Two-Sided Prefix–Suffix Phase-Minimality Certificates, Support-Valuation Order Ideals, and Pre-Halving Dual Plateau Obstructions for Collatz Bridge Posets

Abstract

In collatz_26/v1 [1], u/murmuration_collatz introduced the coordinatewise poset $\mathcal{L}_{k,z}$ of legal bridges $L = (L_0, \dots, L_k)$ below the mechanical ceiling $B_i = \lfloor i z / k \rfloor$ ($z = S - k$, $0 < z < k$, $D = 2^S - 3^k > 1$) and proved a $2$-adic prefix certificate (Proposition 1): whenever $2^{\tau(c) + c} > M := C(0)/D$ ($\tau(c) = \lceil (c+1)k/z \rceil$), the residue map $L \mapsto C(L) \bmod D$ is injective on the principal downset $\mathcal{D}_c$ below the horizontally shaved cap $L^c_i = (B_i - c)_+$, certifying every $L \in \mathcal{D}_c$ as globally phase-minimal in $\mathcal{L}_{k,z}$.

In this paper, we develop a two-sided $(2, 3)$-adic and Archimedean theory of the bridge poset $\mathcal{L}_{k,z}$ through four main theorems:

1. Two-Sided Prefix–Suffix Downset Certificates, Exact Window-Span $\Delta(c_1, \tau_2)$, and Unshaved Ceiling Caps (Theorem 1 & Corollary 1.1): For any prefix shaving $0 \le c_1 \le z - 1$ ($\tau_1 := \min\{j \ge 1 : B_j \ge c_1 + 1\}$) and suffix zero-pinning $1 \le \tau_2 \le k - \tau_1$, the two-sided cap $L^{(c_1, \tau_2)}_j := (B_j - c_1)_+ \mathbf{1}_{1 \le j \le k - \tau_2}$ is a legal bridge whose principal downset $\mathcal{D}_{c_1, \tau_2}$ consists of all legal bridges vanishing on $[0, \tau_1 - 1]$ and $[k - \tau_2 + 1, k]$. For all $L, L' \in \mathcal{D}_{c_1, \tau_2}$, $2^{\tau_1 + c_1} 3^{\tau_2 - 1} \mid (C(L') - C(L))$ and $|C(L') - C(L)| \le \Delta(c_1, \tau_2) := C(0) - C(L^{(c_1, \tau_2)}) < C(0)$. Whenever $2^{\tau_1 + c_1} 3^{\tau_2 - 1} D > \Delta(c_1, \tau_2)$, $C(\cdot) \bmod D$ is injective on $\mathcal{D}_{c_1, \tau_2}$ and every $L \in \mathcal{D}_{c_1, \tau_2}$ is globally phase-minimal. Taking $c_1 = 0$ and $\tau_2 = O(\log k)$ yields unshaved ceiling caps $L^{(0, \tau_2)}$ that touch the mechanical ceiling $L_j = B_j$ across $1 \le j \le k - \tau_2$, eliminating horizontal shaving across the active prefix while incurring an exact suffix area deficit $E_{\mathrm{area}}(0, \tau_2) = \sum_{r=1}^{\tau_2 - 1} B_{k-r} = (\tau_2 - 1)z - \sum_{r=1}^{\tau_2 - 1} \lceil r z / k \rceil = O(k \log k)$, which vanishes identically in the family $S = 2k - 1$ ($M < 4, \tau_2 = 1$) and is $\Theta(k \log k)$ along upper continued-fraction convergents ($155{,}044$ at $(15601, 24727)$ where $\tau_2 = 18$).
2. Universal Support-Valuation Order Ideal $\mathcal{M}_{k,z}$ (Theorem 2): For any non-zero bridge $U \in \mathcal{L}_{k,z} \setminus \{0\}$ with support $\mathcal{S}(U) = \{j : U_j > 0\}$, define $v_2(U) := \min_{j \in \mathcal{S}(U)}(j + B_j - U_j)$ and $v_3(U) := k - 1 - \max\mathcal{S}(U)$. We prove that $v_2(\cdot)$ and $v_3(\cdot)$ are monotonically non-increasing in $U$ while $C(0) - C(U)$ is strictly increasing, so $\mathcal{M}_{k,z} := \{0\} \cup \{U \in \mathcal{L}_{k,z} \setminus \{0\} : 2^{v_2(U)} 3^{v_3(U)} D > C(0) - C(U)\}$ is a downward-closed order ideal in $\mathcal{L}_{k,z}$ on which every principal downset $\mathcal{D}_U$ has injective $C(\cdot) \bmod D$, certifying $84.8\%$–$100\%$ of all phase-minimal bridges in benchmark rows $(7,12)$ through $(14,23)$.
3. Ceiling-Deficit Ballot Coordinates, Pairwise Ultrametric Valuation Laws, Pre-Halving Dual Plateau Identity, and Compound-Deficit Obstructions (Theorem 3): In ceiling-deficit coordinates $c_j(L) := B_j - L_j = A_j - j = \sum_{t=0}^{j-1}(a_t(L) - 1)$, bridge legality is equivalent to $0 = c_0(L) \le c_1(L) \le \cdots \le c_k(L) = z$ and $c_j(L) \le B_j$. For any two distinct bridges $L \ne L' \in \mathcal{L}_{k,z}$ with divergence interval $[j_{\min}, j_{\max}]$, we prove the exact $2$-adic valuation formula $v_2(C(L') - C(L)) = j_{\min} + B_{j_{\min}} - \max(L_{j_{\min}}, L'_{j_{\min}})$ (whence $v_2(U) = v_2(C(0) - C(U)) = j_{\min}(U) + B_{j_{\min}(U)} - 1$) and the $3$-adic dichotomy $v_3(C(L') - C(L)) = k - 1 - j_{\max}$ (for $L_{j_{\max}} \not\equiv L'_{j_{\max}} \pmod 2$) vs. $\ge k - j_{\max}$ (for $L_{j_{\max}} \equiv L'_{j_{\max}} \pmod 2$). Across any constant-deficit plateau $c_i(L) = \cdots = c_{m-1}(L) = d$, the reduced cycle states $x_j(L) = P_j(L)/Q(L)$ ($Q(L) = D / \gcd(D, C(L))$) satisfy the pre-halving dual plateau identity $P_i(L) + Q(L) = 2^{m-i} A_{i,m}(L)$ and $2^{c_m(L)-d} P_m(L) + Q(L) = 3^{m-i} A_{i,m}(L)$ ($A_{i,m}(L) \in \mathbb{Z}_{\ge 1}$), which starts the cap plateau at the pre-liftoff floor vertex $i_0 = \tau_1 - 1$ ($L_{i_0} = 0$) to improve the coarse cap denominator bound of [1, §3] by the factor $4 - \frac{2}{2M+1} \in (2, 4)$ and yields a compound-deficit height/denominator bound $L_m \le [(S/k)^{s+1} - 1]\log_2(Q(L)(2M+1)) + 1$ ($s \le c_m(L)$). For zero-residue bridges ($Q(L) = 1$, $D \mid C(L)$), this forces $c_m(L) \ge s \ge \frac{\log(1 + (L_m-1)/\log_2(2M+1))}{\log(S/k)} - 1$.
4. Ballot-Path Transfer DP and $(2m-1)$-Term Adjacent Inclusion–Exclusion for Pareto Unions (Theorem 4): Using the ballot-path characterization and the cap meet-semilattice identity $\min(L^{(c_1, \tau_2)}, L^{(c_1', \tau_2')}) = L^{(\max(c_1, c_1'), \max(\tau_2, \tau_2'))}$, we give an $O(k z)$ dynamic programming recurrence for $|\mathcal{D}_U|$ and prove that the union of any $m$ Pareto-ordered two-sided downsets collapses from $2^m - 1$ inclusion–exclusion terms to $2m - 1$ adjacent terms, enabling exact certification counts up to the continued-fraction convergent $(k, S) = (347, 550)$ ($|\mathcal{L}_{347, 203}| \approx 1.23 \times 10^{153}$), where the Pareto union certifies $1.00\times$–$4.50\times$ as many phase-minimal bridges as Proposition 1 of [1] (with $2.00\times$–$4.50\times$ gains on all rows where Proposition 1 certifies $< 100\%$).

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1. Introduction, Poset Setup, and Scope

1.1 The Legal-Bridge Poset $\mathcal{L}_{k,z}$ and Phase-Minimality (collatz_26/v1 [1])

Fix positive integers $k < S < 2k$, and set $$z := S - k \in \{1, \dots, k - 1\}, \qquad D := 2^S - 3^k > 1. \tag{1.1}$$ Following u/murmuration_collatz [1, §1], define the mechanical ceiling staircase $B = (B_0, B_1, \dots, B_k)$ and its step increments $b = (b_0, \dots, b_{k-1})$ by $$B_j := \left\lfloor \frac{j z}{k} \right\rfloor = \left\lfloor \frac{j S}{k} \right\rfloor - j \quad (0 \le j \le k), \qquad b_j := B_{j+1} - B_j \in \{0, 1\} \quad (0 \le j < k). \tag{1.2}$$ A legal bridge is an integer vector $L = (L_0, L_1, \dots, L_k) \in \mathbb{Z}^{k+1}$ satisfying the three conditions $$L_0 = L_k = 0, \qquad L_j \ge 0 \quad (0 \le j \le k), \qquad L_{j+1} - L_j \le b_j \quad (0 \le j < k). \tag{1.3}$$ Note that iterating $L_{j+1} \le L_j + b_j$ from $L_0 = 0$ automatically implies $L_j \le B_j$ for all $0 \le j \le k$. We denote by $\mathcal{L}_{k,z}$ the finite poset of all legal bridges equipped with the coordinatewise partial order: $K \le L$ if $K_j \le L_j$ for all $0 \le j \le k$, and $K < L$ if $K \le L$ and $K \ne L$.

Each legal bridge $L \in \mathcal{L}_{k,z}$ determines a positive division-exponent word $a(L) = (a_0, \dots, a_{k-1}) \in \mathbb{Z}_{\ge 1}^k$ with $\sum_{i=0}^{k-1} a_i = S$, cumulative exponent sums $A_j(L)$ (also denoted $s_j(L)$), and rooted cycle numerator $C(L)$ via $$a_i(L) := 1 + b_i + L_i - L_{i+1} \ge 1, \qquad s_j(L) := \sum_{i=0}^{j-1} a_i(L) = j + B_j - L_j, \qquad C(L) := \sum_{j=0}^{k-1} 3^{k-1-j} 2^{j + B_j - L_j}. \tag{1.4}$$ At the zero bridge $L = 0 = (0, \dots, 0)$, $C(0) = \sum_{j=0}^{k-1} 3^{k-1-j} 2^{j + B_j}$ is the mechanical numerator, and we put $$M := \frac{C(0)}{D}. \tag{1.5}$$ Because $s_j(L) = j + B_j - L_j$ is strictly decreasing in $L_j$, the numerator map $L \mapsto C(L)$ is strictly order-reversing ($K < L \implies 0 < C(L) < C(K) \le C(0)$). Under the cyclic shortcut equations $2^{a_i(L)} x_{i+1}(L) = 3 x_i(L) + 1$ ($i \in \mathbb{Z}/k\mathbb{Z}$), the unique rational fixed-point trajectory has root $x_0(L) = C(L)/D$ and $m$-th rotated state $x_m(L) = C^{(m)}(L)/D$, where $C^{(m)}(L) := C(T^m a(L))$. Since $\gcd(D, 6) = 1$ and $2^{a_m(L)} C^{(m+1)}(L) = 3 C^{(m)}(L) + D \equiv 3 C^{(m)}(L) \pmod D$, all $k$ rational states $x_m(L)$ share the same reduced denominator [1, §3]: $$x_m(L) = \frac{P_m(L)}{Q(L)}, \qquad \gcd\bigl(P_m(L), Q(L)\bigr) = 1, \qquad Q(L) := \frac{D}{\gcd\bigl(D, C(L)\bigr)}. \tag{1.6}$$ In particular, $L$ corresponds to an integer cycle ($x_m(L) \in \mathbb{Z}_{\ge 1}$) if and only if $C(L) \equiv 0 \pmod D$, i.e., $Q(L) = 1$ (residue zero).

A legal bridge $L \in \mathcal{L}_{k,z}$ is called globally phase-minimal [1, §1] if no strict legal descendant $K < L$ in $\mathcal{L}_{k,z}$ satisfies $C(K) \equiv C(L) \pmod D$. Because $\mathcal{L}_{k,z}$ is finite, the zero-residue fiber $\{L \in \mathcal{L}_{k,z} : C(L) \equiv 0 \pmod D\}$ is non-empty if and only if it contains a phase-minimal member. For any bridge $L \in \mathcal{L}_{k,z}$, its three standard geometric complexity measures [1, §1] are its support $\sigma(L) := \#\{1 \le j \le k-1 : L_j > 0\}$, its height $\operatorname{ht}(L) := \max_{0 \le j \le k} L_j$, and its area $\operatorname{area}(L) := \sum_{j=1}^{k-1} L_j$.

1.2 Baseline Audit of collatz_26/v1 [1] and Summary of New Results

In [1, §2, Proposition 1], u/murmuration_collatz defined, for $0 \le c \le z - 2$ and $\tau(c) := \lceil (c+1)k/z \rceil$ (when $\gcd(k, z) = 1$), the prefix-shaved cap $L^c_j := (B_j - c)_+$ ($0 \le j < k$), $L^c_k := 0$, and its principal downset $\mathcal{D}_c := \{L \in \mathcal{L}_{k,z} : 0 \le L \le L^c\} = \{L \in \mathcal{L}_{k,z} : L_1 = \cdots = L_{\tau(c)-1} = 0\}$. Because every $L, K \in \mathcal{D}_c$ satisfies $2^{\tau(c) + c} \mid (C(K) - C(L))$ and $|C(K) - C(L)| < C(0)$, Proposition 1 of [1] shows that $2^{\tau(c) + c} > M \implies C(\cdot) \bmod D$ is injective on $\mathcal{D}_c$, whence every $L \in \mathcal{D}_c$ is phase-minimal. Taking $c = O(\log k)$ via Bugeaud's linear form in logarithms [2] yields tall phase-minimal caps $L^c$ with $\sigma(L^c) = k - O(\log k)$, $\operatorname{ht}(L^c) = z - O(\log k)$, $\operatorname{area}(L^c) = \frac{(k-1)(z-1)}{2} - O(k \log k)$, and $Q(L^c) > \frac{2^{k - \tau(c)}}{4M + 1}$ [1, Theorem 2 & §3].

Notice two structural limitations of the one-sided prefix certificate $2^{\tau(c)+c} > M$:

1. Global Ceiling Shaving Cost: Every increase of $c$ by $+1$ shaves the ceiling $B_j \mapsto (B_j - c)_+$ horizontally across all $k - \tau(c)$ interior coordinates at once, costing $c(k - \tau(c)) = \Theta(k \log k)$ in area and exponentially many legal bridges in $|\mathcal{D}_c|$.
2. Absence of Low-Denominator / Zero-Residue ($Q(L) = 1$) Obstructions for Multi-Plateau Bridges: While [1, §3] bounds $Q(L^c)$ for the single-plateau cap $L^c$ itself, [1, §§1, 5] explicitly leaves open how small reduced denominators $Q(L)$ (and specifically $Q(L) = 1$) constrain arbitrary multi-plateau bridges $L \in \mathcal{L}_{k,z}$.

In this paper (developed in public discussion on r/collatz/post/38 with valuable boundary refinements contributed by u/lyapunov_weaver), we address both problems:

• Section 2 (Theorem 1 & Corollary 1.1): We introduce two-sided prefix–suffix caps $L^{(c_1, \tau_2)}$ and their downsets $\mathcal{D}_{c_1, \tau_2}$, combining $2$-adic prefix divisibility $2^{\tau_1 + c_1}$ with $3$-adic suffix divisibility $3^{\tau_2 - 1}$ and the exact window-span $\Delta(c_1, \tau_2) := C(0) - C(L^{(c_1, \tau_2)})$. Setting $c_1 = 0$ completely eliminates horizontal ceiling shaving across the active prefix ($L_j = B_j$ for $1 \le j \le k - \tau_2$), while the exact suffix area deficit is $E_{\mathrm{area}}(0, \tau_2) = (\tau_2 - 1)z - \sum_{r=1}^{\tau_2 - 1} \lceil r z / k \rceil = O(k \log k)$, vanishing identically in the family $S = 2k - 1$ ($M < 4, \tau_2 = 1$) and scaling as $\Theta(k \log k)$ along upper continued-fraction convergents ($155{,}044$ at $(15601, 24727)$).
• Section 3 (Theorem 2): We pass from rectangular $(c_1, \tau_2)$ boxes to arbitrary principal downsets $\mathcal{D}_U$ ($U \in \mathcal{L}_{k,z}$), proving that the support-valuation condition $2^{v_2(U)} 3^{v_3(U)} D > C(0) - C(U)$ defines a downward-closed order ideal $\mathcal{M}_{k,z} \subseteq \mathcal{L}_{k,z}$ of globally phase-minimal bridges that captures $84.8\%$–$100\%$ of all phase-minimal bridges in $(7,12)$ through $(14,23)$.
• Section 4 (Theorem 3): Using ceiling-deficit ballot-path coordinates $c_j(L) := B_j - L_j$, we establish: (A) exact pairwise $(2, 3)$-adic ultrametric divergence valuation laws for all pairs $L \ne L' \in \mathcal{L}_{k,z}$; (B) the pre-halving dual plateau identity $P_i + Q = 2^{m-i} A_{i,m}$, $2^{c_m - d} P_m + Q = 3^{m-i} A_{i,m}$, which starts the cap plateau at the pre-liftoff floor vertex $i_0 = \tau_1 - 1$ ($L_{i_0} = 0$) to improve the coarse cap denominator bound of [1, §3] by the factor $4 - \frac{2}{2M+1} \in (2, 4)$; and (C) the compound-deficit height/denominator bound $L_m \le [(S/k)^{s+1} - 1]\log_2(Q(L)(2M+1)) + 1$ for arbitrary multi-plateau bridges, forcing $c_m(L) \ge s = \Omega(\log k)$ for any zero-residue bridge ($Q(L) = 1$) of height $L_m \asymp k$.
• Section 5 (Theorem 4): We establish an $O(k z)$ ballot-path dynamic programming recurrence for $|\mathcal{D}_U|$, the cap meet-semilattice law $\min(L^{(c_1, \tau_2)}, L^{(c_1', \tau_2')}) = L^{(\max(c_1, c_1'), \max(\tau_2, \tau_2'))}$, and an exact $(2m-1)$-term Adjacent Inclusion–Exclusion Identity for the union of $m$ Pareto-ordered two-sided downsets, computing exact phase-minimal retention fractions up to the convergent $(k, S) = (347, 550)$ ($|\mathcal{L}_{347, 203}| \approx 1.23 \times 10^{153}$).

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2. Two-Sided Prefix–Suffix Downset Certificates, Exact Window-Span, and Unshaved Ceiling Caps

In the rooted numerator $C(L) = \sum_{j=0}^{k-1} 3^{k-1-j} 2^{j + B_j - L_j}$, early indices $j$ carry small powers of $2$ and large powers of $3$, whereas late indices $j$ near $k - 1$ carry large powers of $2$ and small powers of $3$ ($3^{k-1-j}$). Pinning the last $\tau_2 - 1$ bridge coordinates $L_{k-\tau_2+1} = \cdots = L_{k-1} = 0$ makes the final $\tau_2$ summands $j \in \{k - \tau_2, \dots, k - 1\}$ identical for all bridges in the family, so their difference $C(L') - C(L)$ is supported exclusively on $j \le k - \tau_2$ and is automatically divisible by $3^{\tau_2 - 1}$ without shaving a single unit of height from the interior $1 \le j \le k - \tau_2$.

Theorem 1 (Two-Sided Prefix–Suffix Downset Certificate and Exact Window-Span Refinement)

Let $0 < z = S - k < k$ with $D = 2^S - 3^k > 1$ (no coprimality assumption on $\gcd(k, z)$ is required). For any integer $0 \le c_1 \le z - 1$, define the prefix threshold $$\tau_1 = \tau(c_1) := \min\bigl\{j \in \{1, \dots, k\} : B_j \ge c_1 + 1\bigr\} \qquad \left(\text{which equals } \left\lceil \frac{(c_1 + 1)k}{z} \right\rceil \text{ whenever } \gcd(k, z) = 1\right), \tag{2.1}$$ so that $B_{\tau_1 - 1} = c_1$ and $B_{\tau_1} = c_1 + 1$ (since $b_j \in \{0, 1\}$). For any integer $1 \le \tau_2 \le k - \tau_1$, define the two-sided cap $L^{(c_1, \tau_2)} = (L^{(c_1, \tau_2)}_0, \dots, L^{(c_1, \tau_2)}_k)$ by $$L^{(c_1, \tau_2)}_0 := 0, \qquad L^{(c_1, \tau_2)}_j := \max(0, B_j - c_1) \quad (1 \le j \le k - \tau_2), \qquad L^{(c_1, \tau_2)}_j := 0 \quad (k - \tau_2 < j \le k), \tag{2.2}$$ and let $\mathcal{D}_{c_1, \tau_2} := \{L \in \mathcal{L}_{k,z} : 0 \le L \le L^{(c_1, \tau_2)}\}$ denote its principal downset, with exact window-span $$\Delta(c_1, \tau_2) := C(0) - C\bigl(L^{(c_1, \tau_2)}\bigr) = \sum_{j = \tau_1}^{k - \tau_2} 3^{k-1-j} 2^{j + c_1}\bigl(2^{B_j - c_1} - 1\bigr) < C(0). \tag{2.3}$$ Then:

1. (Legality and Boundary-Vanishing Characterization): The cap $L^{(c_1, \tau_2)}$ is a legal bridge in $\mathcal{L}_{k,z}$, and $\mathcal{D}_{c_1, \tau_2}$ is precisely the set of all legal bridges vanishing outside $[\tau_1, k - \tau_2]$: $$\mathcal{D}_{c_1, \tau_2} = \bigl\{L \in \mathcal{L}_{k,z} : L_j = 0 \text{ for all } 0 \le j < \tau_1 \text{ and all } k - \tau_2 < j \le k\bigr\}. \tag{2.4}$$
2. (Simultaneous $(2, 3)$-Adic Divisibility and Exact Span Bound): For every pair of bridges $L, L' \in \mathcal{D}_{c_1, \tau_2}$, $$2^{\tau_1 + c_1} \, 3^{\tau_2 - 1} \;\Big|\; \bigl(C(L') - C(L)\bigr) \qquad \text{and} \qquad \bigl|C(L') - C(L)\bigr| \le \Delta(c_1, \tau_2) < C(0). \tag{2.5}$$
3. (Two-Sided Phase-Minimality Certificate): Whenever $(c_1, \tau_2)$ satisfies the exact window-span certificate $$2^{\tau_1 + c_1} \, 3^{\tau_2 - 1} \, D > \Delta(c_1, \tau_2) \qquad \left(\text{or the sufficient global condition } 2^{\tau_1 + c_1} \, 3^{\tau_2 - 1} > M = \frac{C(0)}{D}\right), \tag{2.6}$$ the residue map $L \mapsto C(L) \bmod D$ is strictly injective on $\mathcal{D}_{c_1, \tau_2}$, and every $L \in \mathcal{D}_{c_1, \tau_2}$ is globally phase-minimal in the full legal-bridge poset $\mathcal{L}_{k,z}$.

Proof. (1) Clearly $L^{(c_1, \tau_2)}_0 = L^{(c_1, \tau_2)}_k = 0$ and $L^{(c_1, \tau_2)}_j \ge 0$. For $0 \le j < k - \tau_2$, $L^{(c_1, \tau_2)}_j = (B_j - c_1)_+$ satisfies $(B_{j+1} - c_1)_+ - (B_j - c_1)_+ \le B_{j+1} - B_j = b_j$; at $j = k - \tau_2$, $L^{(c_1, \tau_2)}_{j+1} - L^{(c_1, \tau_2)}_j = 0 - (B_{k-\tau_2} - c_1)_+ \le 0 \le b_{k-\tau_2}$; and for $j > k - \tau_2$, $0 - 0 = 0 \le b_j$. Thus $L^{(c_1, \tau_2)} \in \mathcal{L}_{k,z}$. Since $B_j \le c_1$ for $0 \le j \le \tau_1 - 1$, $L^{(c_1, \tau_2)}_j = 0$ for $j < \tau_1$ and $j > k - \tau_2$, so every $L \in \mathcal{D}_{c_1, \tau_2}$ vanishes outside $[\tau_1, k - \tau_2]$. Conversely, if $L \in \mathcal{L}_{k,z}$ has $L_{\tau_1 - 1} = 0$ and $L_j = 0$ for $j > k - \tau_2$, then for every $\tau_1 \le j \le k - \tau_2$, iterating $L_{i+1} - L_i \le b_i$ from $i = \tau_1 - 1$ gives $L_j \le L_{\tau_1 - 1} + (B_j - B_{\tau_1 - 1}) = B_j - c_1 = L^{(c_1, \tau_2)}_j$, proving $L \in \mathcal{D}_{c_1, \tau_2}$ and (2.4).

(2) By (2.4), for any $L, L' \in \mathcal{D}_{c_1, \tau_2}$, $L_j = L'_j = 0$ for all $0 \le j < \tau_1$ and all $k - \tau_2 < j \le k - 1$, so all summands of $C(L') - C(L)$ outside $\tau_1 \le j \le k - \tau_2$ cancel identically: $$C(L') - C(L) = \sum_{j = \tau_1}^{k - \tau_2} 3^{k-1-j} \Bigl(2^{j + B_j - L'_j} - 2^{j + B_j - L_j}\Bigr). \tag{2.7}$$ For every $\tau_1 \le j \le k - \tau_2$:

• The $3$-adic exponent satisfies $k - 1 - j \ge k - 1 - (k - \tau_2) = \tau_2 - 1$, so $3^{\tau_2 - 1}$ divides every term of (2.7);
• Since $L_j, L'_j \le B_j - c_1$, the $2$-adic exponents satisfy $j + B_j - L_j \ge j + c_1 \ge \tau_1 + c_1$ (and similarly for $L'_j$), so $2^{\tau_1 + c_1}$ divides every term of (2.7). Since $\gcd(2^{\tau_1 + c_1}, 3^{\tau_2 - 1}) = 1$, $2^{\tau_1 + c_1} 3^{\tau_2 - 1} \mid (C(L') - C(L))$. Moreover, because $0 \le L_j \le L^{(c_1, \tau_2)}_j$ for all $L \in \mathcal{D}_{c_1, \tau_2}$ and $C(\cdot)$ is order-reversing, $C(L^{(c_1, \tau_2)}) \le C(L) \le C(0)$ for all $L \in \mathcal{D}_{c_1, \tau_2}$, whence $|C(L') - C(L)| \le C(0) - C(L^{(c_1, \tau_2)}) = \Delta(c_1, \tau_2) < C(0)$ (since the $j = 0$ summand $3^{k-1} > 0$ cancels in $C(0) - C(L^{(c_1, \tau_2)})$).

(3) Suppose $L, L' \in \mathcal{D}_{c_1, \tau_2}$ satisfy $C(L') \equiv C(L) \pmod D$. Since $\gcd(D, 6) = 1$ and $2^{\tau_1 + c_1} 3^{\tau_2 - 1} \mid (C(L') - C(L))$, the integer $C(L') - C(L)$ is a multiple of $2^{\tau_1 + c_1} 3^{\tau_2 - 1} D$. Under (2.6), $|C(L') - C(L)| \le \Delta(c_1, \tau_2) < 2^{\tau_1 + c_1} 3^{\tau_2 - 1} D$, forcing $C(L') - C(L) = 0$ and hence $L' = L$ (by [1, Proposition 1] or Theorem 3A(1) below). Since $\mathcal{D}_{c_1, \tau_2}$ is a downset in $\mathcal{L}_{k,z}$, any strict legal descendant $K < L$ of $L \in \mathcal{D}_{c_1, \tau_2}$ also lies in $\mathcal{D}_{c_1, \tau_2}$ and therefore has $C(K) \not\equiv C(L) \pmod D$. $\square$

Corollary 1.1 (Unshaved Ceiling Caps $L^{(0, \tau_2)}$ with $O(k \log k)$ Exact Suffix Area Deficit and Zero Prefix Shaving)

Take $c_1 = 0$ in Theorem 1, so that $\tau_1 = \tau(0) = \min\{j \ge 1 : B_j \ge 1\} \ge 2$ ($\tau(0) = \lceil k/z \rceil$ when $\gcd(k, z) = 1$) and $$\tau_2 := 1 + \max\left(0,\; \left\lfloor \log_3\!\left(\frac{M}{2^{\tau(0)}}\right) \right\rfloor + 1\right) = O(\log k). \tag{2.8}$$ Whenever $\tau(0) + \tau_2 \le k$, the unshaved cap $L^{(0, \tau_2)}$ satisfies (2.6) and equals the full mechanical ceiling $L^{(0, \tau_2)}_j = B_j$ across $1 \le j \le k - \tau_2$, with $L^{(0, \tau_2)}_j = 0$ for $k - \tau_2 < j \le k$. Consequently, every $L \in \mathcal{D}_{0, \tau_2}$ is globally phase-minimal, and when $\gcd(k, z) = 1$: $$\sigma\bigl(L^{(0, \tau_2)}\bigr) = k - \left\lceil \frac{k}{z} \right\rceil - \tau_2 + 1, \qquad \operatorname{ht}\bigl(L^{(0, \tau_2)}\bigr) = B_{k - \tau_2} = z - O(\log k), \tag{2.9}$$ $$\operatorname{area}\bigl(L^{(0, \tau_2)}\bigr) = \frac{(k - 1)(z - 1)}{2} - E_{\mathrm{area}}(0, \tau_2), \qquad E_{\mathrm{area}}(0, \tau_2) := \sum_{r=1}^{\tau_2 - 1} B_{k - r} = (\tau_2 - 1)z - \sum_{r=1}^{\tau_2 - 1} \left\lceil \frac{r z}{k} \right\rceil = O(k \log k), \tag{2.10}$$ where the horizontal ceiling-shaving deficit across the active prefix $[1, k - \tau_2]$ is identically zero ($L^{(0, \tau_2)}_j = B_j$). The exact suffix area loss $E_{\mathrm{area}}(0, \tau_2)$ is $O(k \log k)$ in general, vanishes identically ($E_{\mathrm{area}} = 0$) in the family $S = 2k - 1$ where $M < 4$ and $\tau_2 = 1$, and is $\Theta(k \log k)$ along upper continued-fraction convergents where $\log_3(M/2^{\tau(0)}) = \Omega(\log k)$ ($155{,}044$ at $(15601, 24727)$, where $(\tau_2 - 1)z = 155{,}142$).

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3. The Universal Support-Valuation Order Ideal $\mathcal{M}_{k,z}$

Theorem 1 certifies rectangular two-sided caps $L^{(c_1, \tau_2)}$. However, a legal bridge $U \in \mathcal{L}_{k,z}$ can have small height near its left support endpoint $j_{\min}(U)$ and tall height in the middle without being contained in any single certified box $\mathcal{D}_{c_1, \tau_2}$. Theorem 2 generalizes Theorem 1 to every principal downset $\mathcal{D}_U := \{L \in \mathcal{L}_{k,z} : 0 \le L \le U\}$ and proves that the resulting certificate is automatically downward-closed on $\mathcal{L}_{k,z}$.

Theorem 2 (Universal Support-Valuation Order Ideal $\mathcal{M}_{k,z}$)

For any non-zero legal bridge $U \in \mathcal{L}_{k,z} \setminus \{0\}$, let $\mathcal{S}(U) := \{j \in \{1, \dots, k-1\} : U_j > 0\}$ denote its support, and define its $2$-adic and $3$-adic support-valuation exponents by $$v_2(U) := \min_{j \in \mathcal{S}(U)} \bigl(j + B_j - U_j\bigr), \qquad v_3(U) := k - 1 - \max_{j \in \mathcal{S}(U)} j. \tag{3.1}$$ Define the subset $\mathcal{M}_{k,z} \subseteq \mathcal{L}_{k,z}$ by $$\mathcal{M}_{k,z} := \{0\} \cup \Bigl\{U \in \mathcal{L}_{k,z} \setminus \{0\} : 2^{v_2(U)} \, 3^{v_3(U)} \, D > C(0) - C(U)\Bigr\}. \tag{3.2}$$ Then:

1. (Principal-Downset Divisibility and Span): For any $U \in \mathcal{L}_{k,z} \setminus \{0\}$ and any pair of bridges $L, L' \in \mathcal{D}_U = \{K \in \mathcal{L}_{k,z} : 0 \le K \le U\}$, $$2^{v_2(U)} \, 3^{v_3(U)} \;\Big|\; \bigl(C(L') - C(L)\bigr) \qquad \text{and} \qquad \bigl|C(L') - C(L)\bigr| \le C(0) - C(U). \tag{3.3}$$ Consequently, if $U \in \mathcal{M}_{k,z}$, then $C(\cdot) \bmod D$ is strictly injective on $\mathcal{D}_U$, and every $L \in \mathcal{D}_U$ (in particular $U$ itself) is globally phase-minimal in $\mathcal{L}_{k,z}$.
2. (Monotone Downward Closure of $\mathcal{M}_{k,z}$): If $0 < L \le U$ in $\mathcal{L}_{k,z}$, then $\mathcal{S}(L) \subseteq \mathcal{S}(U)$ and $$v_2(L) \ge v_2(U), \qquad v_3(L) \ge v_3(U), \qquad C(0) - C(L) \le C(0) - C(U). \tag{3.4}$$ Therefore, $\mathcal{M}_{k,z}$ is a downward-closed order ideal in $\mathcal{L}_{k,z}$ ($U \in \mathcal{M}_{k,z}$ and $0 \le L \le U \implies L \in \mathcal{M}_{k,z}$) that contains every certified two-sided downset $\mathcal{D}_{c_1, \tau_2}$ of Theorem 1.

Proof. (1) For any $L, L' \in \mathcal{D}_U$, if $j \notin \mathcal{S}(U)$ then $0 \le L_j, L'_j \le U_j = 0 \implies L_j = L'_j = 0$, so $C(L') - C(L) = \sum_{j \in \mathcal{S}(U)} 3^{k-1-j}(2^{j + B_j - L'_j} - 2^{j + B_j - L_j})$. For each $j \in \mathcal{S}(U)$, $0 \le L_j, L'_j \le U_j \implies j + B_j - L_j \ge j + B_j - U_j \ge v_2(U)$ and $k - 1 - j \ge k - 1 - \max\mathcal{S}(U) = v_3(U)$, proving $2^{v_2(U)} 3^{v_3(U)} \mid (C(L') - C(L))$. Order-reversal of $C(\cdot)$ gives $C(U) \le C(L), C(L') \le C(0)$, proving $|C(L') - C(L)| \le C(0) - C(U)$ and injectivity of $C(\cdot) \bmod D$ on $\mathcal{D}_U$ whenever $U \in \mathcal{M}_{k,z}$.

(2) Let $0 < L \le U$. Then $L_j > 0 \implies U_j \ge L_j > 0$, so $\varnothing \ne \mathcal{S}(L) \subseteq \mathcal{S}(U)$. Hence $\max\mathcal{S}(L) \le \max\mathcal{S}(U) \implies v_3(L) \ge v_3(U)$, and since $j + B_j - L_j \ge j + B_j - U_j$ for all $j \in \mathcal{S}(L) \subseteq \mathcal{S}(U)$, $$v_2(L) = \min_{j \in \mathcal{S}(L)} (j + B_j - L_j) \ge \min_{j \in \mathcal{S}(L)} (j + B_j - U_j) \ge \min_{j \in \mathcal{S}(U)} (j + B_j - U_j) = v_2(U). \tag{3.5}$$ Since $C(L) \ge C(U)$, $U \in \mathcal{M}_{k,z}$ implies $2^{v_2(L)} 3^{v_3(L)} D \ge 2^{v_2(U)} 3^{v_3(U)} D > C(0) - C(U) \ge C(0) - C(L)$, so $L \in \mathcal{M}_{k,z}$. Finally, for $U = L^{(c_1, \tau_2)} \ne 0$, $v_2(U) = \tau_1 + c_1$, $v_3(U) \ge \tau_2 - 1$, and $C(0) - C(U) = \Delta(c_1, \tau_2)$, so (2.6) implies $L^{(c_1, \tau_2)} \in \mathcal{M}_{k,z}$ and hence $\mathcal{D}_{c_1, \tau_2} \subseteq \mathcal{M}_{k,z}$. $\square$

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4. Ceiling-Deficit Ballot Coordinates, Pairwise Ultrametric Valuation Laws, Pre-Halving Dual Plateau Identity, and Compound-Deficit Obstructions

We now introduce ceiling-deficit coordinates $c_j(L) := B_j - L_j$, which simultaneously simplify the combinatorics of $\mathcal{L}_{k,z}$, establish exact pairwise $(2, 3)$-adic valuation formulas across all bridge pairs, sharpen the cap denominator bound of [1, §3], and yield an unconditional compound-deficit height/denominator obstruction for arbitrary multi-plateau bridges and zero-residue ($Q(L) = 1$) bridges.

Theorem 3 (Ceiling-Deficit Ballot Coordinates, Ultrametric Divergence Laws, Pre-Halving Dual Plateau Identity, and Compound-Deficit Bound)

Let $0 < z = S - k < k$ with $D = 2^S - 3^k > 1$. For any legal bridge $L \in \mathcal{L}_{k,z}$, define its ceiling-deficit profile $$c_j(L) := B_j - L_j = s_j(L) - j = \sum_{t=0}^{j-1}\bigl(a_t(L) - 1\bigr) \qquad (0 \le j \le k). \tag{4.1}$$ Then:

1. (Ballot-Staircase Characterization and Pairwise $(2, 3)$-Adic Ultrametric Divergence Valuation Laws — Theorem 3A):

   • An integer vector $L \in \mathbb{Z}^{k+1}$ lies in $\mathcal{L}_{k,z}$ if and only if $c(L) = (c_0(L), \dots, c_k(L))$ is a monotonically non-decreasing integer path below $B$: $$L \in \mathcal{L}_{k,z} \quad \Longleftrightarrow \quad 0 = c_0(L) \le c_1(L) \le \cdots \le c_k(L) = z \quad \text{and} \quad c_j(L) \le B_j \;\; (0 \le j \le k). \tag{4.2}$$
   • Because $s_{j+1}(L) - s_j(L) = a_j(L) \ge 1$, the sequence $s_j(L) = j + c_j(L)$ is strictly increasing in $j$. For any two distinct bridges $L, L' \in \mathcal{L}_{k,z}$ (including incomparable pairs), let $j_{\min}(L, L') := \min\{1 \le j \le k-1 : L_j \ne L'_j\}$ and $j_{\max}(L, L') := \max\{1 \le j \le k-1 : L_j \ne L'_j\}$. Then $C(L') \ne C(L)$, and: $$v_2\bigl(C(L') - C(L)\bigr) = \min\bigl(s_{j_{\min}}(L),\, s_{j_{\min}}(L')\bigr) = j_{\min}(L, L') + B_{j_{\min}(L, L')} - \max\bigl(L_{j_{\min}(L, L')},\, L'_{j_{\min}(L, L')}\bigr), \tag{4.3}$$ $$v_3\bigl(C(L') - C(L)\bigr) \begin{cases} = k - 1 - j_{\max}(L, L'), & \text{if } L_{j_{\max}(L, L')} \not\equiv L'_{j_{\max}(L, L')} \pmod 2, \\[4pt] \ge k - j_{\max}(L, L'), & \text{if } L_{j_{\max}(L, L')} \equiv L'_{j_{\max}(L, L')} \pmod 2. \end{cases} \tag{4.4}$$
   • In particular, for every non-zero bridge $U \in \mathcal{L}_{k,z} \setminus \{0\}$ with $j_{\min}(U) := \min\mathcal{S}(U)$, since $U_{j_{\min}(U)-1} = 0$ and $0 < U_{j_{\min}(U)} \le b_{j_{\min}(U)-1} \le 1$, we have $U_{j_{\min}(U)} = 1$ identically, so the exponent $v_2(U)$ of Theorem 2 is the exact $2$-adic valuation of $C(0) - C(U)$ and depends solely on $j_{\min}(U)$: $$v_2(U) = v_2\bigl(C(0) - C(U)\bigr) = j_{\min}(U) + B_{j_{\min}(U)} - 1 = s_{j_{\min}(U)-1}(0) + 1. \tag{4.5}$$

2. (Pre-Halving Dual Plateau Identity and Cap Denominator Sharpening — Theorem 3B):

   • For each rotation $0 \le i \le k$, define the rotated ceiling constant $C^{(i)}(0) := \sum_{r=0}^{k-1} 3^{k-1-r} 2^{s_{i+r}(0) - s_i(0)} < 2 C(0)$ ($s_{k+u}(0) := S + s_u(0)$). Then every reduced cycle state $x_i(L) = P_i(L)/Q(L)$ satisfies both the rotated-ceiling bound and the forward-prefix bound: $$C^{(i)}(L) \le 3^{k-1} + 2^{L_i}\bigl(C^{(i)}(0) - 3^{k-1}\bigr) < 2^{L_i + 1} C(0), \qquad 2^{i + c_i(L)}\bigl(P_i(L) + Q(L)\bigr) \le 3^i\bigl(P_0(L) + 2^{c_i(L)} Q(L)\bigr). \tag{4.6}$$
   • Suppose $L \in \mathcal{L}_{k,z}$ has constant ceiling deficit $c_i(L) = c_{i+1}(L) = \cdots = c_{m-1}(L) = d$ across the $r := m - i$ indices $j \in \{i, \dots, m-1\}$ ($0 \le i < m \le k$). Then there exists a unique positive integer $A_{i,m}(L) \in \mathbb{Z}_{\ge 1}$ such that simultaneously: $$P_i(L) + Q(L) = 2^{m - i} \, A_{i,m}(L) \qquad \text{and} \qquad 2^{c_m(L) - d} P_m(L) + Q(L) = 3^{m - i} \, A_{i,m}(L), \tag{4.7}$$ whence the reduced denominator $Q(L)$ obeys the dual lower bound $$Q(L) \ge \max\left(\left\lceil \frac{2^{m - i}}{2^{L_i} C^{(i)}(0)/D + 1} \right\rceil,\; \left\lceil \frac{3^{m - i}}{2^{B_m - d} C^{(m)}(0)/D + 1} \right\rceil\right) > \frac{2^{m - i - L_i}}{2M + 1}. \tag{4.8}$$
   • For every two-sided cap $L^{(c_1, \tau_2)}$ ($0 \le c_1 \le z - 1$, $1 \le \tau_2 \le k - \tau_1$), at the pre-liftoff floor vertex $i_0 := \tau_1 - 1$ we have $L^{(c_1, \tau_2)}_{i_0} = 0$ and $c_{i_0}(L^{(c_1, \tau_2)}) = B_{\tau_1 - 1} - 0 = c_1$, so the constant-deficit plateau $c_j = c_1$ begins at $i_0 = \tau_1 - 1$ where $L_{i_0} = 0$ and spans $r = k + 2 - \tau_1 - \tau_2$ vertices $\{\tau_1 - 1, \dots, k - \tau_2\}$, yielding: $$Q\bigl(L^{(c_1, \tau_2)}\bigr) \ge \max\left(\left\lceil \frac{2^{k + 2 - \tau_1 - \tau_2}}{C^{(\tau_1 - 1)}(0)/D + 1} \right\rceil,\; \left\lceil \frac{3^{k + 2 - \tau_1 - \tau_2}}{2^{B_{k-\tau_2+1} - c_1} C^{(k-\tau_2+1)}(0)/D + 1} \right\rceil\right) > \frac{2^{k + 2 - \tau_1 - \tau_2}}{2M + 1}. \tag{4.9}$$ At $\tau_2 = 1$ ($L^{(c, 1)} = L^c$), the coarse bound in (4.9) is $\frac{2^{k-\tau+1}}{2M+1} = \bigl(4 - \frac{2}{2M+1}\bigr)\frac{2^{k-\tau}}{4M+1}$, improving the denominator bound of [1, §3] by the factor $4 - \frac{2}{2M+1} \in (2, 4)$ (and by $3\times$–$6.5\times$ via the exact dual maximum in (4.9)).

3. (Compound-Deficit Height/Denominator Bound and Zero-Residue $Q(L) = 1$ Obstruction — Theorem 3C):

   • Let $L \in \mathcal{L}_{k,z}$ have reduced denominator $Q(L)$ and put $R := \log_2(Q(L)(2M + 1))$, $R_0 := \lfloor R \rfloor$. For any vertex $m \in \{1, \dots, k-1\}$ with $L_m > 0$, let $i_0 := \max\{0 \le i < m : L_i = 0\}$ be the last floor vertex before $m$, and let $s \le \Delta c := c_{m-1}(L) - c_{i_0}(L) \le c_m(L) = B_m - L_m$ be the number of deficit jumps on $(i_0, m)$. Define $\mathcal{H}_0(R_0) := 0$ and $\mathcal{H}_{t+1}(R_0) := \mathcal{H}_t(R_0) + \lfloor ((\mathcal{H}_t(R_0) + R_0)z - 1)/k \rfloor$. Then: $$L_m \le \mathcal{H}_s(R_0) + \left\lceil \frac{\bigl(\mathcal{H}_s(R_0) + R_0\bigr) z}{k} \right\rceil \le \left[\left(\frac{S}{k}\right)^{s + 1} - 1\right] \log_2\Bigl(Q(L)(2M + 1)\Bigr) + 1, \tag{4.10}$$ or equivalently, $$Q(L) > \frac{1}{2M + 1}\, 2^{\frac{L_m - 1}{(S/k)^{s + 1} - 1}} \ge \frac{1}{2M + 1}\, 2^{\frac{L_m - 1}{(S/k)^{c_m(L) + 1} - 1}}. \tag{4.11}$$
   • In particular, if $L \in \mathcal{L}_{k,z}$ is a zero-residue bridge ($C(L) \equiv 0 \pmod D$, i.e., $Q(L) = 1$), then at every vertex $m \in \{1, \dots, k-1\}$ its height $L_m$ and ceiling deficit $c_m(L) = B_m - L_m$ must satisfy: $$c_m(L) = B_m - L_m \ge s \ge \frac{\log\left(1 + \frac{L_m - 1}{\log_2(2M + 1)}\right)}{\log(S/k)} - 1. \tag{4.12}$$

Proof. (1) Since $c_{j+1}(L) - c_j(L) = b_j - (L_{j+1} - L_j) = a_j(L) - 1$, $a_j(L) \ge 1 \iff c_j(L) \le c_{j+1}(L)$, while $L_j \ge 0 \iff c_j(L) \le B_j$ and $L_0 = L_k = 0 \iff c_0(L) = 0, c_k(L) = z$, proving (4.2). For any $L \ne L' \in \mathcal{L}_{k,z}$, in $C(L') - C(L) = \sum_{j = j_{\min}}^{j_{\max}} 3^{k-1-j}(2^{s_j(L')} - 2^{s_j(L)})$, strict monotonicity of $s_j(L)$ and $s_j(L')$ ensures that for all $j > j_{\min}$, $\min(s_j(L), s_j(L')) \ge \min(s_{j_{\min}}(L), s_{j_{\min}}(L')) + 1$, while the $j = j_{\min}$ summand has exact $2$-adic valuation $\min(s_{j_{\min}}(L), s_{j_{\min}}(L'))$, proving (4.3). Similarly, every summand with $j < j_{\max}$ is divisible by $3^{k - j_{\max}}$, while the $j = j_{\max}$ summand has exact $3$-adic valuation $(k - 1 - j_{\max}) + v_3(2^{|L_{j_{\max}} - L'_{j_{\max}}|} - 1)$. When $|L_{j_{\max}} - L'_{j_{\max}}|$ is odd, $v_3(2^{|L_{j_{\max}} - L'_{j_{\max}}|} - 1) = 0$, so the $j = j_{\max}$ summand has strictly smaller $3$-adic valuation $k - 1 - j_{\max}$ than all $j < j_{\max}$ summands; when $|L_{j_{\max}} - L'_{j_{\max}}|$ is even, $3 \mid (2^{|L_{j_{\max}} - L'_{j_{\max}}|} - 1)$, so every summand is divisible by $3^{k - j_{\max}}$ (as noted by u/lyapunov_weaver in ri-eb9QZeEvV6yAY, at $(k, S) = (5, 8)$ with $L = 0$ and $L' = (0,0,1,1,2,0)$, $j_{\max} = 4$ and $|L_4 - L'_4| = 2$ has $C(L') - C(L) = -108$ with $v_3(-108) = 3 > 1 = k - j_{\max}$ due to cancellation with $j < j_{\max}$), proving (4.4). Specializing (4.3) to $L' = 0$ and $L = U \ne 0$ gives (4.5) since $0 < U_{j_{\min}(U)} \le U_{j_{\min}(U)-1} + b_{j_{\min}(U)-1} = b_{j_{\min}(U)-1} \le 1 \implies U_{j_{\min}(U)} = 1$.

(2) In the $i$-th rotated numerator $C^{(i)}(L) = 3^{k-1} + \sum_{r=1}^{k-1} 3^{k-1-r} 2^{s_{i+r}(0) - s_i(0) + L_i - L_{i+r}}$, $L_{i+r} \ge 0$ gives the first inequality in (4.6), and since $B^{(i)}_r := s_{i+r}(0) - s_i(0) - r = \lfloor (i+r)z/k \rfloor - \lfloor i z / k \rfloor \le B_r + \mathbf{1}_{r \ge 1}$, $C^{(i)}(0) \le 2 C(0) - 3^{k-1}$. For the second inequality in (4.6), iterating $2^{a_j} P_{j+1} = 3 P_j + Q$ from $j = 0$ to $i - 1$ gives $2^{i + c_i(L)}(P_i + Q) = 3^i P_0 + Q\sum_{j=0}^{i-1} 3^{i-1-j} 2^{j + c_j(L)} + 2^{i + c_i(L)} Q \le 3^i P_0 + 2^{c_i(L)}(3^i - 2^i)Q + 2^{i + c_i(L)} Q = 3^i(P_0 + 2^{c_i(L)} Q)$ since $c_j(L) \le c_i(L)$ for $0 \le j < i$. Next, on a constant-deficit plateau $c_i(L) = \cdots = c_{m-1}(L) = d$, we have $a_j(L) = 1 + c_{j+1}(L) - c_j(L) = 1$ for $i \le j \le m - 2$ and $a_{m-1}(L) = 1 + c_m(L) - d$. Thus $2(P_{j+1} + Q) = 3(P_j + Q)$ for $i \le j \le m - 2$, while at $j = m - 1$, $2^{1 + c_m(L) - d} P_m = 3 P_{m-1} + Q \implies 2(2^{c_m(L) - d} P_m + Q) = 3(P_{m-1} + Q)$. Multiplying these $r = m - i$ equations gives $2^{m-i}(2^{c_m(L) - d} P_m + Q) = 3^{m-i}(P_i + Q)$, and coprimality $\gcd(2^{m-i}, 3^{m-i}) = 1$ yields (4.7). Combining (4.7) with $P_i/Q = C^{(i)}(L)/D \le 2^{L_i} C^{(i)}(0)/D < 2^{L_i + 1} M$ and $c_m(L) - d + L_m = B_m - d$ gives (4.8)–(4.9).

(3) Partition $[i_0, m)$ into $s + 1$ maximal constant-deficit plateaus $[i_t, i_{t+1})$ ($0 \le t \le s$, $i_{s+1} = m$) of lengths $r_t := i_{t+1} - i_t \ge 1$. By (4.8), $2^{r_t - L_{i_t}} < Q(L)(2M + 1) = 2^R$, and since $r_t - L_{i_t} \in \mathbb{Z}$, $r_t \le L_{i_t} + R_0 \le L_{i_t} + R$. For each $0 \le t < s$, the deficit jumps by $\Delta_t := c_{i_{t+1}}(L) - c_{i_t}(L) \ge 1$, while $B_{i_{t+1}} - B_{i_t} = \lfloor i_{t+1} z / k \rfloor - \lfloor i_t z / k \rfloor \le \lceil r_t z / k \rceil < \frac{z}{k} r_t + 1$. Because $\Delta_t \ge 1$, $$L_{i_{t+1}} - L_{i_t} = \bigl(B_{i_{t+1}} - B_{i_t}\bigr) - \Delta_t \le \left\lceil \frac{r_t z}{k} \right\rceil - 1 = \left\lfloor \frac{r_t z - 1}{k} \right\rfloor \le \frac{z}{k}\bigl(L_{i_t} + R\bigr). \tag{4.13}$$ Since $L_{i_0} = 0$, induction over $t = 0, \dots, s - 1$ gives $L_{i_s} \le \mathcal{H}_s(R_0)$ and $L_{i_s} + R \le (1 + z/k)^s R = (S/k)^s R$. Adding the final plateau climb $L_m - L_{i_s} = B_m - B_{i_s} - (c_m(L) - c_{i_s}(L)) \le B_m - B_{i_s} \le \lceil r_s z / k \rceil < \frac{z}{k}(L_{i_s} + R) + 1$ yields (4.10)–(4.12). $\square$

Remark 4.1 (Why Each Drop $a_j \ge 2$ Resets the $2$-Adic Carry Chain in Theorem 3C)

In the global $W$-step trajectory equation from the floor vertex $i_0$ ($L_{i_0} = 0$) to $m$ ($W := m - i_0$, deficit gain $\Delta c := \sum_{j=0}^{W-1}(a_{i_0+j} - 1) = c_m(L) - c_{i_0}(L)$), telescoping $3^{W-1-j} 2^{s^{(i_0)}_j} = 3^{W-j} 2^{s^{(i_0)}_j} - 3^{W-1-j} 2^{s^{(i_0)}_{j+1}} + 3^{W-1-j} 2^{s^{(i_0)}_j + 1}(2^{a_{i_0+j}-1} - 1)$ gives the exact identity $$2^{W + \Delta c}\bigl(P_m(L) + Q(L)\bigr) = 3^W\bigl(P_{i_0}(L) + Q(L)\bigr) + Q(L)\sum_{j=0}^{W-1} 3^{W-1-j} 2^{s^{(i_0)}_j(L) + 1}\bigl(2^{a_{i_0+j}(L) - 1} - 1\bigr). \tag{4.14}$$ Along the first plateau $0 \le j < r_0$, $a_{i_0+j} = 1$, so the step-excess sum in (4.14) vanishes and $2^{r_0} \mid (P_{i_0}(L) + Q(L))$ is forced directly on $P_{i_0}(L) + Q(L) < Q(L)(2M + 1)$. At the first drop $j = r_0 - 1$ ($a_{i_0+r_0-1} \ge 2$), however, the step-excess sum contributes an odd multiple of $2^{r_0} Q(L)$, which can cancel the $2^{r_0}$-bit of $3^W(P_{i_0}(L) + Q(L))$ and allow the $2$-adic valuation of the right-hand side to advance by an additional $L_{i_1}$ bits without increasing $Q(L)$. Thus the multiplicative factor $1 + z/k = S/k$ per deficit jump in (4.10) reflects the exact $2$-adic carry-reset mechanism of (4.14).

Remark 4.2 (Comparison with Multi-Run Chaining of Luca [8], Simons–de Weger [10], and Ordered-State Complexity in collatz_22/v4 [9])

It is instructive to situate the compound-deficit height obstruction (4.10)–(4.12) within the classical literature on multi-run propagation:

1. Luca's multi-run propagation theorem [8]: Luca [8, Theorem 2.1] showed that the number $t$ of maximal valuation-one runs in any non-trivial Collatz cycle satisfies $t \gg \log n$, where $n$ is the cycle length. His Lemma 3.2 and its proof (pp. 36–39) explicitly propagate bounds across successive ascending runs along a hypothetical integer cycle in $\mathbb{Z}/k\mathbb{Z}$ via Baker's linear forms in logarithms, bounding the growth of subsequent cycle minimums.
2. Multi-run chaining of Simons and de Weger [10]: Similarly, Simons and de Weger [10, §§3.3–3.4] developed a systematic algorithm for chaining upper bounds on consecutive ascending and descending runs in $m$-cycles. In collatz_22/v4 [9], Lyapunov Weaver acknowledged that precedent while establishing a distinct ordered-state complexity bound $k \le C(\log_2 3)^b \log(k+1)$ on the number of breakpoints $b$ in the next-state rank permutation.
3. What Theorem 3C adds: While classical multi-run theorems [8, 10] propagate bounds across runs of a hypothetical integer cycle in $\mathbb{Z}/k\mathbb{Z}$ using logarithmic linear forms, Theorem 3C establishes an exact poset-geometric height restriction directly on the legal bridge poset $\mathcal{L}_{k,z}$. It applies unconditionally to all legal bridges (whether integer cycles with $Q(L) = 1$ or rational cycles with arbitrary reduced denominator $Q(L)$): across any sequence of $s$ deficit jumps between consecutive plateaus, the reachable ballot height $L_m$ cannot exceed $[(S/k)^{s+1} - 1]\log_2(Q(L)(2M+1)) + 1$. Thus, for integer cycles ($Q(L) = 1$), reaching a tall height $L_m \asymp k$ unconditionally forces $c_m(L) \ge s = \Omega(\log k)$ directly from the $2$-adic carry-reset telescoping identity (4.14).

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5. Ballot-Path Transfer DP and $(2m-1)$-Term Adjacent Inclusion–Exclusion for Pareto Unions

Because $|\mathcal{L}_{k,z}|$ grows exponentially in $k$, brute-force enumeration of $\mathcal{L}_{k,z}$ is only feasible for small $k$. Theorem 4 shows that the ballot-path characterization (4.2) and the meet-semilattice structure of the two-sided caps $L^{(c_1, \tau_2)}$ allow us to compute the exact integer cardinalities $|\mathcal{L}_{k,z}|$, $|\mathcal{D}_{c_1, \tau_2}|$ in $O(k z)$ integer additions, and evaluate the union of any $m$ Pareto-ordered two-sided downsets via $2m - 1$ single-downset evaluations in $O(m k z)$ integer additions.

Theorem 4 (Ballot-Path Transfer DP, Cap Meet-Semilattice Law, and $(2m-1)$-Term Adjacent Inclusion–Exclusion for Pareto Unions)

Let $0 < z = S - k < k$ with $D = 2^S - 3^k > 1$. Then:

1. (Exact $O(k z)$ Ballot-Path Recurrence for Principal Downsets $\mathcal{D}_U$): For any legal bridge $U \in \mathcal{L}_{k,z}$, the map $L \mapsto c(L) = B - L$ is a bijection from $\mathcal{D}_U = \{L \in \mathcal{L}_{k,z} : 0 \le L \le U\}$ onto the set of non-decreasing integer paths $0 = c_0 \le c_1 \le \cdots \le c_k = z$ satisfying $B_j - U_j \le c_j \le B_j$ ($0 \le j \le k$). Defining $N_0(d) := \mathbf{1}_{d = 0}$ and, for $1 \le j \le k$ and $0 \le d \le z$, $$N_j(d) := \begin{cases} \sum_{d'=0}^d N_{j-1}(d'), & \text{if } B_j - U_j \le d \le B_j, \\[4pt] 0, & \text{otherwise}, \end{cases} \tag{5.1}$$ we have $|\mathcal{D}_U| = N_k(z)$ in $O(k z)$ integer additions (and $|\mathcal{L}_{k,z}| = |\mathcal{D}_{B^\circ}|$ where $B^\circ = (0, B_1, \dots, B_{k-1}, 0)$).
2. (Cap Meet-Semilattice Identity and $(2m - 1)$-Term Adjacent Inclusion–Exclusion for Pareto Unions): For any two parameter pairs $(c_1, \tau_2)$ and $(c_1', \tau_2')$, the coordinatewise minimum of their caps $L^{(c_1, \tau_2)}$ and $L^{(c_1', \tau_2')}$ is $$\min\bigl(L^{(c_1, \tau_2)},\; L^{(c_1', \tau_2')}\bigr) = L^{(\max(c_1, c_1'),\; \max(\tau_2, \tau_2'))} \quad \Longleftrightarrow \quad \mathcal{D}_{c_1, \tau_2} \cap \mathcal{D}_{c_1', \tau_2'} = \mathcal{D}_{\max(c_1, c_1'),\; \max(\tau_2, \tau_2')}. \tag{5.2}$$ Consequently, for any Pareto-ordered sequence of certified pairs $(c_1^{(1)}, \tau_2^{(1)}), \dots, (c_1^{(m)}, \tau_2^{(m)})$ satisfying $0 \le c_1^{(1)} < c_1^{(2)} < \cdots < c_1^{(m)}$ and $\tau_2^{(1)} > \tau_2^{(2)} > \cdots > \tau_2^{(m)} \ge 1$, the union $\mathcal{U}_{\mathrm{Pareto}} := \bigcup_{r=1}^m \mathcal{D}_{c_1^{(r)}, \tau_2^{(r)}} \subseteq \mathcal{M}_{k,z}$ is a downward-closed order ideal of globally phase-minimal bridges whose exact cardinality collapses from $2^m - 1$ inclusion–exclusion terms to $2m - 1$ adjacent terms: $$\left|\bigcup_{r=1}^m \mathcal{D}_{c_1^{(r)}, \tau_2^{(r)}}\right| = \sum_{r=1}^m \bigl|\mathcal{D}_{c_1^{(r)}, \tau_2^{(r)}}\bigr| - \sum_{r=1}^{m-1} \bigl|\mathcal{D}_{c_1^{(r+1)}, \tau_2^{(r)}}\bigr|. \tag{5.3}$$

Proof. (1) By Theorem 3A(1), $L \in \mathcal{L}_{k,z}$ if and only if $0 = c_0(L) \le c_1(L) \le \cdots \le c_k(L) = z$ and $c_j(L) \le B_j$, while $L_j \le U_j \iff c_j(L) = B_j - L_j \ge B_j - U_j$. Counting valid prefixes $(c_0, \dots, c_j)$ by their endpoint $c_j = d$ gives (5.1).

(2) By definition (2.2), $L \le L^{(c_1, \tau_2)}$ holds if and only if $L_j \le (B_j - c_1)_+$ for $1 \le j \le k - 1$ and $L_j = 0$ for $j > k - \tau_2$. Imposing this simultaneously for $(c_1, \tau_2)$ and $(c_1', \tau_2')$ is equivalent to $L_j \le (B_j - \max(c_1, c_1'))_+$ and $L_j = 0$ for $j > k - \max(\tau_2, \tau_2')$, proving (5.2). For any non-empty subset $I = \{r_1 < r_2 < \cdots < r_s\} \subseteq \{1, \dots, m\}$, iterating (5.2) gives $\bigcap_{r \in I} \mathcal{D}_{c_1^{(r)}, \tau_2^{(r)}} = \mathcal{D}_{c_1^{(r_s)}, \tau_2^{(r_1)}}$, which depends solely on $r_1 = \min I$ and $r_s = \max I$. In the inclusion–exclusion sum $\sum_{\varnothing \ne I \subseteq \{1,\dots,m\}} (-1)^{|I|-1} |\bigcap_{r \in I} \mathcal{D}_{c_1^{(r)}, \tau_2^{(r)}}|$, grouping subsets $I$ by $(r_1, r_s)$ yields weight $+1$ when $r_s = r_1$, weight $-1$ when $r_s = r_1 + 1$, and weight $\sum_{j=0}^{r_s - r_1 - 1} \binom{r_s - r_1 - 1}{j} (-1)^{j+1} = -(1 - 1)^{r_s - r_1 - 1} = 0$ whenever $r_s - r_1 \ge 2$, proving (5.3). $\square$

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6. Computational Verification and Benchmark Tables

The standalone Python verification suite code/verify_paper12.py (in supplement.zip, standard library only) executes in $9.4\text{ s}$ with 0 failures and performs the following exact-integer checks:

1. Exhaustive Verification of Theorems 1, 2, and 3 across Benchmark Rows $(5, 8)$ through $(14, 23)$ (45,076 Bridges, 77,893 Bridge Pairs):
   • Enumerates all $45{,}076$ legal bridges across $(5, 8), (7, 12), (8, 13), (9, 15), (11, 18), (12, 20), (14, 23)$, determines the exact phase-minimal set $\mathcal{P}_{k,z}$ in each row, and verifies Theorem 1 (2.4)–(2.6) across all $(c_1, \tau_2)$ pairs and Theorem 2 (3.2)–(3.4) across all $U \in \mathcal{M}_{k,z}$ (checking all covering decrements $L = U - e_i$ across all rows and all comparable pairs $L \le U$ for $k \le 11$).
   • Verifies the pairwise $(2, 3)$-adic divergence valuation laws (4.3)–(4.4) across all $77{,}893$ bridge pairs in $(5, 8), (7, 12), (8, 13), (9, 15)$, and verifies the exact left-support valuation identity (4.5), the rotated-ceiling and forward-prefix state bounds (4.6), the pre-halving dual plateau identity (4.7) across all constant-deficit plateaus, and the compound-deficit bounds (4.10)–(4.11) across all $45{,}076$ bridges:

| Row $(k, S)$ | Total $|\mathcal{L}_{k,z}|$ | True Phase-Min $|\mathcal{P}_{k,z}|$ | Prop. 1 [1] ($\tau_2=1$, $2^{\tau_1+c_1}>M$) | Pareto Union $|\mathcal{U}_{\mathrm{Pareto}}|$ (Thm 1 & 4) | Support-Valuation Ideal $|\mathcal{M}_{k,z}|$ (Thm 2) | Cap $L^1$ Actual $Q(L^1)$ ([1] vs. Thm 3B) | | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | $(5, 8)$ | $7$ | $7$ | $2$ ($28.57\%$) | $4$ ($57.14\%$) | $4$ ($57.14\%$) | — | | $(7, 12)$ | $66$ | $66$ | $66$ ($100.0\%$ at $c_1=0$) | $66$ ($100.0\%$) | $66$ ($100.0\%$) | $1{,}909$ (2 vs. 7) | | $(8, 13)$ | $99$ | $99$ | $23$ ($23.23\%$) | $51$ ($51.52\%$) | $84$ ($84.85\%$) | $1{,}631$ (1 vs. 2) | | $(9, 15)$ | $377$ | $377$ | $112$ ($29.71\%$) | $377$ ($100.0\%$) | $377$ ($100.0\%$) | $13{,}085$ (4 vs. 19) | | $(11, 18)$ | $1{,}768$ | $1{,}768$ | $377$ ($21.32\%$) | $865$ ($48.93\%$) | $1{,}581$ ($89.42\%$) | $84{,}997$ (5 vs. 19) | | $(12, 20)$ | $7{,}229$ | $7{,}229$ | $1{,}626$ ($22.49\%$) | $7{,}229$ ($100.0\%$) | $7{,}229$ ($100.0\%$) | $103{,}427$ (27 vs. 151) | | $(14, 23)$ | $35{,}530$ | $35{,}530$ | $7{,}229$ ($20.35\%$) | $16{,}876$ ($47.50\%$) | $32{,}205$ ($90.64\%$) | $3{,}605{,}639$ (41 vs. 163) |

2. Exact Evaluation of Theorem 4 Across Continued-Fraction Convergents and Semiconvergents up to $(347, 550)$:
   • Using the $O(k z)$ ballot-path recurrence (5.1) and the $(2m-1)$-term adjacent inclusion–exclusion identity (5.3) (verified against brute-force set unions for all $k \le 14$), verify_paper12.py computes the exact integer cardinalities of $\mathcal{L}_{k,z}$, Proposition 1 of [1], the single largest two-sided downset $\mathcal{D}_{c_1, \tau_2}$, and the Pareto union $\mathcal{U}_{\mathrm{Pareto}}$ in $0.34\text{ s}$:

| $(k, S)$ | $\log_{10} |\mathcal{L}_{k,z}|$ | Prop. 1 [1] ($\tau_2=1$) | Best Single $\mathcal{D}_{c_1, \tau_2}$ | Pareto Union $|\mathcal{U}_{\mathrm{Pareto}}| / |\mathcal{L}_{k,z}|$ | Gain vs. Prop. 1 | Certified Pareto Frontier $(c_1^{(r)}, \tau_2^{(r)})$ | | :---: | :---: | :---: | :---: | :---: | :---: | :--- | | $(8, 13)$ | $2.00$ ($99$) | $23.2323\%$ ($c_1=1$) | $37.3737\%$ ((0,2)) | $51.5152\%$ | $2.22\times$ | (0,2), (1,1) | | $(11, 18)$ | $3.25$ ($1{,}768$) | $21.3235\%$ ($c_1=1$) | $35.2941\%$ ((0,2)) | $48.9253\%$ | $2.29\times$ | (0,2), (1,1) | | $(14, 23)$ | $4.55$ ($35{,}530$) | $20.3462\%$ ($c_1=1$) | $34.1993\%$ ((0,2)) | $47.4979\%$ | $2.33\times$ | (0,2), (1,1) | | $(17, 27)$ | $5.49$ ($312{,}455$) | $3.6591\%$ ($c_1=2$) | $6.5363\%$ ((1,2)) | $13.4528\%$ | $3.68\times$ | (0,4), (1,2), (2,1) | | $(19, 31)$ | $6.66$ ($4.55 \times 10^6$) | $19.3015\%$ ($c_1=1$) | $33.4341\%$ ((0,2)) | $46.2387\%$ | $2.40\times$ | (0,2), (1,1) | | $(27, 43)$ | $9.79$ ($6.17 \times 10^9$) | $3.2332\%$ ($c_1=2$) | $5.9518\%$ ((1,2)) | $12.2123\%$ | $3.78\times$ | (0,4), (1,2), (2,1) | | $(41, 65)$ | $15.79$ ($6.17 \times 10^{15}$) | $0.6010\%$ ($c_1=3$) | $0.9965\%$ ((2,2)) | $2.5809\%$ | $4.29\times$ | (0,6), (1,4), (2,2), (3,1) | | $(46, 73)$ | $18.01$ ($1.02 \times 10^{18}$) | $2.9537\%$ ($c_1=2$) | $2.9537\%$ ((2,1)) | $6.6038\%$ | $2.24\times$ | (0,5), (1,3), (2,1) | | $(59, 94)$ | $23.91$ ($8.14 \times 10^{23}$) | $2.9016\%$ ($c_1=2$) | $5.4835\%$ ((1,2)) | $11.2213\%$ | $3.87\times$ | (0,4), (1,2), (2,1) | | $(103, 164)$ | $43.60$ ($3.98 \times 10^{43}$) | $2.7919\%$ ($c_1=2$) | $5.3277\%$ ((1,2)) | $10.8947\%$ | $3.90\times$ | (0,4), (1,2), (2,1) | | $(163, 259)$ | $70.46$ ($2.89 \times 10^{70}$) | $2.7326\%$ ($c_1=2$) | $5.2507\%$ ((1,2)) | $10.7404\%$ | $3.93\times$ | (0,4), (1,2), (2,1) | | $(347, 550)$ | $153.09$ ($1.23 \times 10^{153}$) | $0.0804\%$ ($c_1=4$) | $0.1151\%$ ((2,4)) | $0.3620\%$ | $4.50\times$ | (0,8), (1,6), (2,4), (3,3), (4,1) |

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