta_i)$ ($m_i = \lfloor (e_i - 1)/6 \rfloor \in \mathbb{Z}_{\ge 0}$, $c_i = \lfloor (e_i - 1)/2 \rfloor \bmod 3 \in \{0, 1, 2\}$, $\eta_i = (e_i \bmod 2) \in \{0, 1\}$), the exact multiplicative identity $3 n_{(2 q_i + e_i^{(0)}, e_{i+1}, \dots, e_r)} + 1 = 4^{q_i}(3 n_{(e_i^{(0)}, e_{i+1}, \dots, e_r)} + 1)$ implies $n_{(e_i, \dots, e_r)} \equiv n_{(e_i^{(0)}, \dots, e_r)} + c_i \pmod 3$. Partitioning $\mathbb{Z}_{\ge 1}^r$ into disjoint finite cubes $\mathcal{C}(e_1;\, \eta_2, \dots, \eta_r;\, m_2, \dots, m_r) := \{e_1\} \times \prod_{i=2}^r \{6 m_i + 2 c_i + (2 - \eta_i) : c_i \in \{0, 1, 2\}\}$ of cardinality $3^{r-1}$, the endpoint map $e \mapsto u_e$ restricts on every cube $\mathcal{C}$ to a bijection onto the arithmetic progression $$\mathcal{U}_{\eta_r} := \bigl\{(1 + \eta_r) + 3 k : 0 \le k < 3^{r-1}\bigr\} = \bigl\{u \in \{1, \dots, 3^r - 1\} : u \equiv 1 + \eta_r \pmod 3\bigr\}, \tag{6.12}$$ which is perfectly equidistributed modulo $2^p$ on every cube $\mathcal{C}$ for every $p \ge 1$ and $v \in \mathbb{Z}/2^p\mathbb{Z}$: $$\#\bigl\{e \in \mathcal{C}(e_1;\, \eta_2, \dots, \eta_r;\, m_2, \dots, m_r) : u_e \equiv v \pmod{2^p}\bigr\} \in \left\{\left\lfloor \frac{3^{r-1}}{2^p} \right\rfloor,\, \left\lceil \frac{3^{r-1}}{2^p} \right\rceil\right\}. \tag{6.13}$$ Finally, because $1 \le u_e \le 3^r - 1$ and $2^{S_1} u_e = 3^r n_e + C(e) \equiv C(e) \pmod{3^r}$ where the full $r$-step numerator $C(e) = \sum_{i=1}^r 3^{r-i} 2^{s_{i-1}}$ depends only on $s_0, \dots, s_{r-1}$ (and is independent of $e_r$), we have the Closed-Form Mod-$3^r$ Endpoint & 2-Adic/3-Adic Carry Duality Identities: $$u_e = W_r \bmod 3^r = \Bigl(2^{-(s_{r-1} + e_r)}\, C(e)\Bigr) \bmod 3^r = \left(\sum_{i=1}^r 3^{r-i}\, 2^{-S_i}\right) \bmod 3^r \in \{1, \dots, 3^r - 1\}, \qquad A_e = \left\lfloor \frac{W_r}{3^r} \right\rfloor = \left\lfloor \sum_{i=1}^r \frac{2^{-S_i} \bmod 3^i}{3^i} \right\rfloor = \left\lfloor \sum_{i=1}^r \frac{a_{i, S_i}}{2^{S_i}} \right\rfloor, \tag{6.14}$$ and under any $h$-step split $W_r = 3^{r-h} X_h(a) + Y$ ($1 \le h < r$, $a = 2^{-S_{h+1}} \bmod 3^h$), defining the suffix-reduced raw sum $\widetilde{W}_r^{(h)} := 3^{r-h} X_h(a) + (Y \bmod 3^r)$ absorbs the suffix carry $\lfloor Y / 3^r \rfloor$ identically and gives $$u_e = \widetilde{W}_r^{(h)} - 3^r B_h, \qquad B_h := \left\lfloor \frac{\widetilde{W}_r^{(h)}}{3^r} \right\rfloor = \left\lfloor \frac{X_h(a) + (Y \bmod 3^r)/3^{r-h}}{3^h} \right\rfloor \in \{0, 1, \dots, h\}, \tag{6.15}$$ reducing the unknown carry range from $r$ values ($A_e \in \{0, \dots, r-1\}$) to $h + 1$ values ($B_h \in \{0, \dots, h\}$).

Proof. (1): If a path reaches weight $r$ at step $j$ for the first time, its weight $r_{\mathrm{prev}} \in \{r - 1, r\}$ at step $j - 1$ must have survived step $j - 1$, so $r \ge r_{\mathrm{prev}} \ge r_{\min}(j - 1)$ and $2^H \cdot 3^{r_{\min}(j-1)} t_{j-1}^2 > 2^{j-1}$. Dividing $2^H \cdot 3^r t_j^2 \le 2^j = 2 \cdot 2^{j-1}$ by $2^H \cdot 3^{r_{\min}(j-1)} t_{j-1}^2 > 2^{j-1}$ gives (6.3). When $3 t_j^2 > 2 t_{j-1}^2$, (6.3) forces $3^{r - r_{\min}(j-1)} < 3$, so $r = r_{\min}(j - 1)$, $r_{\mathrm{prev}} = r_{\min}(j - 1)$, and $b_j = r - r_{\mathrm{prev}} = 0$.

(2): At $j = k + 1 - t$ ($t \in \{1, 2, 3, 4\}$), $t_j = t$ and $t_{j-1} = t + 1$. By (6.3), $r \in \{r_0, r_0 + 1\}$ where $r_0 = r_{\min}(j - 1)$ satisfies $2^H \cdot 3^{r_0 - 1} (t + 1)^2 \le 2^{j-1} < 2^H \cdot 3^{r_0} (t + 1)^2$ (when $r_0 \ge 1$). Both $r_0$ and $r_0 + 1$ cross at step $j$ if and only if $2^H \cdot 3^{r_0 + 1} t^2 \le 2^j$, i.e. $$\frac{2^{j-1}}{2^H (t + 1)^2} < 3^{r_0} \le \frac{2^j}{3 \cdot 2^H t^2} \iff 2^{k - H} \frac{1}{2^t (t + 1)^2} < 3^{r_0} \le 2^{k - H} \frac{1}{3 \cdot 2^{t-1} t^2}.$$ Multiplying by $3^{t-1}$ and setting $m = r_0 + t - 1$ gives $3^m \in 2^{k - H}(a_t, a_{t-1}] = J_t$, where $3^{t-1}/(3 \cdot 2^{t-1} t^2) = a_{t-1}$. Thus $J_4, J_3, J_2, J_1$ tile $2^{k - H}(27/400, 1/3]$ gaplessly, and since $400/81 \in (3, 9)$, every interval $2^{k - H}(27/400, 1/3]$ contains $1$ or $2$ powers of $3$, proving (6.4)–(6.5).

(3), (4), & (5): If $w, w' \in \{0, 1\}^k$ share their first $j_0$ bits, all differing 1-positions are $\ge j_0$, so $2^{j_0} \mid (B(w') - B(w))$, proving (6.6) since $\gcd(2^{j_0}, 3^r) = 1$. Substituting $a_{i, S_i + p} = 2^{S_i} \lfloor a_{i, S_i + p}/2^{S_i} \rfloor + a_{i, S_i}$, $\frac{1}{3^i 2^{S_i}} = \frac{q_{i, p} 2^p}{3^i} - \lfloor \frac{a_{i, S_i + p}}{2^{S_i}} \rfloor - \frac{a_{i, S_i}}{2^{S_i}}$, and $\frac{n_e}{2^{S_1}} = \sum_{i=1}^r \frac{a_{i, S_i}}{2^{S_i}} - A_e$ into $u_e = 3^r(n_e/2^{S_1} + \sum_{i=1}^r \frac{1}{3^i 2^{S_i}})$ cancels $\sum_{i=1}^r a_{i, S_i}/2^{S_i}$ identically and yields (6.7)–(6.8). Since $(3 n_{(e_i, \dots, e_r)} + 1)/2^{e_i} \in (0, 3 \cdot 2^{S_{i+1}})$ has suffix valuation word $(e_{i+1}, \dots, e_r)$, it equals $n_{(e_{i+1}, \dots, e_r)} + (\kappa_i + \mathbf{1}_{i=r}) 2^{S_{i+1}}$ for a unique $\kappa_i \in \{0, 1, 2\}$; substituting this into $2^{S_{i+1}} u_{(e_i, \dots, e_r)} = 3^{r-i}((3 n_{(e_i, \dots, e_r)} + 1)/2^{e_i}) + C(e_{i+1}, \dots, e_r)$ and reducing $3 n_{(e_i, \dots, e_r)} + 1 \equiv 1 \pmod 3$ gives (6.9)–(6.11). Writing $e_i = 2 q_i + e_i^{(0)}$ ($e_i^{(0)} = 2 - \eta_i \in \{1, 2\}$), invariance of $\kappa_i$ under $q_i$ factors $4^{q_i}$ out of $3 n_{(e_i, \dots, e_r)} + 1 = 4^{q_i}(3 n_{(e_i^{(0)}, \dots, e_r)} + 1)$, so $(4^{q_i} - 1)/3 \equiv q_i \equiv c_i \pmod 3$ shifts $n_{(e_i, \dots, e_r)} \bmod 3$ by $+c_i \pmod 3$ and makes $c_{i+1} \in \{0, 1, 2\} \mapsto \kappa_i \in \{0, 1, 2\}$ a triangular bijection on every cube $\mathcal{C}$, proving (6.12)–(6.13). Finally, since $q_{i, 0} = (3^i a_{i, S_i} + 1)/2^{S_i} \in \{1, \dots, 3^i - 1\}$ satisfies $q_{i, 0} \equiv 2^{-S_i} \pmod{3^i}$, we have $q_{i, 0} = (2^{-S_i} \bmod 3^i)$ and $W_r = \sum_{i=1}^r 3^{r-i} q_{i, 0} = u_e + 3^r A_e$. Since $1 \le u_e \le 3^r - 1$ ((6.9) and $3 \nmid u_e$), $u_e = W_r \bmod 3^r = (2^{-S_1} C(e)) \bmod 3^r$ and $A_e = \lfloor W_r / 3^r \rfloor$, proving (6.14); subtracting $3^r \lfloor Y / 3^r \rfloor$ from $W_r = 3^{r-h} X_h(a) + Y$ and noting $0 \le X_h(a) < h 3^h$ and $0 \le Y \bmod 3^r < 3^r$ proves $0 \le B_h \le h$ in (6.15). $\blacksquare$

6.1 Scope and Limitations

While Theorems 1–5 provide exact structural identities and certified scale cutoffs, their scope has three essential boundaries:

1. Order-statistics and state-ordering restrictions: Theorem 1 characterizes the Eulerian rank-descent count $d(a) = \mathrm{ascents}(\alpha)$ (when $3 y_1 > c$) and decomposes $b_{\mathrm{gap}}(a) = (\mathrm{runs}(\alpha) - 1) + |\mathcal{J}_{\mathrm{eq}}(a)|$, while Theorem 3 transfers $\Phi$-order bounds ($b_{\mathrm{gap}}^\Phi(a) \le 2 m_+(a) - 1$ and $d_\Phi(a) \le m_+(a)$ on binary words) to actual state order only on the Stern–Brocot upper spine, $(4, 7), (5, 9), (8, 13)$, or cycles satisfying the small-drift condition $x_{\min}(a) > k^2 / (3\ln 2)$. High-complexity cycles with large $E_{\ge 2}(a)$ and $m_+(a)$ outside these regimes are not excluded by these bounds alone.
2. Fixed-slope scale cutoffs vs. arbitrary slopes: Theorem 2 and Table 1 certify finite scale cutoffs $D_{\mathrm{stop}}(r_0, s_0)$ for fixed reduced slope pairs $(r_0, s_0)$, reducing the bounded search range needed to close a given fixed-slope family. They do not bound arbitrary-slope growth as $r_0 \to \infty$ with $2^{s_0} - 3^{r_0}$ small relative to $3^{r_0}$.
3. Finite-word carry/digit decompositions vs. linear dyadic atom bounds: Theorem 5(4)–(5) removes the carry uncertainty via the unrounded suffix-sum floor (6.8), establishes unconditional 3-adic and pointwise atom bounds (6.11) and mod-$6$ cube equidistribution (6.13), and reduces the $h$-step split carry from $r$ values to $h + 1$ values (6.15). Proving a one-sided dyadic atom bound at positive linear resolution $p = \lfloor \delta k \rfloor$ under the geometric measure (Rees [17, Conjecture 4.1]) and ruling out divergent Collatz orbits remain open.

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