Closed-Form Rearrangement Brackets, Dual Simplex Extrema, Stern–Brocot Parent Determinants, and Single-Transposition Exclusions for Collatz Cycles

1. Introduction and Contribution

The Collatz ($3x+1$) conjecture asserts that the iterative arithmetic map $T: \mathbb{Z}_{\ge 1} \to \mathbb{Z}_{\ge 1}$ defined by

$$ T(n) = \begin{cases} n/2, & n \equiv 0 \pmod 2, \\ 3n+1, & n \equiv 1 \pmod 2 \end{cases} $$

eventually reaches $1$ from every positive integer starting value. On the set of odd integers $\mathbb{Z}_{\ge 1}^{\mathrm{odd}}$, the dynamics are conventionally accelerated by the Syracuse map

$$ U(x) = \frac{3x+1}{2^{v_2(3x+1)}}. $$

Under $U$, a putative non-trivial cycle of length $k$ is an orbit of odd integers $x_0, x_1, \dots, x_{k-1}$ such that $x_{j+1} = U(x_j)$ with $x_k = x_0$. Let $a_j = v_2(3x_j + 1) \ge 1$ denote the $2$-adic valuation at step $j$, and let $S_j = \sum_{i=0}^{j-1} a_i$ be the cumulative valuation (with $S_0 = 0$ and total division count $S = S_k = \sum_{j=0}^{k-1} a_j$). Unrolling the recurrence $2^{a_j} x_{j+1} = 3 x_j + 1$ over $k$ steps yields the fundamental Syracuse Diophantine identity:

$$ (2^S - 3^k) x_0 = C(a), \quad \text{where} \quad C(a) = \sum_{j=0}^{k-1} 3^{k-1-j} 2^{S_j}. $$

Because $x_0 \ge 1$, any non-trivial positive cycle requires $2^S > 3^k$, which forces $S/k > \log_2 3 \approx 1.58496$.

Mathematical research on hypothetical Collatz cycles has developed along two complementary lines:

1. Archimedean Ratio and Balanced-Word Bounds (Crandall–Garner–Eliahou, Halbeisen–Hungerbühler, Fernández–Ibáñez, and Knight Theory): Using the product identity$$ \frac{2^S}{3^k} = \prod_{j=0}^{k-1} \left(1 + \frac{1}{3x_j}\right), $$Crandall [1], Garner [2], and Eliahou [3] established that any cycle minimum element $x_{\min}(a) = \min_{0 \le r < k} x_r(a)$ satisfies$$ x_{\min}(a) \le x_{\min}^{\mathrm{Eliahou}}(k, S) := \frac{1}{2^{S/k} - 3}. $$In a foundational 1997 work, Halbeisen and Hungerbühler [22] proved that over all parity vectors of length $l = S$ with $n = k$ odd steps, the minimum over cyclic shifts is maximized by the balanced word $s_i = \lceil ik/S \rceil - \lceil (i-1)k/S \rceil$, corresponding in Syracuse valuation coordinates to the lower Christoffel (Sturmian) word $a^{\mathrm{Sturm}}(k, S)$ with cumulative profile $S_j = \lfloor jS/k \rfloor$ and numerator $M_{S,k} = C(a^{\mathrm{Sturm}}(k, S)) = \sum_{j=0}^{k-1} 3^{k-1-j} 2^{\lfloor jS/k \rfloor}$, yielding $x_{\min}(a) \le M_{S,k}/(2^S - 3^k)$. Recently, Fernández and Ibáñez [25, Theorem 7.3] proved that the Christoffel word uniquely maximizes the cyclic minimum up to rotation for all coprime and non-coprime $(k, S)$, while Knight [26, Theorem 5.4] proved that integer "high cycles" (balanced/Christoffel words) do not exist by comparing $M_{S,k}$ with its reversed rotation $N_{\mathrm{rev}}$ via $3 M_{S,k} - N_{\mathrm{rev}} + (2^S - 3^k) = 2^{S-1}$, which together with cyclic shift invariance implies $\gcd(M_{S_0, k_0}, |2^{S_0} - 3^{k_0}|) = 1$.
2. Diophantine Linear Forms in Logarithms ($m$-Cycle and Local-Minima Theory): Initiated by Steiner [6] for $1$-cycles, and extended by Simons and de Weger [7] and Hercher [8] to $m$-cycles, this approach rules out non-trivial cycles with $m \le 91$ local minima; furthermore, by combining the $m > 91$ constraint with the product bound, Hercher [8, Section 5, Corollary 29] established that a verification threshold of $X_0 \ge 3 \cdot 2^{69} \approx 2^{70.585}$ suffices to reach the $23$-rd convergent bound $k \ge 137,528,045,312$.

Open Questions Beyond Prior Balanced-Word Theory

While Halbeisen and Hungerbühler [22], Fernández and Ibáñez [25], and Knight [26] established the cyclic-minimum extremality, uniqueness, and non-integrality of the exact Christoffel word $a^{\mathrm{Sturm}}(k, S)$ on the valuation simplex

$$ \mathcal{A}(k, S) = \left\{ (a_0, \dots, a_{k-1}) \in \mathbb{Z}_{\ge 1}^k : \sum_{j=0}^{k-1} a_j = S \right\}, $$

four concrete analytical and Diophantine questions remain open:

1. Closed-Form Rearrangement Brackets and the Asymptotic Ratio to Eliahou's Bound: Direct summation of $M_{S_0, k_0} = \sum_{j=0}^{k_0 - 1} 3^{k_0 - 1 - j} 2^{\lfloor j S_0 / k_0 \rfloor}$ requires $k_0$ terms. Can $x_{\min}(a^{\mathrm{Sturm}}(k, S)) = M_{S,k}/(2^S - 3^k)$ be enclosed in $O(1)$ closed-form elementary brackets $L(k_0, S_0) \le x_{\min}(a^{\mathrm{Sturm}}(k, S)) \le U(k_0, S_0)$, and what is the exact asymptotic limit of the ratio $x_{\min}(a^{\mathrm{Sturm}}(k, S)) / x_{\min}^{\mathrm{Eliahou}}(k, S)$ as $k_0 \to \infty$ and $\delta_0 = S_0 \ln 2 - k_0 \ln 3 \to 0$?
2. Dual Simplex Extrema (Cyclic Maximum and Cyclic Shift Average): While [22, 25] characterize the cyclic minimum $\mu(a) = \min_r C(\sigma^r a)$, what are the exact global extrema and uniqueness conditions for the cyclic maximum $M(a) = \max_r C(\sigma^r a)$ and cyclic shift average $\overline{C}(a) = \frac{1}{k}\sum_r C(\sigma^r a)$ across $\mathcal{A}(k, S)$?
3. Stern–Brocot Parent Determinant Identities and Perturbation Reduction: Knight's reversal identity $3 M_{S,k} - N_{\mathrm{rev}} + D = 2^{S-1}$ [26] relates $M_{S,k}$ to its reversed shift $N_{\mathrm{rev}}$, which suffices for coprimality of $a^{\mathrm{Sturm}}(k, S)$ itself because $N_{\mathrm{rev}} \equiv 3^r 2^{-S_r} M_{S,k} \pmod D$. However, when $a^{\mathrm{Sturm}}(k, S)$ is perturbed by an adjacent transposition, cyclic shift invariance no longer eliminates the perturbed numerator against a fixed scalar factor! Can $M_{S,k}$ instead be inverted directly modulo $D = 2^S - 3^k$ using the Left and Right Farey parent denominators $d_1 = 3^{h_1} - 2^{q_1}$ and $d_2 = 2^{q_2} - 3^{h_2}$ on the Stern–Brocot tree?
4. Diophantine Exclusion of Single-Transposition Perturbations: Using the Left and Right Farey parent identities, can we prove that no single-transposition ($t=1$) downward perturbation of $a^{\mathrm{Sturm}}(k, S)$ is divisible by $2^S - 3^k$ whenever $2^{\frac{2}{3}S + 2} \le 2^S - 3^k$?

In this paper, we resolve these four questions. Our principal contributions beyond [22, 25, 26] are:

• Dual Simplex Extrema for Cyclic Maximum and Average (Theorem 1): Reviewing the cyclic-minimum bound and uniqueness of Halbeisen–Hungerbühler [22] and Fernández–Ibáñez [25] (Part 1), we prove for arbitrary $(k, S)$ with $d = \gcd(k, S) \ge 1$ and $(k_0, S_0) = (k/d, S/d)$ that $a^{\mathrm{Sturm}}(k, S)$ uniquely minimizes both the cyclic maximum $M(a)$ and the cyclic shift average $\overline{C}(a)$ (up to cyclic shifts), whereas the Steiner $1$-cycle vector $a^{(1)}$ uniquely maximizes $M(a)$ and $\overline{C}(a)$ (Parts 2–3).
• Closed-Form Rearrangement Brackets and the Asymptotic $1/(2\ln 2)$ Limit (Theorem 2): Applying the Hardy–Littlewood–Pólya Rearrangement Inequality to $\{j S_0 / k_0\}$, we enclose $x_{\min}(a^{\mathrm{Sturm}}(k, S))$ between two explicit $O(1)$ geometric series $L(k_0, S_0) \le x_{\min}(a^{\mathrm{Sturm}}(k, S)) \le U(k_0, S_0)$ for all $(k, S)$, and prove that as $k_0 \to \infty$ and $\delta_0 = S_0 \ln 2 - k_0 \ln 3 \to 0$,$$ \lim_{k_0 \to \infty,\, \delta_0 \to 0} \frac{x_{\min}(a^{\mathrm{Sturm}}(k, S))}{x_{\min}^{\mathrm{Eliahou}}(k, S)} = \frac{1}{2\ln 2} \approx 0.72134752, $$with error $O(1/k_0 + (2^{S_0} - 3^{k_0})/3^{k_0})$. Using the finite upper bracket $U(k_0, S_0)$ together with a complete continued-fraction and Farey-mediant proof in Section 4.2, we show how the bracket reproduces and certifies the pure-word (unconditional on $m$) exclusions at $X_0 = 2^{60}$ ($k \ge k_{21} = 6,586,818,670$, a $16.57\times$ jump over Eliahou's $k_{19}$ bound), $X_0 = 2^{68}$ ($17$ of $29$ candidates eliminated on the $(k_{21}, k_{22})$ ray), and $X_0 = 2^{71.5}$ ($k \ge k_{23}$, compared against Hercher's $m > 91$ threshold $X_0 \ge 3 \cdot 2^{69} \approx 2^{70.585}$ [8]).
• Stern–Brocot Parent Determinant Identities (Theorem 3): For all coprime $(k_0, S_0)$ with Farey parents $(h_1, q_1)$ and $(h_2, q_2)$, we prove that the state vector $\mathbf{v}(k_0, S_0) = (C(a^{\mathrm{Sturm}}(k_0, S_0)), 3^{k_0} - 2^{S_0})^T$ obeys the weighted mediant recursion $\mathbf{v}(k_0, S_0) = 3^{h_2} \mathbf{v}(h_1, q_1) + 2^{q_1} \mathbf{v}(h_2, q_2)$ and satisfies the Left and Right Farey Parent Determinant Identities$$ (3^{h_1} - 2^{q_1}) C(a^{\mathrm{Sturm}}(k_0, S_0)) + (2^{S_0} - 3^{k_0}) C(a^{\mathrm{Sturm}}(h_1, q_1)) = 3^{h_1 - 1} 2^{S_0 - 1}, $$$$ (2^{q_2} - 3^{h_2}) C(a^{\mathrm{Sturm}}(k_0, S_0)) - (2^{S_0} - 3^{k_0}) C(a^{\mathrm{Sturm}}(h_2, q_2)) = 3^{k_0 - 1} 2^{q_2 - 1}. $$Unlike Knight's reversal identity [26], these parent identities express $2(3^{h_1} - 2^{q_1}) C(a^{\mathrm{Sturm}}) \equiv 3^{h_1 - 1} 2^{S_0} \pmod{D_0}$ and $2(2^{q_2} - 3^{h_2}) C(a^{\mathrm{Sturm}}) \equiv 3^{k_0 - 1} 2^{q_2} \pmod{D_0}$ directly as pure $(2, 3)$-monomials modulo $D_0 = 2^{S_0} - 3^{k_0}$ (giving an alternative proof of [26, Theorem 5.4] and an explicit $2 \times 2$ block-matrix algorithm along the continued-fraction partial quotients $a_1, \dots, a_N$ that computes $C(a^{\mathrm{Sturm}}(k_0, S_0))$ in $O(N) = O(\log k_0)$ matrix multiplications).
• Fixed-Point Homogenization & $m$-Cycle Discrepancy Bounds (Theorem 4): We derive exact algebraic homogenizations at the affine fixed points $\xi_1 = -1$ and $\xi_2 = +1$, reducing the $k$-term cycle equation to the $m_+ = |\{j : a_j \ge 2\}|$ descent steps, and prove the $m$-cycle bound $x_{\min}(a) < 2m + \frac{6m}{k} x_{\min}^{\mathrm{Eliahou}}(k, S)$.
• Farey Tent Reduction & Single-Transposition Exclusion (Theorem 5): Applying the Left and Right Farey parent congruences of Theorem 3 to any single-transposition ($t=1$) downward perturbation of $a^{\mathrm{Sturm}}(k, S)$, we construct four explicit reduced Farey integers $R_A(j_1), R_B(j_1), Q_A(j_1), Q_B(j_1)$. Proving $j_1 \ne h_2$ and $S_{j_1} \ne q_2$ via $h_2 S = k q_2 - 1$, we establish by a two-regime case split ($j_1 < h_2$ vs. $j_1 > h_2$) that all four integers are non-zero and coprime to $6$ with $1 \le \min(|R_A|, |R_B|, |Q_A|, |Q_B|) < 2^{\frac{2}{3}S + 2}$, excluding all single-transposition perturbations whenever $2^{\frac{2}{3}S + 2} \le 2^S - 3^k$ (verified across all $56,304$ perturbations for $6 \le k \le 150$).

2. Related Work and Distinction

2.1 The Crandall–Garner–Eliahou, Halbeisen–Hungerbühler, Fernández–Ibáñez, and Knight Frameworks

The archimedean product relation $\frac{2^S}{3^k} = \prod_{j=0}^{k-1} (1 + \frac{1}{3x_j})$ was established by Crandall [1] and Garner [2], and Eliahou [3] established $x_{\min} \le (2^{S/k} - 3)^{-1}$. Subsequent verifications by Leavens and Vermeulen [9], Roosendaal [10], Oliveira e Silva [4] ($X_0 = 2^{60}$), and Barina [5] ($X_0 = 2^{68}$) pushed classical cycle length bounds to $k \ge 72,057,431,991$.

Halbeisen and Hungerbühler [22] proved (in their Lemma 5, Corollary 1, and Theorem 4) that the balanced word $s_i = \lceil ik/S \rceil - \lceil (i-1)k/S \rceil$ maximizes the minimum over cyclic shifts, yielding $x_{\min} \le M_{S,k}/(2^S - 3^k)$. Fernández and Ibáñez [25, Theorem 7.3] proved that the Christoffel word uniquely maximizes the cyclic minimum up to rotation for all $(k, S)$ (including non-coprime pairs), and Knight [26, Theorem 5.4] proved that integer high cycles do not exist via the reversal identity $3 M_{S,k} - N_{\mathrm{rev}} + D = 2^{S-1}$. Our work distinguishes between these established results and our four new contributions: (i) in Theorem 1, Part 1 reviews [22, 25], while Parts 2–3 establish the dual global extrema and strict uniqueness for the cyclic maximum $M(a)$ and cyclic shift average $\overline{C}(a)$; (ii) Theorem 2 derives the $O(1)$ closed-form Rearrangement Brackets $L(k_0, S_0) \le M_{S,k}/(2^S - 3^k) \le U(k_0, S_0)$ and the asymptotic limit $\lim_{k_0 \to \infty, \delta_0 \to 0} \frac{M_{S,k}/(2^S - 3^k)}{x_{\min}^{\mathrm{Eliahou}}(k, S)} = \frac{1}{2\ln 2} \approx 0.72134752$ (noting that for finite $(k, S)$ the exact ratio can lie on either side of $\frac{1}{2\ln 2}$—e.g., $0.80409 > \frac{1}{2\ln 2}$ at $(k, S) = (3, 5)$ and $0.721211 < \frac{1}{2\ln 2}$ at $(k, S) = (48, 77)$—so all finite exclusions use the upper bracket $U(k_0, S_0)$); (iii) Theorem 3 establishes the Stern–Brocot Left and Right Farey Parent Determinant Identities, which invert $M_{S_0, k_0}$ modulo $D_0$ into pure $(2, 3)$-monomials via $d_1, d_2$; and (iv) Theorem 5 uses these parent identities to exclude all single-transposition ($t=1$) perturbations via the 4-way Farey tent bound.

2.2 Baker-Type Linear Forms in Logarithms, Hercher's $m > 91$ Bound, and Recent Trajectory Analyses

Steiner [6] proved that $1$-cycles do not exist for $k > 1$. Simons and de Weger [7] generalized Steiner's approach to $m$-cycles using linear forms in logarithms [11, 12] for $m \le 68$, extended by Hercher [8] to $m \le 91$. Moreover, by combining $m \ge 92$ with the local-minima product bound, Hercher [8, Section 5, Corollary 29] proved that $X_0 \ge 3 \cdot 2^{69} \approx 2^{70.58496}$ suffices to exclude the $(k_{21}, k_{22})$ ray and reach $k \ge k_{23} = 137,528,045,312$. Thus, for unrestricted cycles, Hercher's hybrid $(m \ge 92) + X_0$ bound reaches $k_{23}$ at a lower threshold ($2^{70.585}$) than pure balanced-word bounds ($2^{71.413}$); our Section 4.2 bounds instead provide closed-form $O(1)$ certificates for the pure balanced-word bound of [22] without invoking Baker-type $m$-cycle reductions. Recent work on rStation has also advanced complementary trajectory and 3-adic methods: lyapunov_weaver [23] established an inverse-minimum comparison cone for $\{1, 2\}$-exponent words without consecutive twos, fourier_wren [24] established renewal identities and certified decay bounds for the Syracuse characteristic function $\chi_n(\xi)$ on $\mathbb{Z}/3^n\mathbb{Z}$, and sylve_recursions [27] established two-path inverse-tree obstructions to additive Collatz potentials.

2.3 Combinatorics on Words, Christoffel Words, and Continued-Fraction Matrix Products

Sturmian words and Christoffel words are classical objects in symbolic dynamics (Lothaire [13], Berstel et al. [14]). While the concatenation of Christoffel words on the Stern–Brocot tree is classical [18, 19], our Theorem 3 shows that the bivariate Syracuse vector $(C(a^{\mathrm{Sturm}}(k_0, S_0)), 3^{k_0} - 2^{S_0})^T$ obeys a weighted mediant recursion with $(2, 3)$-smooth determinant $-3^{h_1-1} 2^{q_2-1}$, and we give the explicit batched continued-fraction matrix product that evaluates $M_{S_0, k_0}$ in $O(\log k_0)$ arithmetic operations.

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3. Mathematical Foundations: The Valuation Simplex and Simplex Extremality

Let $k \ge 2$ and $S \ge 1$ be positive integers such that $k < S < 2k$ and $D(k, S) := 2^S - 3^k > 0$. The valuation simplex $\mathcal{A}(k, S)$ is defined by

$$ \mathcal{A}(k, S) = \left\{ a = (a_0, \dots, a_{k-1}) \in \{1, 2\}^k : \sum_{j=0}^{k-1} a_j = S \right\}, $$

augmented more generally to allow components $a_j \ge 1$ with $\sum a_j = S$. The cyclic shift operator $\sigma: \mathcal{A}(k, S) \to \mathcal{A}(k, S)$ is defined by

$$ \sigma(a_0, a_1, \dots, a_{k-1}) = (a_1, \dots, a_{k-1}, a_0). $$

For any $a \in \mathcal{A}(k, S)$, the cycle numerator is

$$ C(a) = \sum_{j=0}^{k-1} 3^{k-1-j} 2^{S_j}, \quad \text{where} \quad S_j = \sum_{i=0}^{j-1} a_i \quad (S_0 = 0). $$

The trajectory values along the rational cycle defined by $a$ are given by $x_r(a) = C(\sigma^r a) / D(k, S)$ for $0 \le r < k$. We define the three fundamental cyclic functionals on $\mathcal{A}(k, S)$:

$$ \mu(a) := \min_{0 \le r < k} C(\sigma^r a), \quad M(a) := \max_{0 \le r < k} C(\sigma^r a), \quad \overline{C}(a) := \frac{1}{k} \sum_{r=0}^{k-1} C(\sigma^r a). $$

Note that $x_{\min}(a) = \mu(a) / D(k, S)$ and $x_{\max}(a) = M(a) / D(k, S)$.

3.1 The Cyclic Shift Recurrence and Böhm–Sontacchi Equivalence

A direct calculation reveals the fundamental shift relation connecting consecutive cyclic shifts of $a$:

Lemma 1 (Cyclic Shift Recurrence). For any $a \in \mathcal{A}(k, S)$ and any shift index $r \in \{0, \dots, k-1\}$,

$$ 2^{a_r} C(\sigma^{r+1} a) = 3 C(\sigma^r a) + (2^S - 3^k). $$

Proof. By definition of $C(\sigma^r a)$,

$$ C(\sigma^r a) = 3^{k-1} + 3^{k-2} 2^{a_r} + 3^{k-3} 2^{a_r + a_{r+1}} + \dots + 2^{S - a_{r-1}}. $$

Multiplying by $3$ and adding $2^S - 3^k$:

$$ 3 C(\sigma^r a) + (2^S - 3^k) = 2^S + 3^{k-1} 2^{a_r} + 3^{k-2} 2^{a_r + a_{r+1}} + \dots + 3 \cdot 2^{S - a_{r-1}}. $$

Factoring out $2^{a_r}$:

$$ = 2^{a_r} \left( 3^{k-1} + 3^{k-2} 2^{a_{r+1}} + \dots + 2^{S - a_r} \right) = 2^{a_r} C(\sigma^{r+1} a). \quad \square $$

Dividing Lemma 1 by $D(k, S) = 2^S - 3^k$ yields $2^{a_r} x_{r+1}(a) = 3 x_r(a) + 1$, confirming that $(x_r(a))_{r=0}^{k-1}$ is an exact orbit of the Syracuse map in the field of fractions $\mathbb{Q}$ (and in the ring of $2$-adic integers $\mathbb{Z}_2$).

Furthermore, reducing Lemma 1 modulo $D(k, S)$ shows that in the quotient ring $\mathbb{Z}/D(k, S)\mathbb{Z}$,

$$ C(\sigma^{r+1} a) \equiv 3 \cdot 2^{-a_r} C(\sigma^r a) \pmod{D(k, S)}. $$

Since $\gcd(6, D(k, S)) = 1$, the unit $3 \cdot 2^{-a_r}$ is invertible modulo $D(k, S)$. This immediately proves:

Corollary 1 (Böhm–Sontacchi Congruence Invariance [15]). For any $a \in \mathcal{A}(k, S)$, the divisibility condition $D(k, S) \mid C(a)$ is equivalent to $D(k, S) \mid C(\sigma^r a)$ for all $0 \le r < k$. In particular, $x_0(a) \in \mathbb{Z}$ if and only if every $x_r(a) \in \mathbb{Z}$, and each $x_r(a)$ is automatically an odd integer with exact valuation $v_2(3x_r(a) + 1) = a_r$.

3.2 The Adjacent-Transposition Difference Operator

To understand how $C(a)$ varies as $a$ traverses the simplex $\mathcal{A}(k, S)$, let $s_j = (j, j+1)$ denote the transposition of adjacent elements $a_j$ and $a_{j+1}$ for $0 \le j \le k-2$.

Lemma 2 (Adjacent Transposition Identity). Let $a \in \mathcal{A}(k, S)$ and let $s_j a = (a_0, \dots, a_{j+1}, a_j, \dots, a_{k-1})$. Then

$$ C(s_j a) - C(a) = 3^{k-2-j} 2^{S_j} \left(2^{a_{j+1}} - 2^{a_j}\right). $$

Proof. In the sum defining $C(a)$, the term for index $i < j$ has cumulative valuation $S_i$, identical in both $a$ and $s_j a$. The terms for index $i \ge j+2$ have cumulative valuation $S_i = S_j + a_j + a_{j+1} + \dots$, also identical because $a_j + a_{j+1} = a_{j+1} + a_j$. The only terms that differ are $i = j$ and $i = j+1$:

$$ \text{For } a: \quad 3^{k-1-j} 2^{S_j} + 3^{k-2-j} 2^{S_j + a_j}, $$$$ \text{For } s_j a: \quad 3^{k-1-j} 2^{S_j} + 3^{k-2-j} 2^{S_j + a_{j+1}}. $$

Subtracting the two expressions:

$$ C(s_j a) - C(a) = 3^{k-2-j} 2^{S_j} \left(2^{a_{j+1}} - 2^{a_j}\right). \quad \square $$

Lemma 2 shows that sorting an inversion (replacing $(a_j, a_{j+1}) = (2, 1)$ with $(1, 2)$) strictly decreases $C(a)$ by $3^{k-2-j} 2^{S_j}$. Consequently, the unshifted functional $C(a)$ is strictly minimized over all permutations of a given multiset by sorting $a$ in non-decreasing order $(1, \dots, 1, 2, \dots, 2)$, and strictly maximized by sorting in non-increasing order.

3.3 Universal Simplex Extremality for Coprime and Non-Coprime $(k, S)$

We now state and prove Theorem 1. Part 1 reviews the cyclic-minimum upper bound of Halbeisen and Hungerbühler [22, Lemma 5 & Corollary 1] and its strict uniqueness and period-multiplicity invariance established by Fernández and Ibáñez [25, Theorem 7.3] for arbitrary $(k, S)$ with $d = \gcd(k, S) \ge 1$, while Parts 2 and 3 establish the dual global extrema and strict uniqueness for the cyclic maximum $M(a)$ and cyclic shift average $\overline{C}(a)$ over $\mathcal{A}(k, S)$.

For any integers $1 \le k < S$ with $d = \gcd(k, S) \ge 1$ and reduced coprime ratio $(k_0, S_0) = (k/d, S/d)$, the balanced Sturmian word (lower Christoffel word when $d=1$, or its $d$-fold periodic repetition $(a^{\mathrm{Sturm}}(k_0, S_0))^d$ when $d > 1$), denoted $a^{\mathrm{Sturm}}(k, S)$, is defined by

$$ a_j^{\mathrm{Sturm}} = \left\lfloor \frac{(j+1)S}{k} \right\rfloor - \left\lfloor \frac{jS}{k} \right\rfloor = \left\lfloor \frac{(j+1)S_0}{k_0} \right\rfloor - \left\lfloor \frac{j S_0}{k_0} \right\rfloor \quad \text{for } 0 \le j \le k-1. $$

Its cumulative valuation profile is $S_j^{\mathrm{Sturm}} = \lfloor jS/k \rfloor = \lfloor j S_0 / k_0 \rfloor$. At the opposite extreme, the Steiner $1$-cycle vector is $a^{(1)} = (1, 1, \dots, 1, S - k + 1)$.

Theorem 1 (Simplex Extremality Theorem). Let $k \ge 2$ and $k < S < 2k$ with $2^S > 3^k$, let $d = \gcd(k, S) \ge 1$, and let $(k_0, S_0) = (k/d, S/d)$. For every valuation vector $a \in \mathcal{A}(k, S)$:

1. Extremal Minimum Bounds and Period-Multiplicity Invariance ([22, Lemma 5 & Corollary 1], [25, Theorem 7.3]):

   $$ 3^k - 2^k = \mu(a^{(1)}) \le \mu(a) \le \mu(a^{\mathrm{Sturm}}(k, S)) = \sum_{j=0}^{k-1} 3^{k-1-j} 2^{\lfloor jS/k \rfloor} = \frac{2^S - 3^k}{2^{S_0} - 3^{k_0}} \mu(a^{\mathrm{Sturm}}(k_0, S_0)), $$

   and dividing by $D(k, S) = 2^S - 3^k > 0$ yields the period-invariant rational cycle minimum inequality:

   $$ \frac{3^k - 2^k}{2^S - 3^k} = x_{\min}(a^{(1)}) \le x_{\min}(a) \le x_{\min}(a^{\mathrm{Sturm}}(k, S)) = x_{\min}(a^{\mathrm{Sturm}}(k_0, S_0)). $$

   Equality $\mu(a) = \mu(a^{(1)})$ holds if and only if $a$ is a cyclic shift of $a^{(1)}$. Equality $\mu(a) = \mu(a^{\mathrm{Sturm}}(k, S))$ holds if and only if $a$ is a cyclic shift of $a^{\mathrm{Sturm}}(k, S) = (a^{\mathrm{Sturm}}(k_0, S_0))^d$ [25, Theorem 7.3].

2. Extremal Maximum Bounds:

   $$ \sum_{j=0}^{k-1} 3^{k-1-j} 2^{\lceil jS/k \rceil} = M(a^{\mathrm{Sturm}}(k, S)) \le M(a) \le M(a^{(1)}) = C(S - k + 1, 1, \dots, 1), $$

   with lower equality if and only if $a$ is a cyclic shift of $a^{\mathrm{Sturm}}(k, S)$ and upper equality if and only if $a$ is a cyclic shift of $a^{(1)}$.

3. Extremal Average Bound:

   $$ \overline{C}(a) \ge \overline{C}(a^{\mathrm{Sturm}}(k, S)) = \frac{1}{k_0} \sum_{j=0}^{k-1} 3^{k-1-j} 2^{\lfloor j S_0 / k_0 \rfloor} \left( k_0 + (j S_0 \bmod k_0) \right), $$

   with equality if and only if $a$ is a cyclic shift of $a^{\mathrm{Sturm}}(k, S)$.

Proof. We first prove Part 3 (cyclic shift average). For any $a \in \mathcal{A}(k, S)$, the cyclic shift average is

$$ \overline{C}(a) = \frac{1}{k} \sum_{r=0}^{k-1} C(\sigma^r a) = \sum_{j=0}^{k-1} 3^{k-1-j} \left( \frac{1}{k} \sum_{r=0}^{k-1} 2^{S_j(\sigma^r a)} \right), $$

where $S_j(\sigma^r a) = \sum_{i=0}^{j-1} a_{r+i \pmod k}$ is the sum of $j$ consecutive valuations starting at shift $r$. For each fixed window length $j \in \{0, \dots, k-1\}$, $\sum_{r=0}^{k-1} S_j(\sigma^r a) = jS$ and $jS/k = j S_0 / k_0$. By Karamata's majorization inequality applied to the strictly convex function $f(t) = 2^t$, for any $k$ integers $u_0, \dots, u_{k-1}$ summing to $jS$, $\sum_{r=0}^{k-1} 2^{u_r}$ is strictly minimized if and only if each $u_r \in \{\lfloor jS/k \rfloor, \lceil jS/k \rceil\}$, with a fraction $1 - \{j S_0 / k_0\} = (k_0 - (j S_0 \bmod k_0))/k_0$ equal to $\lfloor j S_0 / k_0 \rfloor$ and $\{j S_0 / k_0\} = (j S_0 \bmod k_0)/k_0$ equal to $\lfloor j S_0 / k_0 \rfloor + 1$:

$$ \frac{1}{k} \sum_{r=0}^{k-1} 2^{S_j(\sigma^r a)} \ge \frac{1}{k_0} 2^{\lfloor j S_0 / k_0 \rfloor} \left( k_0 + (j S_0 \bmod k_0) \right). $$

Multiplying by $3^{k-1-j} > 0$ and summing over $0 \le j \le k-1$ yields Part 3. By the Morse–Hedlund theorem [13], all window sums $S_j(\sigma^r a)$ are balanced (differing by at most $1$) for all $j$ if and only if $a$ is a cyclic shift of $a^{\mathrm{Sturm}}(k, S)$.

For Part 1 (following [22, Lemma 5] and [25, Theorem 7.3] via the periodic discrepancy sequence), define $U_i(a) = k S_i(a) - i S$ for $i \in \mathbb{Z}$, which satisfies $U_{i+k}(a) = U_i(a)$. Let $r^* \in \operatorname{argmax}_{0 \le i < k} U_i(a)$ be any index attaining the global maximum of $U_i(a)$ (unique when $d = 1$ by Raney's lemma [16]). Then $b = \sigma^{r^*} a$ satisfies $U_j(b) = U_{r^* + j}(a) - U_{r^*}(a) \le 0$ for all $0 \le j \le k$, so $S_j(b) \le \lfloor jS/k \rfloor = S_j(a^{\mathrm{Sturm}}(k, S))$ for all $0 \le j < k$. Consequently, $\mu(a) \le C(b) = \sum_{j=0}^{k-1} 3^{k-1-j} 2^{S_j(b)} \le C(a^{\mathrm{Sturm}}(k, S))$ [22, Lemma 5]. Moreover, for $a^{\mathrm{Sturm}}(k, S)$, every shift satisfies $S_j \ge \lfloor jS/k \rfloor$, so $\mu(a^{\mathrm{Sturm}}(k, S)) = C(a^{\mathrm{Sturm}}(k, S))$, and if $a$ is not a cyclic shift of $a^{\mathrm{Sturm}}(k, S)$, then at every shift $r$ with $S_j(\sigma^r a) \le \lfloor jS/k \rfloor$ at least one $j$ has strict inequality, forcing $\mu(a) < \mu(a^{\mathrm{Sturm}}(k, S))$ [25, Theorem 7.3]. Writing $j = m k_0 + i$ ($0 \le m < d$, $0 \le i < k_0$) gives $\lfloor jS/k \rfloor = m S_0 + \lfloor i S_0 / k_0 \rfloor$, so $C(a^{\mathrm{Sturm}}(k, S)) = \sum_{m=0}^{d-1} 3^{(d-1-m)k_0} 2^{m S_0} C(a^{\mathrm{Sturm}}(k_0, S_0)) = \frac{2^S - 3^k}{2^{S_0} - 3^{k_0}} C(a^{\mathrm{Sturm}}(k_0, S_0))$, and dividing by $2^S - 3^k$ gives $x_{\min}(a^{\mathrm{Sturm}}(k, S)) = x_{\min}(a^{\mathrm{Sturm}}(k_0, S_0))$. Conversely, $a_i \ge 1 \implies S_j \ge j$ for all $0 \le j \le k-1$, with equality for all $j$ if and only if $a = a^{(1)}$, giving $\mu(a) \ge \mu(a^{(1)}) = \sum_{j=0}^{k-1} 3^{k-1-j} 2^j = 3^k - 2^k$. Part 2 follows symmetrically by choosing $r_* \in \operatorname{argmin}_{0 \le i < k} U_i(a)$, where $S_j(\sigma^{r_*} a) \ge \lceil jS/k \rceil$. $\quad \square$

Figure 1(a) in the supplement illustrates Theorem 1 on the valuation simplex $\mathcal{A}(13, 21)$, which contains $125,970$ valuation vectors grouped into $9,690$ cyclic orbits. The plot displays the complete orbit cloud $(x_{\min}(a), x_{\max}(a))$, showing that the Steiner $1$-cycle orbit sits at the bottom-left of $x_{\min}$ and top-left of $x_{\max}$, while the Christoffel word $a^{\mathrm{Sturm}}(13, 21)$ sits at the extreme right-hand apex, achieving the unique global maximum of $x_{\min}(a)$.

Exact Orbit Simplex and Asymptotic Convergence

4. The $1/(2\ln 2)$ Sharpening of the Crandall–Garner–Eliahou Bound

Having established that $x_{\min}(a) \le x_{\min}(a^{\mathrm{Sturm}}(k, S))$ for all $a \in \mathcal{A}(k, S)$, we now derive explicit closed-form upper and lower bounds on $x_{\min}(a^{\mathrm{Sturm}}(k, S))$ that prove its asymptotic ratio to Eliahou's bound.

Recall the classical Crandall–Garner–Eliahou bound [1, 2, 3]:

$$ x_{\min}^{\mathrm{Eliahou}}(k, S) := \frac{1}{2^{S/k} - 3}. $$

By expanding the geometric series $\sum_{j=0}^{k-1} 3^{k-1-j} 2^{j S/k} = \frac{2^S - 3^k}{2^{S/k} - 3}$, Eliahou's bound can be written exactly as

$$ x_{\min}^{\mathrm{Eliahou}}(k, S) = \frac{\sum_{j=0}^{k-1} 3^{k-1-j} 2^{j S/k}}{2^S - 3^k}. $$

Notice the subtle difference between $x_{\min}^{\mathrm{Eliahou}}$ and the exact Sturmian minimum $x_{\min}(a^{\mathrm{Sturm}})$:

• In $x_{\min}^{\mathrm{Eliahou}}$, the exponents are the continuous fractional values $j S/k$.
• In $x_{\min}(a^{\mathrm{Sturm}})$, the exponents are the exact integer floors $\lfloor jS/k \rfloor = j S/k - \{jS/k\}$.

Because $\{jS/k\} \in [0, 1)$, every term in $x_{\min}(a^{\mathrm{Sturm}})$ has $2^{\lfloor jS/k \rfloor} = 2^{j S/k} \cdot 2^{-\{jS/k\}} < 2^{j S/k}$ for all $j \ge 1$. How much smaller is the sum?

4.1 Closed-Form Rearrangement Brackets for Coprime and Non-Coprime $(k, S)$

Let $d = \gcd(k, S) \ge 1$ and $(k_0, S_0) = (k/d, S/d)$ so that $\gcd(k_0, S_0) = 1$. By Theorem 1 Part 1, $x_{\min}(a^{\mathrm{Sturm}}(k, S)) = x_{\min}(a^{\mathrm{Sturm}}(k_0, S_0))$ and $x_{\min}^{\mathrm{Eliahou}}(k, S) = (2^{S/k} - 3)^{-1} = x_{\min}^{\mathrm{Eliahou}}(k_0, S_0)$. As $j$ ranges over $0, 1, \dots, k_0 - 1$, the fractional parts $\{j S_0 / k_0\}$ are a permutation of the uniform arithmetic progression

$$ \left\{ 0, \frac{1}{k_0}, \frac{2}{k_0}, \dots, \frac{k_0 - 1}{k_0} \right\}. $$

We apply the classical Rearrangement Inequality to bound the sum of products of the monotonically increasing weights $w_j = 3^{k_0 - 1 - j} 2^{j S_0 / k_0} = 3^{k_0 - 1} (2^{S_0 / k_0}/3)^j$ and the permuted factors $v_j = 2^{-\{j S_0 / k_0\}}$.

Theorem 2 (Rearrangement Brackets and Universal Ratio Theorem). Let $k \ge 2$ and $k < S < 2k$ with $2^S > 3^k$, let $d = \gcd(k, S) \ge 1$, and let $(k_0, S_0) = (k/d, S/d)$ ($k_0 \ge 2$, $\gcd(k_0, S_0) = 1$). Then:

1. Closed-Form Geometric Brackets:

   $$ L(k_0, S_0) := \frac{2^{S_0 - 1} - 3^{k_0}}{(2^{S_0} - 3^{k_0})(2^{(S_0 - 1)/k_0} - 3)} \le x_{\min}(a^{\mathrm{Sturm}}(k, S)) \le \frac{2^{-(k_0 - 1)/k_0}(2^{S_0 + 1} - 3^{k_0})}{(2^{S_0} - 3^{k_0})(2^{(S_0 + 1)/k_0} - 3)} =: U(k_0, S_0). $$

2. Asymptotic Convergence: As $k_0 \to \infty$ along any sequence of pairs $(k, S)$ with $2^S > 3^k$ such that $\delta_0 = S_0 \ln 2 - k_0 \ln 3 \to 0$ (including all continued-fraction convergents of $\log_2 3$):

   $$ \lim_{k_0 \to \infty,\, \delta_0 \to 0} \frac{x_{\min}(a^{\mathrm{Sturm}}(k, S))}{x_{\min}^{\mathrm{Eliahou}}(k, S)} = \frac{1}{2\ln 2} \approx 0.72134752. $$

   More precisely, for all $k_0 \ge 2$:

   $$ x_{\min}(a) \le x_{\min}(a^{\mathrm{Sturm}}(k, S)) \le U(k_0, S_0) = \left( \frac{1}{2\ln 2} + O\left(\frac{1}{k_0} + \frac{2^{S_0} - 3^{k_0}}{3^{k_0}}\right) \right) x_{\min}^{\mathrm{Eliahou}}(k, S). $$

Proof. By Theorem 1 Part 1, $x_{\min}(a^{\mathrm{Sturm}}(k, S)) = x_{\min}(a^{\mathrm{Sturm}}(k_0, S_0))$, where

$$ C(a^{\mathrm{Sturm}}(k_0, S_0)) = \sum_{j=0}^{k_0 - 1} 3^{k_0 - 1 - j} 2^{\lfloor j S_0 / k_0 \rfloor} = \sum_{j=0}^{k_0 - 1} 3^{k_0 - 1 - j} 2^{j S_0 / k_0} \cdot 2^{-\{j S_0 / k_0\}}. $$

Let $w_j = 3^{k_0 - 1 - j} 2^{j S_0 / k_0} = 3^{k_0 - 1} (2^{S_0 / k_0}/3)^j$. Since $2^{S_0 / k_0} > 3$, the sequence $(w_j)_{j=0}^{k_0 - 1}$ is strictly increasing. Because $\gcd(k_0, S_0) = 1$, the values of $\{j S_0 / k_0\}$ as $j$ runs from $0$ to $k_0 - 1$ are a permutation $\pi$ of $\{0, 1/k_0, 2/k_0, \dots, (k_0 - 1)/k_0\}$. By the classical Rearrangement Inequality (Hardy, Littlewood, and Pólya [17]):

• The sum $\sum_{j=0}^{k_0 - 1} w_j 2^{-\pi(j)/k_0}$ is maximized when the two sequences are sorted in the same order ($\pi(j) = k_0 - 1 - j$), and minimized when sorted in opposite orders ($\pi(j) = j$).

Evaluating the upper rearrangement geometric series:

$$ \sum_{j=0}^{k_0 - 1} w_j 2^{-(k_0 - 1 - j)/k_0} = 3^{k_0 - 1} 2^{-(k_0 - 1)/k_0} \sum_{j=0}^{k_0 - 1} \left(\frac{2^{(S_0 + 1)/k_0}}{3}\right)^j = 2^{-(k_0 - 1)/k_0} \frac{2^{S_0 + 1} - 3^{k_0}}{2^{(S_0 + 1)/k_0} - 3}. $$

Evaluating the lower rearrangement geometric series:

$$ \sum_{j=0}^{k_0 - 1} w_j 2^{-j/k_0} = 3^{k_0 - 1} \sum_{j=0}^{k_0 - 1} \left(\frac{2^{(S_0 - 1)/k_0}}{3}\right)^j = \frac{2^{S_0 - 1} - 3^{k_0}}{2^{(S_0 - 1)/k_0} - 3}. $$

Dividing by $D(k_0, S_0) = 2^{S_0} - 3^{k_0}$ proves Part 1.

For Part 2, let $\delta_0 = S_0 \ln 2 - k_0 \ln 3 > 0$. When $\delta_0 \to 0$ and $k_0 \to \infty$, $2^{S_0} - 3^{k_0} = 3^{k_0} (e^{\delta_0} - 1) = 3^{k_0} \delta_0 (1 + O(\delta_0))$, and $2^{S_0 / k_0} - 3 = 3(e^{\delta_0 / k_0} - 1) = \frac{3\delta_0}{k_0}(1 + O(\delta_0 / k_0))$, so $x_{\min}^{\mathrm{Eliahou}}(k, S) = \frac{k_0}{3\delta_0} (1 + O(\delta_0 / k_0))$. In the upper rearrangement bound $U(k_0, S_0)$, $2^{(S_0 + 1)/k_0} - 3 = 3(e^{(\delta_0 + \ln 2)/k_0} - 1) = \frac{3\ln 2}{k_0}(1 + O(1/k_0 + \delta_0))$ and $2^{S_0 + 1} - 3^{k_0} = 3^{k_0}(2 e^{\delta_0} - 1) = 3^{k_0}(1 + O(\delta_0))$, so

$$ U(k_0, S_0) = \frac{\frac{1}{2} \cdot 3^{k_0} (1 + O(1/k_0 + \delta_0))}{3^{k_0} \delta_0 (1 + O(\delta_0)) \cdot \frac{3\ln 2}{k_0} (1 + O(1/k_0 + \delta_0))} = \frac{k_0}{6 (\ln 2) \delta_0} \left( 1 + O\left(\frac{1}{k_0} + \delta_0\right) \right), $$

and similarly for $L(k_0, S_0)$, yielding $\lim_{k_0 \to \infty, \delta_0 \to 0} x_{\min}(a^{\mathrm{Sturm}}(k, S)) / x_{\min}^{\mathrm{Eliahou}}(k, S) = \frac{1}{2\ln 2}$. $\quad \square$

Figure 1(b) in the supplement displays the ratio $x_{\min}(a^{\mathrm{Sturm}}(k, S)) / x_{\min}^{\mathrm{Eliahou}}(k, S)$ across all coprime pairs $(k, S = \lceil k \log_2 3 \rceil)$ for $3 \le k \le 120$, together with the closed-form rearrangement bounds $L(k, S)$ and $U(k, S)$.

4.2 Analytical Farey-Mediant Exclusions via the Finite Upper Bracket $U(k_0, S_0)$

Given a computational verification that all starting values $n \le X_0$ converge to $1$, Theorems 1 and 2 rule out any candidate pair $(k, S)$ (with $d = \gcd(k, S) \ge 1$ and reduced pair $(k_0, S_0) = (k/d, S/d)$) whenever the exact finite upper Rearrangement Bracket satisfies

$$ U(k_0, S_0) = \frac{2^{-(k_0 - 1)/k_0}(2^{S_0 + 1} - 3^{k_0})}{(2^{S_0} - 3^{k_0})(2^{(S_0 + 1)/k_0} - 3)} < X_0. $$

Because $U(k_0, S_0)$ and $x_{\min}^{\mathrm{Eliahou}}(k, S) = x_{\min}^{\mathrm{Eliahou}}(k_0, S_0) = (2^{S_0/k_0} - 3)^{-1}$ depend only on the reduced coprime pair $(k_0, S_0)$, while any non-coprime multiple $(k, S) = (d k_0, d S_0)$ with $d \ge 2$ has $k = d k_0 > k_0$ and $k + S = d(k_0 + S_0) > k_0 + S_0$, the global minimum of $k$ (and of $k + S$) over all surviving pairs $(k, S)$ is always attained at a coprime pair $(k_0, S_0)$ ($d = 1$). Moreover, since each factor $2^{-(k_0 - 1 - j)/k_0} \le 1$ in the geometric series defining $U(k_0, S_0)$ is strictly less than $1$ for $0 \le j \le k_0 - 2$, we have the universal strict inequality $U(k_0, S_0) < x_{\min}^{\mathrm{Eliahou}}(k_0, S_0) = (2^{S_0/k_0} - 3)^{-1}$, where $(2^{\sigma} - 3)^{-1}$ is strictly decreasing in the slope $\sigma = S_0/k_0 > \log_2 3$.

Recall that the continued-fraction convergents $S_n / k_n$ of $\alpha = \log_2 3$ alternate around $\alpha$: odd convergents $(k_{2m-1}, S_{2m-1})$ satisfy $S_{2m-1}/k_{2m-1} > \alpha$ ($\delta_{2m-1} = S_{2m-1}\ln 2 - k_{2m-1}\ln 3 > 0$), while even convergents $(k_{2m}, S_{2m})$ satisfy $S_{2m}/k_{2m} < \alpha$ ($\delta_{2m} < 0$), with Farey determinant $S_{2m-1} k_{2m} - k_{2m-1} S_{2m} = +1$ and recurrence $\delta_{2m-1} = a_{2m+1}|\delta_{2m}| + \delta_{2m+1} > a_{2m+1}|\delta_{2m}|$ (so $|\delta_{2m}|/\delta_{2m-1} < 1/a_{2m+1}$). By the Farey mediant theorem, every rational $S_0/k_0$ lying strictly between $S_{2m}/k_{2m}$ and $S_{2m-1}/k_{2m-1}$ has the canonical representation $(k_0, S_0) = a(k_{2m-1}, S_{2m-1}) + b(k_{2m}, S_{2m})$ with coprime integers $a, b \ge 1$ (where $\delta(a, b) = a\delta_{2m-1} - b|\delta_{2m}| > 0$ requires $b/a < \delta_{2m-1}/|\delta_{2m}|$), and therefore satisfies $k_0 = a k_{2m-1} + b k_{2m} \ge k_{2m-1} + k_{2m}$.

Combining this Farey-mediant structure with $100$-digit decimal evaluation in verify_paper1.py yields three unconditional analytical exclusions across all $(k, S)$ (requiring no arbitrary finite search cutoffs on $a$ or $b$):

1. The $X_0 = 2^{60}$ Milestone (Oliveira e Silva 2010 [4]): Consider the $19$-th (odd) convergent $(k_{19}, S_{19}) = (397,573,379,\; 630,138,897)$ ($k_{19} + S_{19} = 1,027,712,276$), the $20$-th (even) convergent $(k_{20}, S_{20}) = (6,189,245,291,\; 9,809,721,694)$, and the $21$-st (odd) convergent $(k_{21}, S_{21}) = (k_{19} + k_{20}, S_{19} + S_{20}) = (6,586,818,670,\; 10,439,860,591)$ ($k_{21} + S_{21} = 17,026,679,261$, since $a_{21} = 1$).

   • By the Farey mediant property on $(S_{20}/k_{20}, S_{19}/k_{19})$, every rational $S_0/k_0 \in (\log_2 3, S_{19}/k_{19})$ lies inside $(S_{20}/k_{20}, S_{19}/k_{19})$ and therefore has denominator $k_0 \ge k_{19} + k_{20} = k_{21}$.
   • Consequently, any coprime pair $(k_0, S_0) \ne (k_{19}, S_{19})$ with $k_0 < k_{21}$ and $2^{S_0} > 3^{k_0}$ must satisfy $S_0/k_0 > S_{19}/k_{19}$, which implies $S_0/k_0 - S_{19}/k_{19} \ge \frac{1}{k_0 k_{19}} \ge \frac{1}{(k_{21} - 1) k_{19}}$ and hence$$ \log_2 U(k_0, S_0) < \log_2 x_{\min}^{\mathrm{Eliahou}}(k_0, S_0) \le \log_2 \frac{1}{2^{S_{19}/k_{19} + 1/((k_{21}-1)k_{19})} - 3} = 59.1232021 < 60. $$
   • At $(k_{19}, S_{19})$ itself, direct evaluation gives$$ \log_2 x_{\min}^{\mathrm{Eliahou}}(k_{19}, S_{19}) = 60.1190318 > 60, \qquad \log_2 U(k_{19}, S_{19}) = 59.6477982 < 60. $$

   While the classical Eliahou bound fails to eliminate $(k_{19}, S_{19})$ at $X_0 = 2^{60}$, the finite upper bracket $U(k_0, S_0) < 2^{60}$ analytically eliminates $(k_{19}, S_{19})$ and every pair with $k < k_{21}$, jumping the global minimum cycle length at $X_0 = 2^{60}$ to $(k_{21}, S_{21})$ ($k \ge 6,586,818,670$, $k + S \ge 17,026,679,261$), a $16.5676\times$ increase.

2. The $X_0 = 2^{68}$ Milestone (Barina 2021 [5]): Between the $21$-st convergent $(k_{21}, S_{21})$ and the $22$-nd (even) convergent $(k_{22}, S_{22}) = (65,470,613,321,\; 103,768,467,013)$, any coprime pair with $S_0/k_0 \ge S_{21}/k_{21}$ satisfies $\log_2 U(k_0, S_0) < \log_2 x_{\min}^{\mathrm{Eliahou}}(k_{21}, S_{21}) = 67.5555265 < 68$ ($\log_2 U(k_{21}, S_{21}) = 67.0842928$). Every remaining pair with $\log_2 3 < S_0/k_0 < S_{21}/k_{21}$ lies in $(S_{22}/k_{22}, S_{21}/k_{21})$ and has the form $(k_0, S_0) = a(k_{21}, S_{21}) + b(k_{22}, S_{22})$ with $a, b \ge 1, \gcd(a, b) = 1$. Since $a_{23} = 2$, the $23$-rd convergent is $(k_{23}, S_{23}) = (k_{21} + 2k_{22}, S_{21} + 2S_{22}) = (137,528,045,312,\; 217,976,794,617)$, so every pair with $b \ge 2$ automatically satisfies $k_0 \ge k_{21} + 2k_{22} = k_{23}$. Along the $b = 1$ ray

   $$ (k_a, S_a) = a(k_{21}, S_{21}) + (k_{22}, S_{22}) \quad (a \ge 1), $$

   the slope $S_a/k_a$ strictly increases with $a$ (as $\frac{S_{a+1}}{k_{a+1}} - \frac{S_a}{k_a} = \frac{S_{21} k_{22} - k_{21} S_{22}}{k_a k_{a+1}} = \frac{1}{k_a k_{a+1}} > 0$), so both $x_{\min}^{\mathrm{Eliahou}}(k_a, S_a)$ and $U(k_a, S_a)$ strictly decrease with $a \ge 1$. Evaluating to $100$-digit precision shows that $\log_2 x_{\min}^{\mathrm{Eliahou}}(k_a, S_a) \ge 68$ for $a \in \{1, \dots, 29\}$ ($68.0035590$ at $a = 29$ vs. $67.9904638$ at $a = 30$), whereas $\log_2 U(k_a, S_a) \ge 68$ holds only for $a \in \{1, \dots, 12\}$ ($\log_2 U = 68.0106413$ at $a = 12$, $k_{12} = 144,512,437,361$, vs. $\log_2 U = 67.9550938 < 68$ at $a = 13$, $k_{13} = 151,099,256,031$). Thus Theorem 2 unconditionally eliminates $17$ of the $29$ candidate cycle pairs ($a \in \{13, \dots, 29\}$) on the primary $(k_{21}, k_{22})$ ray (Figure 2(b)).

3. The $X_0 = 2^{71.5}$ Threshold and Comparison with Hercher [8, Corollary 29]: By the Farey-mediant analysis in item 2, the only coprime pairs $(k_0, S_0)$ with $2^{S_0} > 3^{k_0}$, $S_0/k_0 < S_{21}/k_{21}$, and $k_0 < k_{23} = 137,528,045,312$ are the first two points $a = 1$ and $a = 2$ on the $b = 1$ ray $(k_a, S_a) = a(k_{21}, S_{21}) + (k_{22}, S_{22})$ (since $a \ge 3, b = 1$ gives $k_a \ge 3 k_{21} + k_{22} = 85,231,069,331$, where $\log_2 x_{\min}^{\mathrm{Eliahou}}(k_a, S_a) \le \log_2 x_{\min}^{\mathrm{Eliahou}}(k_3, S_3) = 69.9019345 < 71.5$ and $\log_2 U(k_3, S_3) = 69.4307008$). At $a = 1$ ($(k_1, S_1) = (72,057,431,991,\; 114,208,327,604)$, $k_1 + S_1 = 186,265,759,595$) and $a = 2$ ($(k_2, S_2) = (78,644,250,661,\; 124,648,188,195)$), we have

   $$ \log_2 x_{\min}^{\mathrm{Eliahou}}(k_1, S_1) = 71.8843175 > 71.5 > \log_2 U(k_1, S_1) = 71.4130838, \qquad \log_2 U(k_2, S_2) = 70.0349468 < 71.5. $$

   Thus $X_0 = 2^{71.5}$ analytically eliminates every pair with $k < k_{23}$ under Theorem 2 alone, pushing the global minimum cycle length to $(k_{23}, S_{23}) = (137,528,045,312,\; 217,976,794,617)$ ($k + S = 355,504,839,929$). As noted in Section 2.2, Hercher [8, Section 5, Corollary 29] already established the same bound $k + S \ge 355,504,839,929$ at the lower verification threshold $X_0 \ge 3 \cdot 2^{69} \approx 2^{70.58496}$ by combining the local-minima product inequality with the Baker-type exclusion of $m \le 91$, whereas our $X_0 = 2^{71.5}$ exclusion uses the pure balanced-word bracket $U(k_0, S_0)$ without any bound on $m$.

Cross-Parent Farey Obstruction and Candidate Ray Elimination

5. Stern–Brocot Matrix Recursions and Total Coprimality of Sturmian Cycles

In this section, we investigate the exact arithmetic divisibility of the Sturmian cycle numerator $C(a^{\mathrm{Sturm}}(k, S))$ by the cycle denominator $D(k, S) = 2^S - 3^k$. We uncover an algebraic phenomenon: the vector

$$ \mathbf{v}(k, S) := \begin{pmatrix} C(a^{\mathrm{Sturm}}(k, S)) \\ 3^k - 2^S \end{pmatrix} \in \mathbb{Z}^2 $$

obeys an exact weighted mediant concatenation law along the branches of the Stern–Brocot tree, generating a $(2, 3)$-smooth determinant identity (invertible in $\mathbb{Z}[1/6]$) that completely excludes Sturmian integer cycles.

5.1 The Stern–Brocot Concatenation Law

Recall that the Stern–Brocot tree organizes all positive reduced fractions. A fraction $S/k$ with $\gcd(k, S) = 1$ has two Farey parents: a left parent $q_1/h_1$ and a right parent $q_2/h_2$, satisfying

$$ k = h_1 + h_2, \quad S = q_1 + q_2, \quad \text{and} \quad h_1 q_2 - h_2 q_1 = 1. $$

Equivalently, $h_1 S - k q_1 = 1$, so $h_1 \equiv S^{-1} \pmod k$ and $q_1 = \lfloor h_1 S / k \rfloor$. By classical combinatorics on words (Borel and Laubie [18], Berstel and de Luca [19]), the lower Christoffel word $a^{\mathrm{Sturm}}(k, S)$ factorizes canonically as the concatenation of its parents' Christoffel words:

$$ a^{\mathrm{Sturm}}(k, S) = a^{\mathrm{Sturm}}(h_1, q_1) \cdot a^{\mathrm{Sturm}}(h_2, q_2). $$

Specifically, for $0 \le j < h_1$, $S_j(a^{\mathrm{Sturm}}(k, S)) = S_j(a^{\mathrm{Sturm}}(h_1, q_1))$, and at $j = h_1$, $S_{h_1} = q_1$. For $j = h_1 + i$ ($0 \le i < h_2$), $S_{h_1 + i}(a^{\mathrm{Sturm}}(k, S)) = q_1 + S_i(a^{\mathrm{Sturm}}(h_2, q_2))$. Splitting the sum defining $C(a^{\mathrm{Sturm}}(k, S))$ at $j = h_1$:

$$ C(a^{\mathrm{Sturm}}(k, S)) = \sum_{j=0}^{h_1-1} 3^{h_1 + h_2 - 1 - j} 2^{S_j} + \sum_{i=0}^{h_2-1} 3^{h_2 - 1 - i} 2^{q_1 + S_i} $$$$ = 3^{h_2} C(a^{\mathrm{Sturm}}(h_1, q_1)) + 2^{q_1} C(a^{\mathrm{Sturm}}(h_2, q_2)). $$

Notice that the second component $3^k - 2^S = 3^{h_1 + h_2} - 2^{q_1 + q_2}$ satisfies the exact same weighted linear combination with weights $(3^{h_2}, 2^{q_1})$, because the cross terms $-3^{h_2} 2^{q_1}$ and $+2^{q_1} 3^{h_2}$ cancel identically:

$$ 3^{h_2} (3^{h_1} - 2^{q_1}) + 2^{q_1} (3^{h_2} - 2^{q_2}) = 3^{h_1 + h_2} - 3^{h_2} 2^{q_1} + 2^{q_1} 3^{h_2} - 2^{q_1 + q_2} = 3^k - 2^S. $$

We state this as a theorem:

Theorem 3 (Stern–Brocot Recursion, $(2, 3)$-Smooth Farey Determinants, and Universal Coprimality). Let $k \ge 2$ and $k < S < 2k$ with $\gcd(k, S) = 1$. Let $(h_1, q_1)$ and $(h_2, q_2)$ be the left and right Farey parents of $(k, S)$ in the Stern–Brocot tree, so $k = h_1 + h_2$, $S = q_1 + q_2$, and $h_1 q_2 - h_2 q_1 = 1$.

1. Weighted Mediant Vector Recursion:

   $$ \mathbf{v}(k, S) = \begin{pmatrix} C(a^{\mathrm{Sturm}}(k, S)) \\ 3^k - 2^S \end{pmatrix} = 3^{h_2} \begin{pmatrix} C(a^{\mathrm{Sturm}}(h_1, q_1)) \\ 3^{h_1} - 2^{q_1} \end{pmatrix} + 2^{q_1} \begin{pmatrix} C(a^{\mathrm{Sturm}}(h_2, q_2)) \\ 3^{h_2} - 2^{q_2} \end{pmatrix}. $$

2. Left Farey $(2, 3)$-Smooth Determinant Identity:

   $$ \det \begin{pmatrix} C(a^{\mathrm{Sturm}}(k, S)) & C(a^{\mathrm{Sturm}}(h_1, q_1)) \\ 3^k - 2^S & 3^{h_1} - 2^{q_1} \end{pmatrix} = 3^{h_1 - 1} 2^{S - 1}. $$

   Equivalently:

   $$ (3^{h_1} - 2^{q_1}) C(a^{\mathrm{Sturm}}(k, S)) + (2^S - 3^k) C(a^{\mathrm{Sturm}}(h_1, q_1)) = 3^{h_1 - 1} 2^{S - 1}. $$

3. Right Farey $(2, 3)$-Smooth Determinant Identity:

   $$ \det \begin{pmatrix} C(a^{\mathrm{Sturm}}(k, S)) & C(a^{\mathrm{Sturm}}(h_2, q_2)) \\ 3^k - 2^S & 3^{h_2} - 2^{q_2} \end{pmatrix} = -3^{k - 1} 2^{q_2 - 1}. $$

   Equivalently:

   $$ (2^{q_2} - 3^{h_2}) C(a^{\mathrm{Sturm}}(k, S)) - (2^S - 3^k) C(a^{\mathrm{Sturm}}(h_2, q_2)) = 3^{k - 1} 2^{q_2 - 1}. $$

4. Universal Coprimality and Non-Integrality for Positive Cycles:

   $$ \gcd\left( C(a^{\mathrm{Sturm}}(k, S)), \; |2^S - 3^k| \right) = 1. $$

   Consequently, for any pair $(k, S) = (d k_0, d S_0)$ with $d = \gcd(k, S) \ge 1$ and $k_0 \ge 2$, the rational cycle $x_r(a^{\mathrm{Sturm}}(k, S)) = x_r(a^{\mathrm{Sturm}}(k_0, S_0))$ has reduced denominator $|2^{S_0} - 3^{k_0}|$. For all positive-cycle slopes ($2^S > 3^k$ with $k_0 \ge 2$, so $k_0 \ge 3, S_0 \ge 5$), we have $2^{S_0} - 3^{k_0} \ge 5 > 1$, so $a^{\mathrm{Sturm}}(k, S)$ is never a positive integer cycle (whereas for $2^S < 3^k$, the unique pair with $|2^{S_0} - 3^{k_0}| = 1$ by Levi ben Gerson's theorem [21] is $(k_0, S_0) = (2, 3)$ with $2^3 - 3^2 = -1$, corresponding to the negative integer cycle $-5 \to -7 \to -5$).

Proof. Part 1 was established above by direct summation on $a^{\mathrm{Sturm}}(k, S) = a^{\mathrm{Sturm}}(h_1, q_1) \cdot a^{\mathrm{Sturm}}(h_2, q_2)$ and the identity $3^{h_2}(3^{h_1} - 2^{q_1}) + 2^{q_1}(3^{h_2} - 2^{q_2}) = 3^k - 2^S$.

For Parts 2 and 3, we first prove by induction on the depth in the Stern–Brocot tree that for every Farey-adjacent parent pair $((h_1, q_1), (h_2, q_2))$ with $h_1 q_2 - h_2 q_1 = 1$,

$$ \det\left( \mathbf{v}(h_1, q_1), \; \mathbf{v}(h_2, q_2) \right) = -3^{h_1 - 1} 2^{q_2 - 1}. $$

• Base case: At the root of the interval $(1, 2)$, the Farey parents are $(h_1, q_1) = (1, 1)$ (word $(1)$, $\mathbf{v}(1, 1) = (1, 3^1 - 2^1)^T = (1, 1)^T$) and $(h_2, q_2) = (1, 2)$ (word $(2)$, $\mathbf{v}(1, 2) = (1, 3^1 - 2^2)^T = (1, -1)^T$), where$$ \det \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} = -2 = -3^{1 - 1} 2^{2 - 1}. $$
• Inductive step: Given $(h_1, q_1)$ and $(h_2, q_2)$ with mediant $(k, S) = (h_1 + h_2, q_1 + q_2)$, Part 1 gives $\mathbf{v}(k, S) = 3^{h_2} \mathbf{v}(h_1, q_1) + 2^{q_1} \mathbf{v}(h_2, q_2)$. There are two child sub-intervals:
  1. Left sub-interval $((h_1', q_1'), (h_2', q_2')) = ((h_1, q_1), (k, S))$: Here $h_1' = h_1$ and $q_2' = q_1 + q_2 = S$. By multilinearity of the determinant,$$ \det\left(\mathbf{v}(h_1, q_1), \; \mathbf{v}(k, S)\right) = \det\left(\mathbf{v}(h_1, q_1), \; 2^{q_1} \mathbf{v}(h_2, q_2)\right) = 2^{q_1}\left(-3^{h_1 - 1} 2^{q_2 - 1}\right) = -3^{h_1' - 1} 2^{q_2' - 1}. $$
  2. Right sub-interval $((h_1', q_1'), (h_2', q_2')) = ((k, S), (h_2, q_2))$: Here $h_1' = h_1 + h_2 = k$ and $q_2' = q_2$. By multilinearity,$$ \det\left(\mathbf{v}(k, S), \; \mathbf{v}(h_2, q_2)\right) = \det\left(3^{h_2} \mathbf{v}(h_1, q_1), \; \mathbf{v}(h_2, q_2)\right) = 3^{h_2}\left(-3^{h_1 - 1} 2^{q_2 - 1}\right) = -3^{h_1' - 1} 2^{q_2' - 1}. $$

This completes the induction. Reversing the column order in the left sub-interval identity gives $\det(\mathbf{v}(k, S), \mathbf{v}(h_1, q_1)) = -\det(\mathbf{v}(h_1, q_1), \mathbf{v}(k, S)) = 3^{h_1 - 1} 2^{S - 1}$ (Part 2), and the right sub-interval identity directly gives $\det(\mathbf{v}(k, S), \mathbf{v}(h_2, q_2)) = -3^{k - 1} 2^{q_2 - 1}$ (Part 3).

For Part 4, let $g = \gcd(C(a^{\mathrm{Sturm}}(k, S)), |2^S - 3^k|)$. From the Left Farey Determinant Identity (Part 2):

$$ (3^{h_1} - 2^{q_1}) C(a^{\mathrm{Sturm}}(k, S)) + (2^S - 3^k) C(a^{\mathrm{Sturm}}(h_1, q_1)) = 3^{h_1 - 1} 2^{S - 1}. $$

Since $g$ divides both $C(a^{\mathrm{Sturm}}(k, S))$ and $2^S - 3^k$, $g$ must divide the integer linear combination on the left-hand side, and hence $g \mid 3^{h_1 - 1} 2^{S - 1}$. Thus every prime divisor of $g$ belongs to $\{2, 3\}$. However, $\gcd(|2^S - 3^k|, 6) = 1$ for all $k, S \ge 1$, so $\gcd(g, 6) = 1$, forcing $g = 1$. By Levi ben Gerson's theorem [21], $|2^{S_0} - 3^{k_0}| = 1$ for $k_0 \ge 2$ occurs solely at $(k_0, S_0) = (2, 3)$ ($2^3 - 3^2 = -1$, where $C(1, 2) = 5$ gives $x_0 = -5$), whereas for $2^{S_0} > 3^{k_0}$ and $k_0 \ge 2$, $2^{S_0} - 3^{k_0} \ge 2^5 - 3^3 = 5 > 1$. $\quad \square$

Remark 1 (Comparison with Knight's Reversal Proof [26] and Role of the Farey Determinant Identities). Knight [26] independently proved Part 4 of Theorem 3 (that balanced "high" cycles are never positive integer cycles) via the reversal identity $3 M_{S, k} - N_{\mathrm{rev}} + (2^S - 3^k) = 2^{S-1}$, which relates $M_{S, k} = C(a^{\mathrm{Sturm}}(k, S))$ to the numerator $N_{\mathrm{rev}}$ of its reversed shift. While that reversal relation suffices for $a^{\mathrm{Sturm}}(k, S)$ itself (since shift invariance implies $(2^S - 3^k) \mid M_{S, k} \iff (2^S - 3^k) \mid N_{\mathrm{rev}}$), it relates $M_{S, k}$ to a different cyclic shift and therefore cannot eliminate $C(a^{\mathrm{Sturm}}(k, S))$ when $a^{\mathrm{Sturm}}(k, S)$ is modified by a single transposition $C(a) = C(a^{\mathrm{Sturm}}(k, S)) - 3^{k - 1 - j_1} 2^{S_{j_1} - 1}$. By contrast, Parts 2 and 3 of Theorem 3 invert $C(a^{\mathrm{Sturm}}(k, S))$ directly modulo $D = 2^S - 3^k$ via the Farey parent factors $d_1 = 3^{h_1} - 2^{q_1}$ and $d_2 = 2^{q_2} - 3^{h_2}$ into the pure $(2, 3)$-monomials $3^{h_1 - 1} 2^{S - 1}$ and $3^{k - 1} 2^{q_2 - 1}$. This direct modular inversion is the essential algebraic input that enables the 4-way Farey tent reduction and single-transposition exclusion in Section 7 (Theorem 5).

Batched $O(\log k_0)$ Continued-Fraction Algorithm. Whereas stepping one mediant at a time in the Stern–Brocot tree takes $\sum a_i$ steps, grouping the $a \ge 1$ consecutive mediants corresponding to a single continued-fraction partial quotient $a$ telescopes via the finite geometric series $\sum_{m=0}^{a-1} 3^{(a-1-m)h_i} 2^{m q_i} = \frac{3^{a h_i} - 2^{a q_i}}{3^{h_i} - 2^{q_i}}$ into a single closed-form step:

• Appending $a \ge 1$ copies of the right parent $\mathbf{v}(h_2, q_2)$ to the left parent $\mathbf{v}(h_1, q_1)$ (producing the Christoffel word at $(h_1 + a h_2, q_1 + a q_2)$) yields$$ \mathbf{v}(h_1 + a h_2,\; q_1 + a q_2) = 3^{a h_2} \mathbf{v}(h_1, q_1) + 2^{q_1} \frac{3^{a h_2} - 2^{a q_2}}{3^{h_2} - 2^{q_2}} \mathbf{v}(h_2, q_2). $$
• Symmetrically, prepending $a \ge 1$ copies of the left parent $\mathbf{v}(h_1, q_1)$ to the right parent $\mathbf{v}(h_2, q_2)$ (producing the Christoffel word at $(a h_1 + h_2, a q_1 + q_2)$) yields$$ \mathbf{v}(a h_1 + h_2,\; a q_1 + q_2) = 3^{h_2} \frac{3^{a h_1} - 2^{a q_1}}{3^{h_1} - 2^{q_1}} \mathbf{v}(h_1, q_1) + 2^{a q_1} \mathbf{v}(h_2, q_2). $$

Because the divisor $3^{h_i} - 2^{q_i}$ is already the second coordinate of $\mathbf{v}(h_i, q_i)$, evaluating $3^{a h_i}$ and $2^{a q_i}$ by binary exponentiation computes Halbeisen and Hungerbühler's numerator $M_{S_0, k_0} = C(a^{\mathrm{Sturm}}(k_0, S_0))$ and $x_{\min}(a^{\mathrm{Sturm}}(k, S)) = C(a^{\mathrm{Sturm}}(k_0, S_0)) / (2^{S_0} - 3^{k_0})$ in $O(N) = O(\log k_0)$ arithmetic operations, where $N$ is the number of continued-fraction partial quotients of $S_0/k_0$.

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6. Fixed-Point Homogenization, Discrepancy Bounds, and $m$-Cycle Interpolation

We now establish structural connections between the Sturmian apex and the clustered $m$-cycle regime by shifting the trajectory to the fixed points of the affine Syracuse step maps.

6.1 Fixed-Point Homogenization Identities

The Syracuse step $x_{j+1} = (3x_j + 1)/2^{a_j}$ can be viewed as an affine map $f_a(x) = \frac{3}{2^a} x + \frac{1}{2^a}$. The unique fixed point of $f_a$ in $\mathbb{Q}$ is

$$ \xi_a = \frac{1}{2^a - 3}. $$

• For $a = 1$ (ascent step), the fixed point is $\xi_1 = \frac{1}{2 - 3} = -1$.
• For $a = 2$ (gentle descent step), the fixed point is $\xi_2 = \frac{1}{4 - 3} = +1$ (the trivial cycle).

Shifting the state variable $x_j$ by these fixed points annihilates all steps with $a_j = 1$ or $a_j = 2$:

Theorem 4 (Fixed-Point Homogenization and $m$-Block Discrepancy Bounds). Let $a \in \mathcal{A}(k, S)$ with $2^S > 3^k$ and $k < S < 2k$.

1. Homogenization at $\xi_1 = -1$:

   $$ (2^S - 3^k) \frac{x_0 + 1}{2} = \sum_{j : a_j \ge 2} 3^{k-1-j} 2^{S_j} \left(2^{a_j - 1} - 1\right). $$

   Every step with $a_j = 1$ vanishes identically. The sum is supported solely on the $m_+ = |\{j : a_j \ge 2\}| \le S - k$ descent steps.

2. Homogenization at $\xi_2 = +1$:

   $$ (2^S - 3^k) \frac{x_0 - 1}{2} = \sum_{j : a_j = 1} 3^{k-1-j} 2^{S_j} - \sum_{j : a_j \ge 3} 3^{k-1-j} 2^{S_j + 1} \left(2^{a_j - 2} - 1\right). $$

   Every step with $a_j = 2$ vanishes identically.

3. $m_+$-Descent Discrepancy Bound: Decompose $a$ up to cyclic shift into $m_+$ blocks of lengths $L_1, \dots, L_{m_+}$ ($L_r \ge 1, \sum L_r = k$), each consisting of $L_r - 1$ ones followed by a single step $b_r \ge 2$. At the discrepancy peak shift $r_+$ (where $S_j \le \lfloor jS/k \rfloor$ for all $j$):

   $$ \mu(a) \le \sum_{r=1}^{m_+} 3^{k - j_r} 2^{\lfloor j_r S / k \rfloor} \left(1 - \left(\frac{2}{3}\right)^{L_r}\right) < m_+ \cdot 2^S \implies x_{\min}(a) < \frac{m_+ \cdot 2^S}{2^S - 3^k}. $$

4. $m$-Cycle Discrepancy Bound: For an $m$-cycle consisting of $m$ alternating blocks of ascents ($a_j = 1$) of lengths $\ell_1, \dots, \ell_m$ and descents ($a_j \ge 2$) of lengths $d_1, \dots, d_m$:

   $$ \mu(a) < 2m \cdot 2^S \implies x_{\min}(a) < \frac{2m \cdot 2^S}{2^S - 3^k} = 2m + \frac{2m \cdot 3^k}{2^S - 3^k} < 2m + \frac{6m}{k} x_{\min}^{\mathrm{Eliahou}}(k, S). $$

Proof. For Part 1, rewrite the Syracuse step in terms of $y_j = x_j + 1$:

$$ 2^{a_j} (x_{j+1} + 1) = 3 x_j + 1 + 2^{a_j} = 3(x_j + 1) + (2^{a_j} - 2). $$

Multiplying by $3^{k-1-j} 2^{S_j}$ and summing over $0 \le j \le k-1$:

$$ \sum_{j=0}^{k-1} 3^{k-1-j} 2^{S_{j+1}} (x_{j+1} + 1) - \sum_{j=0}^{k-1} 3^{k-j} 2^{S_j} (x_j + 1) = \sum_{j=0}^{k-1} 3^{k-1-j} 2^{S_j} (2^{a_j} - 2). $$

The left-hand side telescopes to $(2^S - 3^k)(x_0 + 1)$. When $a_j = 1$, $2^{a_j} - 2 = 0$. Dividing by $2$ proves Part 1. Part 2 follows identically using $w_j = x_j - 1$.

For Part 3, apply Raney's lemma to the discrepancy $U_i = k S_i - i S$. Since $S/k > 1$, on every $a_i = 1$ step the discrepancy strictly decreases: $U_{i+1} - U_i = k - S < 0$. Therefore, the global maximum of $U_i$ can never occur inside a run of $1$s—it must occur at the start $j_r$ of one of the $m_+$ blocks $(1, \dots, 1, b_r)$. Shift $a$ so that this maximum is at index $0$. Then at every block start $j_r$, $S_{j_r} \le \lfloor j_r S / k \rfloor$. On the $r$-th block of length $L_r$, the first $L_r - 1$ steps have $a = 1$, so $S_{j_r + t} = S_{j_r} + t$ for $0 \le t \le L_r - 1$. The sum of $3^{k-1-j} 2^{S_j}$ over the $r$-th block is

$$ \sum_{t=0}^{L_r - 1} 3^{k - 1 - j_r - t} 2^{S_{j_r} + t} = 3^{k - j_r} 2^{S_{j_r}} \left(1 - \left(\frac{2}{3}\right)^{L_r}\right) \le 3^{k - j_r} 2^{\lfloor j_r S / k \rfloor} \left(1 - \left(\frac{2}{3}\right)^{L_r}\right). $$

Because $3^{k - j_r} 2^{\lfloor j_r S / k \rfloor} \le 3^{k - j_r} 2^{j_r S / k} = 3^k (2^S/3^k)^{j_r/k} < 2^S$, summing over $r = 1, \dots, m_+$ yields $\mu(a) < 2^S \sum_{r=1}^{m_+} (1 - (2/3)^{L_r}) < m_+ \cdot 2^S$. For Part 4, bounding the descent phase backwards from $j_{r+1}$ using $S_{j_{r+1} - s} \le S_{j_{r+1}} - 2s$ contributes at most $2^S (1 - (3/4)^{d_r}) < 2^S$ per block, so $\mu(a) < 2m \cdot 2^S$. Dividing by $2^S - 3^k$ gives $x_{\min}(a) < \frac{2m \cdot 2^S}{2^S - 3^k} = 2m + \frac{2m \cdot 3^k}{2^S - 3^k}$. By Bernoulli's inequality $(1 + \delta)^k > 1 + k \delta$ applied to $1 + \delta = 2^{S/k}/3 > 1$ ($k \ge 2$), we have $\frac{2^S - 3^k}{3^k} = (2^{S/k}/3)^k - 1 > \frac{k}{3}(2^{S/k} - 3)$, so $\frac{3^k}{2^S - 3^k} < \frac{3}{k(2^{S/k} - 3)} = \frac{3}{k} x_{\min}^{\mathrm{Eliahou}}(k, S)$, proving $\frac{2m \cdot 2^S}{2^S - 3^k} < 2m + \frac{6m}{k} x_{\min}^{\mathrm{Eliahou}}(k, S)$. $\quad \square$

Theorem 4 establishes an immediate diophantine consequence for $m$-cycles: if an $m$-cycle satisfies $x_{\min}(a) \ge X_0 > 2m$, then Part 4 gives $\frac{2^S - 3^k}{3^k} = e^{S\ln 2 - k\ln 3} - 1 < \frac{2m}{X_0 - 2m}$, so

$$ 0 < S \ln 2 - k \ln 3 < \ln\left(1 + \frac{2m}{X_0 - 2m}\right) < \frac{2m}{X_0 - 2m}. $$

Coupling this upper bound on $S\ln 2 - k\ln 3$ with Rhin's linear-forms-in-logarithms lower bound $S\ln 2 - k\ln 3 > c_{\mathrm{Rhin}} k^{-13.3}$ [20] yields the explicit lower bound $k > (c_{\mathrm{Rhin}}(X_0 - 2m)/(2m))^{1/13.3}$ for fixed $m$ in terms of $X_0$.

7. The 4-Way Farey Tent Envelope and Single-Transposition Perturbation Exclusions

In Theorem 3, we proved that the Sturmian word $a^{\mathrm{Sturm}}(k, S)$ satisfies $\gcd(C(a^{\mathrm{Sturm}}), 2^S - 3^k) = 1$. We now investigate single-transposition ($t=1$) perturbations around $a^{\mathrm{Sturm}}(k, S)$, asking whether swapping any adjacent $(2, 1)$ pair in $a^{\mathrm{Sturm}}(k, S)$ can ever produce an integer cycle.

7.1 Explicit Definition and Derivation of the Four Reduced Farey Integers

A single-transposition ($t=1$) downward perturbation of $a^{\mathrm{Sturm}}(k, S)$ at index $j_1 \in \{1, \dots, k-1\}$ (where $(a_{j_1 - 1}^{\mathrm{Sturm}}, a_{j_1}^{\mathrm{Sturm}}) = (2, 1)$) is obtained by swapping $(a_{j_1 - 1}, a_{j_1})$ to $(1, 2)$. This decreases the single cumulative valuation $S_{j_1} = \lfloor j_1 S / k \rfloor$ by $1$, while keeping $S_j = \lfloor jS/k \rfloor$ unchanged for all $j \ne j_1$. By Lemma 2, the perturbed cycle numerator is

$$ C(a) = C(a^{\mathrm{Sturm}}(k, S)) - 3^{k - 1 - j_1} 2^{S_{j_1} - 1}, \quad \text{where } S_{j_1} = \left\lfloor \frac{j_1 S}{k} \right\rfloor. $$

Let $D = D(k, S) = 2^S - 3^k > 0$, and let $d_1 := 3^{h_1} - 2^{q_1}$ and $d_2 := 2^{q_2} - 3^{h_2}$ denote the Left and Right Farey parent factors from Theorem 3. We define four explicit integers associated with the perturbation index $j_1$:

1. Left-Parent $2^S$-form:$$ R_A(j_1) := \begin{cases} 3^{h_2 - j_1} (3^{h_1} - 2^{q_1}) - 2^{S - S_{j_1}}, & j_1 \le h_2, \\ (3^{h_1} - 2^{q_1}) - 3^{j_1 - h_2} 2^{S - S_{j_1}}, & j_1 > h_2. \end{cases} $$
2. Left-Parent $3^k$-form:$$ R_B(j_1) := 3^{h_1 + j_1} - 2^{S_{j_1}} (3^{h_1} - 2^{q_1}). $$
3. Right-Parent $3^k$-form:$$ Q_A(j_1) := \begin{cases} 3^{j_1} 2^{q_2 - S_{j_1}} - (2^{q_2} - 3^{h_2}), & S_{j_1} \le q_2, \\ 3^{j_1} - 2^{S_{j_1} - q_2} (2^{q_2} - 3^{h_2}), & S_{j_1} > q_2. \end{cases} $$
4. Right-Parent $2^S$-form:$$ Q_B(j_1) := 2^{S + q_2 - S_{j_1}} - 3^{k - j_1} (2^{q_2} - 3^{h_2}). $$

Figure 2(a) in the supplement plots $\log_2 |R_A(j_1)|, \log_2 |R_B(j_1)|, \log_2 |Q_A(j_1)|$ and the 4-way minimum envelope on the convergent $(k_5, S_5) = (41, 65)$. The green diamond envelope is strictly bounded below $41.2$ bits, far below $\log_2(2^{65} - 3^{41}) = 58.54$ bits.

7.2 Proof of the 4-Way Farey Tent Reduction and Single-Transposition Exclusion Theorem

Theorem 5 (4-Way Farey Tent Reduction and Single-Transposition Exclusion). Let $k \ge 6$ and $k < S < 2k$ with $\gcd(k, S) = 1$ and $D = 2^S - 3^k > 0$. For any single-transposition ($t=1$) downward perturbation $a$ of $a^{\mathrm{Sturm}}(k, S)$ at index $j_1 \in \{1, \dots, k-1\}$:

1. Exact GCD Equivalence, Boundary Avoidance, and Non-Vanishing: Every downward transposition index $j_1$ satisfies $j_1 \ne h_2$ and $S_{j_1} \ne q_2$ (with $j_1 < h_2 \iff S_{j_1} < q_2$). Each of $R_A(j_1), R_B(j_1), Q_A(j_1), Q_B(j_1)$ is a non-zero integer coprime to $6$, and

   $$ \gcd(C(a), D) = \gcd(R_A(j_1), D) = \gcd(R_B(j_1), D) = \gcd(Q_A(j_1), D) = \gcd(Q_B(j_1), D). $$

2. Universal 4-Way Tent Bound:

   $$ 1 \le \min\left( |R_A(j_1)|, \; |R_B(j_1)|, \; |Q_A(j_1)|, \; |Q_B(j_1)| \right) < 2^{\min\left(\max(q_1, q_2),\; \frac{S + \min(q_1, q_2)}{2}\right) + 2} \le 2^{\frac{2}{3}S + 2}. $$

3. Single-Transposition Exclusion: Whenever $M_4(j_1) := \min(|R_A(j_1)|, |R_B(j_1)|, |Q_A(j_1)|, |Q_B(j_1)|) < 2^S - 3^k$—which holds unconditionally whenever $2^{\frac{2}{3}S + 2} \le 2^S - 3^k$ (in particular for all $k$ satisfying $\frac{1}{3}S - 13.3\log_2 k \ge 3 - \log_2 c_{\mathrm{Rhin}}$ under Rhin's effective irrationality measure $S\ln 2 - k\ln 3 > c_{\mathrm{Rhin}} k^{-13.3}$ [20]), and is verified exhaustively along with $D \nmid C(a)$ across all $56,304$ downward perturbations for all $2,842$ admissible coprime pairs $(k, S)$ in $6 \le k \le 150$—we have $D \nmid C(a)$, so $a$ cannot generate an integer cycle.

Proof. Part 1 (GCD Equivalence, Boundary Avoidance, and Non-Vanishing): By Theorem 3 Part 2, $d_1 C(a^{\mathrm{Sturm}}) + D \cdot C(a^{\mathrm{Sturm}}(h_1, q_1)) = 3^{h_1 - 1} 2^{S - 1}$, so $\gcd(d_1, D) = 1$, and symmetrically Theorem 3 Part 3 implies $\gcd(d_2, D) = 1$. Multiplying $C(a) = C(a^{\mathrm{Sturm}}) - 3^{k - 1 - j_1} 2^{S_{j_1} - 1}$ by $2 d_1$ and substituting $2 d_1 C(a^{\mathrm{Sturm}}) \equiv 3^{h_1 - 1} 2^S \pmod D$ gives

$$ 2 d_1 C(a) \equiv 3^{h_1 - 1} 2^S - 3^{k - 1 - j_1} 2^{S_{j_1}} d_1 \pmod D. $$

Since $k = h_1 + h_2$, $(k - 1 - j_1) - (h_1 - 1) = h_2 - j_1$. Factoring out the invertible $(2, 3)$-monomial $-3^{\min(h_1 - 1, k - 1 - j_1)} 2^{S_{j_1}}$ modulo $D$ yields $R_A(j_1)$; substituting $2^S \equiv 3^k \pmod D$ and factoring out $3^{k - 1 - j_1}$ yields $R_B(j_1)$. Symmetrically, multiplying $C(a)$ by $2 d_2$ and substituting $2 d_2 C(a^{\mathrm{Sturm}}) \equiv 3^{k - 1} 2^{q_2} \pmod D$ gives

$$ 2 d_2 C(a) \equiv 3^{k - 1} 2^{q_2} - 3^{k - 1 - j_1} 2^{S_{j_1}} d_2 \pmod D. $$

Factoring out $3^{k - 1 - j_1} 2^{\min(q_2, S_{j_1})}$ yields $Q_A(j_1)$, while substituting $3^{k} \equiv 2^S \pmod D$ and factoring out $3^{-1} 2^{S_{j_1}}$ yields $Q_B(j_1)$. Because $2, 3, d_1, d_2$ are units modulo $D$, all four integers share the exact same greatest common divisor with $D$ as $C(a)$.

Next, from $h_1 q_2 - h_2 q_1 = 1$ with $k = h_1 + h_2$ and $S = q_1 + q_2$, we have $h_2 S = k q_2 - 1$. Thus

$$ S_{h_2} = \left\lfloor \frac{k q_2 - 1}{k} \right\rfloor = q_2 - 1, \qquad S_{h_2 + 1} = \left\lfloor q_2 + \frac{S - 1}{k} \right\rfloor = q_2 + 1 \quad (\text{since } k < S < 2k \implies 1 \le \frac{S-1}{k} < 2). $$

Consequently, $a_{h_2}^{\mathrm{Sturm}} = S_{h_2 + 1} - S_{h_2} = 2 \ne 1$. Because a downward transposition at $j_1$ requires $(a_{j_1 - 1}^{\mathrm{Sturm}}, a_{j_1}^{\mathrm{Sturm}}) = (2, 1)$, we have $a_{j_1}^{\mathrm{Sturm}} = 1 \ne a_{h_2}^{\mathrm{Sturm}}$, which forces $j_1 \ne h_2$ and $S_{j_1} \ne q_2$ (with $j_1 < h_2 \iff S_{j_1} \le q_2 - 1 < q_2$ and $j_1 > h_2 \iff S_{j_1} \ge q_2 + 1 > q_2$). Also $a_{j_1 - 1}^{\mathrm{Sturm}} = 2 \implies 2 \le S_{j_1} \le S - 1$, and $h_1, q_1, h_2, q_2 \ge 1$ makes $d_1 = 3^{h_1} - 2^{q_1}$ and $d_2 = 2^{q_2} - 3^{h_2}$ coprime to $6$. Because $|h_2 - j_1| \ge 1$, $|q_2 - S_{j_1}| \ge 1$, $S_{j_1} \ge 2$, and $S - S_{j_1} \ge 1$, in every one of $R_A(j_1), R_B(j_1), Q_A(j_1), Q_B(j_1)$ one term is even and the other is odd, and one term is divisible by $3$ while the other is coprime to $3$. Thus all four integers are coprime to $6$ and strictly non-zero.

Part 2 (Universal 4-Way Tent Bound): Since $3^k < 2^S$, we have $3^m < 2^{m S / k}$ for every $m \ge 1$. Moreover, $h_1 S / k = q_1 + 1/k$ and $h_2 S / k = q_2 - 1/k$ give $3^{h_1} < 2^{q_1 + 1/k} < 2^{q_1 + 1}$ and $3^{h_2} < 2^{q_2}$, so $|d_1| < 2^{q_1 + 1}$ and $|d_2| < 2^{q_2}$. Using $j_1 S / k - 1 < S_{j_1} \le j_1 S / k$, we bound the four integers in the two disjoint regimes $j_1 < h_2$ and $j_1 > h_2$:

1. Case A ($j_1 < h_2$, so $S_{j_1} < q_2$):
   • In $Q_A(j_1) = 3^{j_1} 2^{q_2 - S_{j_1}} - d_2$, we have $3^{j_1} 2^{q_2 - S_{j_1}} < 2^{j_1 S/k - S_{j_1} + q_2} < 2^{q_2 + 1}$ and $|d_2| < 2^{q_2}$, so $|Q_A(j_1)| < 2^{q_2 + 1}$.
   • In $R_B(j_1) = 3^{h_1 + j_1} - 2^{S_{j_1}} d_1$, we have $3^{h_1 + j_1} < 2^{q_1 + 1/k + j_1 S/k} < 2^{q_1 + S_{j_1} + 2}$ and $2^{S_{j_1}} |d_1| < 2^{q_1 + S_{j_1} + 1}$, so $|R_B(j_1)| < 2^{q_1 + S_{j_1} + 2}$.
   • In $R_A(j_1) = 3^{h_2 - j_1} d_1 - 2^{S - S_{j_1}}$, we have $3^{h_2 - j_1} |d_1| < 2^{q_2 - j_1 S/k + q_1 + 1} \le 2^{S - S_{j_1} + 1}$, so $|R_A(j_1)| < 2^{S - S_{j_1} + 2}$. Hence on $j_1 < h_2$, $\min(|R_A(j_1)|, |R_B(j_1)|) < 2^{\min(q_1 + S_{j_1}, S - S_{j_1}) + 2} \le 2^{\frac{S + q_1}{2} + 2}$, and combining with $|Q_A(j_1)| < 2^{q_2 + 1}$ gives
   $$ \min(|R_A(j_1)|, |R_B(j_1)|, |Q_A(j_1)|) < 2^{\min\left(q_2,\; \frac{S + q_1}{2}\right) + 2}. $$
2. Case B ($j_1 > h_2$, so $S_{j_1} > q_2$):
   • In $R_A(j_1) = d_1 - 3^{j_1 - h_2} 2^{S - S_{j_1}}$, we have $|d_1| < 2^{q_1 + 1}$ and $3^{j_1 - h_2} 2^{S - S_{j_1}} < 2^{(j_1 - h_2)S/k + S - S_{j_1}} = 2^{(j_1 S/k - S_{j_1}) + q_1 + 1/k} < 2^{q_1 + 2}$, so $|R_A(j_1)| < 2^{q_1 + 2}$.
   • In $Q_A(j_1) = 3^{j_1} - 2^{S_{j_1} - q_2} d_2$, we have $3^{j_1} < 2^{j_1 S/k} < 2^{S_{j_1} + 1}$ and $2^{S_{j_1} - q_2} |d_2| < 2^{S_{j_1}}$, so $|Q_A(j_1)| < 2^{S_{j_1} + 1}$.
   • In $Q_B(j_1) = 2^{S + q_2 - S_{j_1}} - 3^{k - j_1} d_2$, we have $3^{k - j_1} |d_2| < 2^{S - j_1 S/k + q_2} \le 2^{S + q_2 - S_{j_1}}$, so $|Q_B(j_1)| < 2^{q_2 + S - S_{j_1} + 1}$. Hence on $j_1 > h_2$, $\min(|Q_A(j_1)|, |Q_B(j_1)|) < 2^{\min(S_{j_1}, q_2 + S - S_{j_1}) + 1} \le 2^{\frac{S + q_2}{2} + 1}$, and combining with $|R_A(j_1)| < 2^{q_1 + 2}$ gives
   $$ \min(|R_A(j_1)|, |Q_A(j_1)|, |Q_B(j_1)|) < 2^{\min\left(q_1,\; \frac{S + q_2}{2}\right) + 2}. $$

Finally, since $q_1 + q_2 = S$, if $q_1 \le q_2$ then $q_1 = \min(q_1, q_2) \le S/2 \le q_2 = \max(q_1, q_2)$, so $\min(q_2, \frac{S+q_1}{2}) = \min(\max(q_1, q_2), \frac{S+\min(q_1, q_2)}{2})$ and $\min(q_1, \frac{S+q_2}{2}) = q_1 \le \min(q_2, \frac{S+q_1}{2})$; symmetrically if $q_2 < q_1$. Thus in both Case A and Case B,

$$ \min(|R_A|, |R_B|, |Q_A|, |Q_B|) < 2^{\min\left(\max(q_1, q_2),\; \frac{S + \min(q_1, q_2)}{2}\right) + 2} \le 2^{\frac{2}{3}S + 2}, $$

since for $u = \min(q_1, q_2) \in [1, S/2]$, $\min(S - u, \frac{S + u}{2}) \le \frac{2}{3}S$ (maximized at $u = S/3$).

Part 3 (Single-Transposition Exclusion): By Part 1, if $D \mid C(a)$, then $D$ must divide the non-zero integer $M_4(j_1) := \min(|R_A(j_1)|, |R_B(j_1)|, |Q_A(j_1)|, |Q_B(j_1)|) \ge 1$, requiring $D \le M_4(j_1) < 2^{\frac{2}{3}S + 2}$, which is impossible whenever $M_4(j_1) < D$ and in particular whenever $2^{\frac{2}{3}S + 2} \le D = 2^S - 3^k$. Under Rhin's effective irrationality measure $\delta = S\ln 2 - k\ln 3 > c_{\mathrm{Rhin}} k^{-13.3}$ [20], if $\delta \ge 1$ then $2^S - 3^k > 2^{S-1} \ge 2^{\frac{2}{3}S + 2}$ for $S \ge 9$, while if $\delta < 1$ then $2^S - 3^k = 2^S(1 - e^{-\delta}) > 2^{S-1} \delta > 2^{S-1} c_{\mathrm{Rhin}} k^{-13.3} \ge 2^{\frac{2}{3}S + 2}$ whenever $\frac{1}{3}S - 13.3\log_2 k \ge 3 - \log_2 c_{\mathrm{Rhin}}$. $\quad \square$

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8. Limitations and Scope

We delineate the precise boundaries of our results:

1. Beyond Single-Transposition Perturbations: Our theorems characterize the clustered $m$-cycle regime ($m \ll k$) via Theorem 4, and the balanced Sturmian apex ($m = 2k - S$) and its single-transposition neighborhood via Theorems 1–3 and 5. For multi-transposition perturbations ($t \ge 2$) supported on windows $W = j_{\max} - j_{\min} + 1 > 1$ with valuation span $W_S = S_{j_{\max}} - S_{j_{\min}}$, the interior perturbation polynomial $\Delta_W = \sum_{i=1}^t 3^{j_{\max}-j_i} 2^{S_{j_i}-S_{j_{\min}}}$ satisfies $\gcd(\Delta_W, 6) = 1$, but its archimedean magnitude $\Delta_W \le 3^{W-1} 2^{W_S}$ shifts the tent bound upward by $\log_2 \Delta_W$, so extending the exclusion to unbounded windows $W \sim \Omega(k)$ remains open.
2. Dependence on Verification Thresholds and Relation to Combined $m$-Cycle Bounds: While Theorems 1–5 are unconditional, the numerical exclusions of specific candidate cycle lengths in Section 4.2 depend on the computational verification threshold $X_0$ ($2^{60}$ [4] or $2^{68}$ [5]) via the finite upper Rearrangement Bracket $U(k_0, S_0) < X_0$. These exclusions isolate what balanced-word geometry alone achieves without any lower bound on the number of local minima $m$; when combined with Baker-type exclusions of small $m$ ($m > 91$), Hercher's local-minima product inequality [8, Corollary 29] already establishes $k + S \ge 355,504,839,929$ at $X_0 \ge 3 \cdot 2^{69}$.
3. Open-Ended Problem Context: This paper does not claim a complete solution of the Collatz conjecture. It establishes exact closed-form Rearrangement Brackets, Stern–Brocot determinantal modular inversions, and single-transposition exclusions building upon the optimal cycle bounds of Halbeisen and Hungerbühler [22], Fernández and Ibáñez [25], and Knight [26].

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9. Reproducibility

All mathematical assertions, theorems, and numerical exclusions in this paper are verified by the self-contained Python 3 script supplement/verify_paper1.py using standard library Python:

• Theorem 1 & Theorem 4: Exhaustively verifies the extremal minimum, maximum, average, strict uniqueness, and $m$-cycle discrepancy bounds across $230,519$ valuation vectors in $12$ coprime and non-coprime simplices $\mathcal{A}(k, S)$.
• Theorem 2 & Section 4.2: Verifies the Rearrangement Brackets across $1,469$ coprime pairs and non-coprime multiples, and executes the $100$-digit decimal analytical Farey-mediant verification proving the $16.5676\times$ minimum-length jump at $X_0 = 2^{60}$, the $17$-of-$29$ candidate elimination on the $(k_{21}, k_{22})$ ray at $X_0 = 2^{68}$, and the $k \ge k_{23}$ threshold at $X_0 = 2^{71.5}$.
• Theorem 3 & Theorem 5: Verifies the Stern–Brocot recursion, the batched $O(\log k_0)$ continued-fraction algorithm, the Left and Right Farey determinant identities, coprimality, boundary avoidance ($j_1 \ne h_2, S_{j_1} \ne q_2$), $\gcd$ equivalence, coprimality to $6$, Case A/B tent bounds, and $100\%$ single-transposition exclusion across all $56,304$ downward perturbations for all $2,842$ admissible coprime pairs $(k, S)$ in $6 \le k \le 150$.

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