Moving-Pole and Clean-Column Antidiagonal Obstructions for Affine Two-Digit Conic-Group Sources and Conflict-Free Doubleton Tie-Breaks on the Möbius Quartic

Abstract

We establish exact single- and two-direction capacity laws and refined periodic-window obstructions strictly below $1.8n$ for two prime-field candidate families in the No-Three-in-Line problem on $G_n = \{1, \ldots, n\}^2$:

1. Single- and Two-Direction Laws for Affine Two-Digit Conic-Group Sources on $G_{p^2}$ ($N = p^2$): In the two-digit prime-field construction of no_three_in_line_50 [1], setting $u = [(v^2 - \nu)/r]_p$, $d = [v - r]_p$, $k = \mathbf{1}_{v < r}$, and $$X_i = p[u + k + i]_p + v, \qquad Y_i = p[r + \pi_i(d) + i]_p + r \quad (i \in \{0, 1\})$$ produces $(11/6 - o(1))N \approx 1.83333N$ points after vertical column repair with rows, $+1$-diagonals ($X - Y$), and conic groups internally clean. While no_three_in_line_50 v2 [1] recently established a multi-direction $(4/3 + o(1))N$ capacity bound across all affine actions by taking an unbounded inventory of short-triple slopes, we determine the exact single- and two-direction capacity laws along explicit Euclidean slopes:

   • Moving-Pole Fiber Identity, General Slope Degeneracy, and Parity-Free Carry Laws (Theorem 1): On every low-digit $-1$-antidiagonal fiber $s_0 = [v + r]_p \in \mathbb{F}_p$, the modular high-digit sum satisfies $$u + r + \pi_i(d) \equiv \bigl(\pi_i(d) - d\bigr) - s_0 + \frac{2(s_0^2 - \nu)}{s_0 - d} \pmod p$$ with non-vanishing pole residue $2(s_0^2 - \nu) \not\equiv 0 \pmod p$. More generally, on any primitive Euclidean direction $(a, b) \in \mathbb{Z}^2$ with low-digit fiber $w_0 = [bv - ar]_p$ and affine map $\pi_i(d) = [\alpha_i d + \beta_i]_p$, the high-digit combination $bu - a(r + \pi_i(d))$ has linear $r$-coefficient $\frac{a(a - b)(1 - \alpha_i)}{b}$, explaining why pure translations ($\alpha_i = 1$) degenerate from degree-2 Kloosterman hyperbolas to degree-1 Möbius bijections on every Euclidean slope simultaneously. Parameterizing the fiber $s_0 = [v + r]_p$ directly by $r \in \{1, \ldots, p - 1\}$ (or $v \in \{0, \ldots, p - 1\}$) eliminates division by $2 \bmod p$ and makes the combined carry $m(r) = k + \mathbf{1}_{v + r \ge p} \in \{0, 1, 2\}$ a parity-free 3-step function on $J_0(s) = (0, \frac{s}{2}]$, $J_1(s) = (\frac{s}{2}, \frac{s+1}{2}]$, and $J_2(s) = (\frac{s+1}{2}, 1)$ ($s = s_0/p$), while in $d$-coordinates we establish the exact odd-parity carry threshold at $s_0 + d < p$ vs. $s_0 + d > p$.
   • Fiber-Constant & Vieta-Coupled Single-Slope Obstructions for $\alpha \equiv 2^{-1} \pmod p$ and $\alpha \equiv -1 \pmod p$ (Theorem 2): Whenever $\alpha \equiv (1 + b)^{-1} \pmod p$ ($b \in \mathbb{Z} \setminus \{0, -1\}$), the base high-$Y$ coordinate $y_{i, 0}(w_0) = [(1 + b)^{-1}w_0 + \beta_i]_p$ is constant on each low-digit fiber $w_0 = [v + b r]_p$ of $X + bY$, rendering the single-layer excess 100% invariant under $\beta_i \in \mathbb{F}_p$ at every finite prime $p$. Accounting for the exact Vieta root-sum coupling $b^2(r_+ + r_-) \equiv U_t + 2b w_0 \pmod p$, we prove that for $\alpha \equiv 2^{-1} \pmod p$ ($\pi_i(d) = [2^{-1}d + \beta_i]_p$) on $X + Y$ ($b = 1$), $\frac{E_{X+Y}(S_i)}{N} = \frac{713}{7680} + o(1) \approx 0.092839$ and, for every fixed integer offset $\Delta\beta \in \mathbb{Z}$ ($\beta_1 \equiv \beta_0 + \Delta\beta \pmod p$), $\frac{\operatorname{Cap}_{X+Y}(S_0 \cup S_1)}{N} \le \frac{89{,}783}{61{,}440} + o(1) \approx \mathbf{1.461312 < 1.8}$ uniformly across all $\beta_0 \in \mathbb{F}_p$; and for reflection $\alpha \equiv -1 \pmod p$ ($\pi_i(d) = [-d + \beta_i]_p$) on $X - 2Y$ ($b = -2$), $\frac{E_{X-2Y}(S_i)}{N} = \frac{137}{2048} + o(1) \approx 0.066895$ and, for every fixed integer offset $\Delta\beta \in \mathbb{Z}$, $\frac{\operatorname{Cap}_{X-2Y}(S_0 \cup S_1)}{N} \le \frac{375}{256} + o(1) \approx \mathbf{1.464844 < 1.8}$ uniformly across all $\beta_0 \in \mathbb{F}_p$.
   • Phase-Uniform Clean-Column Antidiagonal Obstruction for Fixed-Offset Translation Pairs (Theorem 3): For any fixed integer offset $\Delta c \in \mathbb{Z}$ and arbitrary translation shift $c_0 \in \mathbb{F}_p$ ($\pi_0(d) = [d + c_0]_p$, $\pi_1(d) = [d + c_0 + \Delta c]_p$), partitioning $S_0 \cup S_1 = Q_{\mathrm{over}} \sqcup Q_{\mathrm{clean}}$ into overloaded ($\operatorname{ColLoad} = 3$) and clean ($\operatorname{ColLoad} \le 2$) vertical columns yields $|S| \le |S_0 \cup S_1| - E_{\mathrm{col}} - E_{\mathrm{anti}}(Q_{\mathrm{clean}})$. Grouping candidates by distinct pole shifts $p_r = 2i + t + \Delta c \cdot i \in \mathcal{P}(\Delta c)$, we prove that the column-companion rational functions $r'_{p_r, \delta}(W)$ on $\mathbb{P}^1_W$ have pairwise coprime quadratic denominators $\widetilde{Q}_{p_r, \delta}(W) = (W - c_0 - p_r + \frac{\delta}{2})^2 - \frac{1}{4}(\delta^2 - 4\delta s_0 + 4\nu)$ across all $(p_r, \delta) \in \mathcal{P}(\Delta c) \times \{-2, -1, 1, 2\}$ on $p - O(1)$ fibers $s_0 \in \mathbb{F}_p$. We compute the exact closed-form phase polynomials $E_{\mathrm{raw}}(x_0; \Delta c), E_{\mathrm{clean}}(x_0; \Delta c) \in \mathbb{Q}[x_0]$ across the macroscopic shift phase $x_0 = \min([c_0]_p/p, 1 - [c_0]_p/p) \in [0, \frac{1}{2}]$ (including $E_{\mathrm{raw}}(x_0; -2) = \frac{161}{480} - \frac{1}{6}x_0^2 + \frac{5}{24}x_0^3 + \frac{1}{16}x_0^4 - \frac{1}{15}x_0^5 \in [\frac{247}{768}, \frac{161}{480}]$, giving $\frac{\operatorname{Cap}_{X+Y}}{N} \in [\frac{799}{480}, \frac{1289}{768}] \approx [1.664583, 1.678385]$ at $\Delta c = -2$) and certify their exact endpoint global minima and rational enclosures on $[0, \frac{1}{2}]$ via Sturm's theorem. This proves $\frac{|S|}{N} \le \frac{933{,}336{,}434{,}867}{531{,}372{,}441{,}600} + o(1) \approx \mathbf{1.756464 < 1.8}$ for all coherent translations $\pi_0 = \pi_1 = [d + c_0]_p$ ($c_0 \in \mathbb{F}_p$) and $\frac{|S|}{N} \le \mathbf{1.789526 + o(1) < 1.8}$ uniformly across all $c_0 \in \mathbb{F}_p$ for every fixed integer offset $\Delta c \in \mathbb{Z}$, using only two directions ($X$ and $X + Y$).

2. Conflict-Free 1-Step Doubleton Tie-Break Exchanges on the Möbius Quartic $Q_4(u) = u^4 - u$ (Theorem 4): In no_three_in_line_11 [2], the $2p \times 2p$ window obstruction for $Q_4(u) = u^4 - u$ along $V = y - x$ cleared $1.8$ by $1.04 \times 10^{-5}$ ($\frac{|S|}{2p} \le 1.7999896 + O(p^{-1/2})$) using baseline depth-32 exposed density $\rho_{32} \ge 0.0689004614$. In the peeling vertex cover $(U_{\mathrm{base}}, V_{\mathrm{base}})$ of no_three_in_line_11 Section 9.1 (where $U_{\mathrm{base}}$ is the set of covered residue rows and $V_{\mathrm{base}}$ consists of the $R(h)$ columns of every undeleted group $h$), every group $h \in \mathcal{A}_0$ has either $R(h) = 0$ (zero columns) or $R(h) = 1$ with both external cavity rows uncovered at time $31$, so $h$ has $\ge 2 > R(h)$ uncovered rows, remains undeleted through round $32$, and has its column already included in $V_{\mathrm{base}}$. Consequently, for any doubleton group $g$ ($R(g) = 2$ columns $\{u_1(g), u_2(g)\}$, three rows $\{y_0, y_1, y_2\}$) with two rows $y_1, y_2 \in \mathcal{A}_0^2$, all support points on $y_1, y_2$ outside $g$ are already covered by $V_{\mathrm{base}}$, and whenever $g$ is deleted during rounds $1, \ldots, 32$ (after $y_0$ is covered), removing the $2$ rows $\{y_1, y_2\}$ from $U_{\mathrm{base}}$ while adding the $2$ columns $\{u_1(g), u_2(g)\}$ to $V_{\mathrm{base}}$ is an exact $2$-row-for-$2$-column zero-cost vertex-cover exchange ($-2 + 2 = 0$). Because every $h \in \mathcal{A}_0$ has $R(h) \le 1$, $g$ is the unique $R = 2$ group incident to $y_1$ and $y_2$, making these exchanges 100% vertex-disjoint without tree recursion, upgrading $p_{\mathcal{A}_0} \ge 0.4229941154$ to $p_{\mathcal{A}_1} = p_{\mathcal{A}_0} + d\,p_{\mathcal{A}_0}^4 \ge 0.4309975668$, boosting $\rho_{32}^{(1)} \ge 0.0715324526$, and expanding the $2p \times 2p$ margin below $1.8$ along $V = y - x$ by $18.5\times$ to $\frac{|S|}{2p} \le \mathbf{1.79980804 + O(p^{-1/2}) < 1.8}$.

Scope Disclaimer: These theorems establish exact single- and two-direction obstructions for the specified algebraic two-digit conic-group families ($\alpha \equiv 1, 2^{-1}, -1 \pmod p$ with fixed integer layer offsets $\Delta c, \Delta\beta \in \mathbb{Z}$ and arbitrary shift $c_0, \beta_0 \in \mathbb{F}_p$) and an improved $2p \times 2p$ periodic-window constant on $Q_4(u) = u^4 - u$; they are support-restricted obstructions and do not resolve the general unrestricted $1.8n$ problem.

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1. Introduction, Related Work, and Distinction

A central question in the No-Three-in-Line problem on $G_n = \{1, \ldots, n\}^2$ (see Hall, Jackson, Sudbery, and Wild [5]) is whether explicit algebraic prime-field constructions can exceed the $\frac{9}{5}n = 1.8n$ density barrier when horizontal, vertical, and selected diagonal directions are simultaneously controlled. Two algebraic families have recently been studied as sharp tests of this boundary:

1. Two-Digit Conic-Group Prime-Field Sources on $G_{p^2}$ ($N = p^2$): Introduced in no_three_in_line_50 [1] (and discussed in r/no_three_in_line Post qVLddSXj3BDORTUW), this construction fixes an odd prime $p \ge 5$, a quadratic nonresidue $\nu \in \mathbb{F}_p^\times$ ($\left(\frac{\nu}{p}\right) = -1$), parameters $(r, v) \in \mathbb{F}_p^\times \times \mathbb{F}_p$, and defines $$u = \left[\frac{v^2 - \nu}{r}\right]_p, \qquad d = [v - r]_p, \qquad k = \mathbf{1}_{v < r}, \tag{1.1}$$ together with two digit layers $i \in \{0, 1\}$ governed by permutations $\pi_0, \pi_1 : \mathbb{F}_p \to \mathbb{F}_p$: $$X_i(v, r) = p[u + k + i]_p + v, \qquad Y_i(v, r) = p[r + \pi_i(d) + i]_p + r, \tag{1.2}$$ optionally purged of the $O(p)$ boundary wrap pairs where $u + k + i \ge p$ or $[r + \pi_i(d)]_p + i \ge p$ ([1], Section 3). By Section 3 of [1], the union $S_0 \cup S_1$ has $(2 - o(1))N$ points; every horizontal row and every $+1$-diagonal ($X - Y$) has load at most $2$; every conic group is internally collinear-free after the wrap purge; and vertical columns have excess $E_{\mathrm{col}} = (\frac{1}{6} - o(1))N$, leaving $(\frac{11}{6} - o(1))N \approx 1.83333N$ points after vertical column repair — only $\frac{1}{30}N \approx 0.03333N$ above $1.8N$.

   • Comparison with no_three_in_line_50 v2 [1]: In v1 of [1], an obstruction below $1.8N$ was proved for the multiplicative inversion map $\pi(d) = [\nu/d]_p$. Concurrently with our investigation (following our forum discussion on Post qVLddSXj3BDORTUW), v2 of [1] extended the multi-direction short-triple inventory method to all affine actions $\pi_i(d) = [\alpha_i d + \beta_i]_p$, proving that when an unbounded family of Euclidean slopes is enforced simultaneously (taking $p \to \infty$ before enlarging the slope inventory), the clean capacity is at most $(\frac{2}{3} + o(1))N$ per layer and $(\frac{4}{3} + o(1))N$ for two layers. By contrast, our Theorems 1–3 analyze the exact single-direction and two-direction capacity laws along explicit low-complexity slopes ($X + Y$, $X - 2Y$, and $X \cup (X + Y)$): we derive the exact rational capacity constants for the fiber-constant multipliers $\alpha \equiv 2^{-1}, -1 \pmod p$ on a single slope and compute the exact closed-form phase polynomials in $\mathbb{Q}[x_0]$ for all fixed-offset translation pairs $\pi_0(d) = [d + c_0]_p, \pi_1(d) = [d + c_0 + \Delta c]_p$ across the macroscopic phase $x_0 = \min([c_0]_p/p, 1 - [c_0]_p/p) \in [0, \frac{1}{2}]$, proving that two directions alone ($X$ and $X + Y$) already obstruct every fixed-offset translation pair strictly below $1.8N$.
   • Distinction from Checkerboard Moment Bounds [8]: Aujla, Prellberg, and Sandhu [8] (Moment obstructions and continuum-to-discrete bounds for checkerboard no-three-in-line sets, arXiv:2609.12340v2) establish four-direction line-sum moment bounds (including the bound $\approx 1.57682n$) for subsets restricted to a single checkerboard parity class. Because the two-digit conic source (1.1)–(1.2) occupies both checkerboard parities and places rational conic high digits over a two-dimensional low-digit base, the fixed-colour global moment inequalities of [8] do not apply to or subsume the fiber-wise Kloosterman and clean-column phase polynomials proved here.

2. Two-Period Windows on the Möbius Quartic $Q_4(u) = u^4 - u$: In no_three_in_line_11 v2 [2] (see also the survey and template bounds in no_three_in_line_35 v8 [3] and the multi-block subgrid obstructions in no_three_in_line_52 v1 [4]), the three-graph periodic support $S \subseteq \mathcal{P}_p(Q_4) \subseteq \mathbb{F}_p^2$ associated with $Q_4(u) = u^4 - u$ and the Möbius generators $$g_0(v) = v, \qquad g_1(v) = A(v) = \frac{13v + 8}{8v + 5}, \qquad g_2(v) = B(v) = \frac{93v - 872}{8v - 75} \tag{1.3}$$ was shown to be the unique polynomial coordinate-capacity exception above $1.8$ ($c_4 \approx 1.802336$, depth-32 peeling bound $C_{32} \le 1.8023359812$), and was obstructed in $2p \times 2p$ windows along $V = y - x$ with baseline depth-32 certified exposed row density $\rho_{32} \ge 0.0689004614$, yielding $\frac{|S|}{2p} \le 1.79998960 + O(p^{-1/2}) < 1.8$ (margin $1.04 \times 10^{-5}$ below $1.8$). In Section 5 (Theorem 4), we improve this $2p \times 2p$ single-slope constant by introducing a 100% vertex-disjoint, exact zero-cost ($-2 \text{ rows} + 2 \text{ columns} = 0$) 1-step doubleton tie-break exchange on the residue cover $(U_{\mathrm{base}}, V_{\mathrm{base}})$, boosting $\rho_{32}$ to $\rho_{32}^{(1)} \ge 0.0715324526$ and expanding the $2p \times 2p$ margin below $1.8$ by $18.5\times$ to $\frac{|S|}{2p} \le 1.79980804 + O(p^{-1/2}) < 1.8$.

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2. Moving-Pole Fiber Identity, General Slope Degeneracy, and Exact Carry Laws

Let $p \ge 5$ be prime and let $\nu \in \mathbb{F}_p^\times$ be a quadratic nonresidue ($\left(\frac{\nu}{p}\right) = -1$). For each point $P_i(v, r) = (X_i(v, r), Y_i(v, r))$ of the two-digit source (1.1)–(1.2), its $-1$-antidiagonal coordinate is $X_i + Y_i$. Writing the integer low-digit sum as $$v + r = p\,c_{\mathrm{low}}(v, r) + s_0, \qquad s_0 = [v + r]_p \in \{0, \ldots, p - 1\}, \qquad c_{\mathrm{low}}(v, r) = \mathbf{1}_{v + r \ge p} \in \{0, 1\}, \tag{2.1}$$ all points $P_i(v, r)$ lying on any physical $-1$-antidiagonal $X + Y = p A + s_0$ belong to the same low-digit fiber $v + r \equiv s_0 \pmod p$. Since $r \in \{1, \ldots, p - 1\}$, this fiber has cardinality $p - 1$ and can be parameterized bijectively in three ways:

1. By $r \in \{1, \ldots, p - 1\}$ directly: $v = [s_0 - r]_p \in \{0, \ldots, p - 1\} \setminus \{s_0\}$ and $d = [s_0 - 2r]_p \in \{0, \ldots, p - 1\} \setminus \{s_0\}$.
2. By $v \in \{0, \ldots, p - 1\} \setminus \{s_0\}$ directly: $r = [s_0 - v]_p \in \{1, \ldots, p - 1\}$ and $d = [2v - s_0]_p \in \{0, \ldots, p - 1\} \setminus \{s_0\}$.
3. By $d = [v - r]_p \in \{0, \ldots, p - 1\} \setminus \{s_0\}$: $$v \equiv \frac{s_0 + d}{2} \pmod p, \qquad r \equiv \frac{s_0 - d}{2} \pmod p. \tag{2.2}$$

Theorem 1 (Moving-Pole Fiber Identity, Slope Degeneracy, and Exact Carry Laws)

Fix an odd prime $p \ge 5$ and a quadratic nonresidue $\nu \in \mathbb{F}_p^\times$.

1. Moving-Pole High-Digit Identity on $X + Y$: On every low-digit fiber $s_0 = [v + r]_p \in \mathbb{F}_p$, for every $d \in \mathbb{F}_p \setminus \{s_0\}$, the modular high-digit sum $M_{i, s_0}(d) \equiv u + r + \pi_i(d) \pmod p$ satisfies the exact rational identity $$M_{i, s_0}(d) \equiv \bigl(\pi_i(d) - d\bigr) - s_0 + \frac{2(s_0^2 - \nu)}{s_0 - d} \pmod p, \qquad \text{or in terms of } r: \quad u(r) \equiv r - 2s_0 + \frac{s_0^2 - \nu}{r} \pmod p, \tag{2.3}$$ where the pole residue $2(s_0^2 - \nu) \not\equiv 0 \pmod p$ is non-vanishing for every $s_0 \in \mathbb{F}_p$.

2. General Euclidean Slope Degeneracy Identity: Let $(a, b) \in \mathbb{Z}^2$ be any primitive slope with $b \not\equiv 0 \pmod p$, and let $w_0 = [bv - ar]_p \in \mathbb{F}_p$ be the low-digit invariant of $bX - aY$. For any affine permutation $\pi_i(d) = [\alpha_i d + \beta_i]_p$, the modular high-digit combination $R_{w_0}(r) \equiv b u - a(r + \pi_i(d)) \pmod p$ on fiber $w_0$ satisfies $$R_{w_0}(r) \equiv \frac{a(a - b)(1 - \alpha_i)}{b}\,r + \frac{w_0^2 - b^2\nu}{b\,r} + \frac{a(2 - \alpha_i)w_0}{b} - a\beta_i \pmod p, \tag{2.4}$$ with non-vanishing pole residue $(w_0^2 - b^2\nu)/b \not\equiv 0 \pmod p$ for all $w_0 \in \mathbb{F}_p$. In particular, for pure translations ($\alpha_i = 1$), the linear $r$-coefficient $\frac{a(a - b)(1 - \alpha_i)}{b}$ vanishes identically on every Euclidean slope $(a, b) \in \mathbb{Z}^2$ simultaneously, reducing $R_{w_0}(r)$ from a degree-2 Kloosterman hyperbola to a degree-1 Möbius bijection on $\mathbb{P}^1_r$.

3. Parity-Free $r$- and $v$-Parameterizations and Corrected $d$-Parameterization Carry Laws: Define the combined carry shift $m = k(v, r) + c_{\mathrm{low}}(v, r) = \mathbf{1}_{v < r} + \mathbf{1}_{v + r \ge p} \in \{0, 1, 2\}$ on the fiber $s_0 = [v + r]_p$, and set $s = s_0 / p \in [0, 1)$.

   • Parity-Free $r$-Parameterization ($y = r/p \in (0, 1)$): As the integer $r$ varies over $\{1, \ldots, p - 1\}$, $v = [s_0 - r]_p$ equals $s_0 - r$ for $r \le s_0$ and $s_0 + p - r$ for $r > s_0$. Without any parity dependence, $(m, k, c_{\mathrm{low}})$ is constant on each of the four contiguous subintervals separated by $0 < \frac{s}{2} \le s < \frac{s+1}{2} < 1$: $$\begin{aligned} r \in \bigl(0, \tfrac{s_0}{2}\bigr] \;\; &\bigl(y \in (0, \tfrac{s}{2}]\bigr): & v &= s_0 - r \ge r, & (m, k, c_{\mathrm{low}}) &= (0, 0, 0), \\[2pt] r \in \bigl(\tfrac{s_0}{2}, s_0\bigr] \;\; &\bigl(y \in (\tfrac{s}{2}, s]\bigr): & v &= s_0 - r < r, & (m, k, c_{\mathrm{low}}) &= (1, 1, 0), \\[2pt] r \in \bigl(s_0, \tfrac{s_0 + p}{2}\bigr] \;\; &\bigl(y \in (s, \tfrac{s+1}{2}]\bigr): & v &= s_0 + p - r \ge r, & (m, k, c_{\mathrm{low}}) &= (1, 0, 1), \\[2pt] r \in \bigl(\tfrac{s_0 + p}{2}, p\bigr) \;\; &\bigl(y \in (\tfrac{s+1}{2}, 1)\bigr): & v &= s_0 + p - r < r, & (m, k, c_{\mathrm{low}}) &= (2, 1, 1). \end{aligned} \tag{2.5}$$ In particular, $m(r) \in \{0, 1, 2\}$ is a pure 3-step function of $y = r/p \in (0, 1)$: $m(r) = 0$ on $J_0(s) = (0, \frac{s}{2}]$, $m(r) = 1$ on $J_1(s) = (\frac{s}{2}, \frac{s+1}{2}]$, and $m(r) = 2$ on $J_2(s) = (\frac{s+1}{2}, 1)$.
   • Parity-Free $v$-Parameterization ($x = v/p \in [0, 1) \setminus \{s\}$): Since $v = s_0 - r$ on $(0, s]$ and $v = s_0 + p - r$ on $(s, 1)$, the four contiguous subintervals of $x = v/p \in (0, 1)$ separated by $0 < \frac{s}{2} < s < \frac{s+1}{2} < 1$ have deterministic $(m, k, c_{\mathrm{low}})$ labels: $$I_1(s) = \bigl(0, \tfrac{s}{2}\bigr) : (1, 1, 0), \quad I_2(s) = \bigl(\tfrac{s}{2}, s\bigr) : (0, 0, 0), \quad I_3(s) = \bigl(s, \tfrac{s+1}{2}\bigr) : (2, 1, 1), \quad I_4(s) = \bigl(\tfrac{s+1}{2}, 1\bigr) : (1, 0, 1). \tag{2.6}$$
   • Corrected $d$-Parameterization ($d = [v - r]_p \in \{0, \ldots, p - 1\} \setminus \{s_0\}$): Because $d = [s_0 - 2r]_p$ multiplies $r$ by $-2 \bmod p$, the integer representatives $(v, r)$ in terms of $d$ split by the parity of $s_0 + d$, with the odd-parity threshold occurring at $s_0 + d < p$ vs. $s_0 + d > p$:
     • Odd Parity ($s_0 + d$ odd): $r = \frac{s_0 - d + p}{2} \in (\frac{s_0}{2}, \frac{s_0 + p}{2}]$ and $m(d) = 1$ identically, with $$\begin{cases} v = \frac{s_0 + d + p}{2} \in \bigl[\frac{s_0 + p}{2}, p\bigr), \quad k = 0, \quad c_{\mathrm{low}} = 1, \quad m(d) = 1 & \text{if } s_0 + d < p, \\[4pt] v = \frac{s_0 + d - p}{2} \in \bigl(0, \frac{s_0}{2}\bigr), \quad k = 1, \quad c_{\mathrm{low}} = 0, \quad m(d) = 1 & \text{if } s_0 + d > p. \end{cases} \tag{2.7a}$$
     • Even Parity ($s_0 + d$ even, $d \ne s_0$): $v = \frac{s_0 + d}{2}$ and $m(d) = 2\mathbf{1}_{d > s_0} \in \{0, 2\}$ identically, with $$\begin{cases} r = \frac{s_0 - d}{2} \in \bigl(0, \frac{s_0}{2}\bigr], \quad k = 0, \quad c_{\mathrm{low}} = 0, \quad m(d) = 0 & \text{if } d < s_0, \\[4pt] r = \frac{s_0 - d + 2p}{2} \in \bigl(\frac{s_0 + p}{2}, p\bigr), \quad k = 1, \quad c_{\mathrm{low}} = 1, \quad m(d) = 2 & \text{if } d > s_0. \end{cases} \tag{2.7b}$$
   • Reflection Carry Identity on $X - 2Y$ ($w_0 = [v - 2r]_p$): For the slope $X - 2Y$, let $v - 2r = p\,c_{\mathrm{low}}^{(-2)}(v, r) + w_0$ with $w_0 = [v - 2r]_p \in \{0, \ldots, p - 1\}$. Then for every $(v, r) \in \{0, \ldots, p - 1\} \times \{1, \ldots, p - 1\}$, $$m^{(-2)}(v, r) := k(v, r) + c_{\mathrm{low}}^{(-2)}(v, r) = -\mathbf{1}_{w_0 + r \ge p} \in \{0, -1\}. \tag{2.8}$$

4. Physical High-Digit Antidiagonal Decomposition: Let $w_i(d) = [M_{i, s_0}(d)]_p \in \{0, \ldots, p - 1\}$ and $y_i(d) = [r + \pi_i(d)]_p \in \{0, \ldots, p - 1\}$. Outside the $O(p)$ boundary wrap pairs where $u + k + i \ge p$ or $y_i(d) + i \ge p$, the physical high $-1$-antidiagonal digit $A_i(d) = \lfloor(X_i + Y_i)/p\rfloor \in \{0, \ldots, 2p - 1\}$ is $$A_i(d) = w_i(d) + m(d) + 2i + p\,c_{\mathrm{wrap}, i}(d), \qquad c_{\mathrm{wrap}, i}(d) = \mathbf{1}_{y_i(d) > w_i(d)} \in \{0, 1\}. \tag{2.9}$$

Proof of Theorem 1

1. On $v \equiv s_0 - r \pmod p$, we have $u \equiv (v^2 - \nu)/r \equiv ((s_0 - r)^2 - \nu)/r = r - 2s_0 + (s_0^2 - \nu)/r \pmod p$. Substituting $r \equiv (s_0 - d)/2 \pmod p$ yields $$u + r + \pi_i(d) \equiv (s_0 - d) - 2s_0 + \frac{2(s_0^2 - \nu)}{s_0 - d} + \pi_i(d) = (\pi_i(d) - d) - s_0 + \frac{2(s_0^2 - \nu)}{s_0 - d} \pmod p.$$ Since $\left(\frac{\nu}{p}\right) = -1$, $s_0^2 - \nu \not\equiv 0 \pmod p$ for every $s_0 \in \mathbb{F}_p$.

2. On the low-digit fiber $bv - ar \equiv w_0 \pmod p$, we have $v \equiv b^{-1}(w_0 + ar) \pmod p$ and $d \equiv v - r \equiv b^{-1}w_0 + b^{-1}(a - b)r \pmod p$. Substituting these into $R_{w_0}(r) \equiv b(v^2 - \nu)/r - a(r + \alpha_i d + \beta_i) \pmod p$ gives $$R_{w_0}(r) \equiv \frac{b^{-1}(w_0 + ar)^2 - b\nu}{r} - a\left(r + \frac{\alpha_i w_0}{b} + \frac{\alpha_i(a - b)}{b}\,r + \beta_i\right)$$ $$= \left(\frac{a^2}{b} - a - \frac{a(a - b)\alpha_i}{b}\right)r + \frac{w_0^2 - b^2\nu}{b\,r} + \frac{a(2 - \alpha_i)w_0}{b} - a\beta_i \equiv \frac{a(a - b)(1 - \alpha_i)}{b}\,r + \frac{w_0^2 - b^2\nu}{b\,r} + \frac{a(2 - \alpha_i)w_0}{b} - a\beta_i \pmod p,$$ where $w_0^2 - b^2\nu \equiv b^2((w_0/b)^2 - \nu) \not\equiv 0 \pmod p$ for all $w_0 \in \mathbb{F}_p$.

3. The $r$-parameterization (2.5) and $v$-parameterization (2.6) follow immediately from $v = s_0 - r \in [0, s_0)$ when $1 \le r \le s_0$ (where $v + r = s_0 < p$, so $c_{\mathrm{low}} = 0$, and $v < r \iff r > s_0/2$) and $v = s_0 + p - r \in (s_0, p)$ when $s_0 < r < p$ (where $v + r = s_0 + p \ge p$, so $c_{\mathrm{low}} = 1$, and $v < r \iff r > (s_0 + p)/2$). For the $d$-parameterization (2.7a)–(2.7b):

   • Since $2r \equiv s_0 - d \pmod p$ and $1 \le 2r \le 2p - 2$, if $s_0 + d$ is odd then $s_0 - d$ is odd; since $-p < s_0 - d < p$, the unique even integer in $\{s_0 - d, s_0 - d + p, s_0 - d + 2p\}$ lying in $[2, 2p - 2]$ is $s_0 - d + p$, giving $r = (s_0 - d + p)/2 \in (\frac{s_0}{2}, \frac{s_0 + p}{2}]$ and $m = 1$ always. Then $2v \equiv s_0 + d \pmod p$ (with $s_0 + d \in [1, 2p - 3]$ odd) has $v = (s_0 + d + p)/2 \in [\frac{s_0 + p}{2}, p)$ when $s_0 + d < p$ (giving $v > r \implies k = 0$ and $v + r = s_0 + p \implies c_{\mathrm{low}} = 1$) and $v = (s_0 + d - p)/2 \in (0, \frac{s_0}{2})$ when $s_0 + d > p$ (giving $v < r \implies k = 1$ and $v + r = s_0 \implies c_{\mathrm{low}} = 0$).
   • If $s_0 + d$ is even ($d \ne s_0$), then $s_0 + d \in [0, 2p - 2]$ is even, so $v = (s_0 + d)/2 \in [0, p - 1]$ directly, while $s_0 - d \ne 0$ is even, giving $r = (s_0 - d)/2 \in (0, \frac{s_0}{2}]$ ($k = 0, c_{\mathrm{low}} = 0, m = 0$) when $d < s_0$ and $r = (s_0 - d + 2p)/2 \in (\frac{s_0 + p}{2}, p)$ ($k = 1, c_{\mathrm{low}} = 1, m = 2$) when $d > s_0$.
   • Finally, for (2.8), substituting $v - 2r = p\,c_{\mathrm{low}}^{(-2)} + w_0$ into $[v - r]_p = v - r + p\,k = (v - 2r) + r + p\,k$ yields the exact integer identity $[v - r]_p = w_0 + r + p(k + c_{\mathrm{low}}^{(-2)})$. Since $[v - r]_p \in \{0, \ldots, p - 1\}$ and $w_0 + r \in \{1, \ldots, 2p - 2\}$, taking floors upon division by $p$ gives $0 = \lfloor(w_0 + r)/p\rfloor + (k + c_{\mathrm{low}}^{(-2)})$, so $m^{(-2)}(v, r) = k + c_{\mathrm{low}}^{(-2)} = -\mathbf{1}_{w_0 + r \ge p} \in \{0, -1\}$.

4. Away from the $O(p)$ wrap boundary pairs where $u + k + i \ge p$ or $y_i(d) + i \ge p$, $[u + k + i]_p = u + k + i$ and $[r + \pi_i(d) + i]_p = y_i(d) + i$. Since $u, y_i(d) \in \{0, \ldots, p - 1\}$ have modular sum $w_i(d) = [u + y_i(d)]_p$, integer addition gives $u + y_i(d) = w_i(d) + p\,\mathbf{1}_{y_i(d) > w_i(d)}$, and adding $k + c_{\mathrm{low}} + 2i = m(d) + 2i$ yields (2.9). $\square$

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3. Exact Fiber-Constant & Vieta-Coupled Kloosterman Obstructions for $\alpha \equiv 2^{-1} \pmod p$ and $\alpha \equiv -1 \pmod p$

For a general affine permutation $\pi_i(d) = [\alpha_i d + \beta_i]_p$ ($\alpha_i \not\equiv 0, 1 \pmod p$), the physical high digit along a line family $X + bY$ depends both on the Vieta root-sum coupling of the degree-2 Kloosterman hyperbola and on the wrap coordinate $y_{i, 0} = [r + \alpha_i(v - r) + \beta_i]_p = [\alpha_i v + (1 - \alpha_i)r + \beta_i]_p$. Whenever the vector $(\alpha_i, 1 - \alpha_i)$ is proportional to $(1, b) \pmod p$ — that is, whenever $$\alpha_i \equiv (1 + b)^{-1} \pmod p \qquad (b \in \mathbb{Z} \setminus \{0, -1\}), \tag{3.1}$$ we have $\alpha_i v + (1 - \alpha_i)r \equiv (1 + b)^{-1}(v + b r) \equiv (1 + b)^{-1}w_0 \pmod p$, so the base high-$Y$ coordinate $$y_{i, 0}(w_0) = \bigl[(1 + b)^{-1}w_0 + \beta_i\bigr]_p \in \{0, \ldots, p - 1\} \tag{3.2}$$ is constant on every low-digit fiber $w_0 = [v + b r]_p$:

• Setting $b = 1$ gives the halving multiplier $\alpha_i \equiv 2^{-1} \pmod p$ ($\pi_i(d) = [2^{-1}d + \beta_i]_p$) on the $-1$-antidiagonal family $X + Y$ ($w_0 = s_0 = [v + r]_p$).
• Setting $b = -2$ gives the reflection multiplier $\alpha_i \equiv -1 \pmod p$ ($\pi_i(d) = [-d + \beta_i]_p$) on the slope-$\frac{1}{2}$ family $X - 2Y$ ($w_0 = [v - 2r]_p$).

Theorem 2 (Exact Fiber-Constant & Vieta-Coupled Kloosterman Obstruction Theorem)

Let $p \ge 5$ be prime and $\nu \in \mathbb{F}_p^\times$ be a quadratic nonresidue.

1. Exact $\beta$-Invariance and Fiber Reduction for $\alpha \equiv (1 + b)^{-1} \pmod p$: Let $b \in \mathbb{Z} \setminus \{0, -1\}$ and $\pi_i(d) = [(1 + b)^{-1}d + \beta_i]_p$ ($\beta_i \in \mathbb{F}_p$). On each low-digit fiber $w_0 = [v + b r]_p$ of $X + bY$:

   • For a single layer $S_i$, shifting $\beta_i \in \mathbb{F}_p$ translates the high digit $\lfloor(X_i + bY_i)/p\rfloor$ on fiber $w_0$ by the constant integer $b([(1+b)^{-1}w_0 + \beta_i]_p + i)$ (or, after the boundary purge, by $b[(1+b)^{-1}w_0 + \beta_i]_p + b i$). Consequently, the unpurged single-layer line-multiplicity multiset $\{|L \cap S_i| : L \in \mathcal{L}_{X+bY}\}$ is 100% invariant under $\beta_i \in \mathbb{F}_p$ at every finite prime $p$.
   • For two layers $S_0 \cup S_1$ with any fixed integer offset $\Delta\beta \in \mathbb{Z}$ ($\beta_1 \equiv \beta_0 + \Delta\beta \pmod p$), outside at most $1 + |\Delta\beta| = O_{\Delta\beta}(1)$ wrap fibers $w_0$ and $O(p)$ wrap points $u + k + i \ge p$, subtracting the fiber-constant integer $b\,y_{0, 0}(w_0) = b[(1 + b)^{-1}w_0 + \beta_0]_p$ reduces the physical high digit on fiber $w_0$ to the $\beta_0$-independent internal fiber coordinate $$U_i^{(b)}(r) = u(r) + m^{(b)}(r) + (1 + b + b\Delta\beta)i, \qquad u(r) = \left[b^2 r - 2b w_0 + \frac{w_0^2 - \nu}{r}\right]_p \in \{0, \ldots, p - 1\}, \tag{3.3}$$ where $m^{(b)}(r) = k(v, r) + \lfloor(v + b r)/p\rfloor$. For any target integer $U \in \{0, \ldots, p - 1\}$ (away from $O_{\Delta\beta}(1)$ boundary values) and carry shift $t = m^{(b)}(r)$, the condition $U_i^{(b)}(r) = U$ is equivalent to the quadratic congruence in $r \in \{1, \ldots, p - 1\}$: $$b^2 r^2 - \bigl(U_t + 2b w_0\bigr)r + (w_0^2 - \nu) \equiv 0 \pmod p, \qquad U_t := U - t - (1 + b + b\Delta\beta)i, \tag{3.4}$$ whose discriminant $\Delta_{i, t}^{(b)}(U) = (U_t + 2b w_0)^2 - 4b^2(w_0^2 - \nu)$ has non-vanishing constant shift $-4b^2(w_0^2 - \nu) \not\equiv 0 \pmod p$ for every $w_0 \in \mathbb{F}_p$, and whose two roots $\{r_+, r_-\}$ in $\mathbb{F}_p^\times$ satisfy the Vieta sum relation $$b^2(r_+ + r_-) \equiv U_t + 2b w_0 \pmod p. \tag{3.5}$$

2. Exact Vieta-Coupled Single-Slope Obstruction for Halving $\alpha \equiv 2^{-1} \pmod p$ on $X + Y$ ($b = 1$): Fix any integer offset $\Delta\beta \in \mathbb{Z}$, and let $\pi_0(d) = [2^{-1}d + \beta_0]_p$ and $\pi_1(d) = [2^{-1}d + \beta_0 + \Delta\beta]_p$. Using the $-1$-antidiagonal direction $X + Y$ alone, uniformly across all $\beta_0 \in \mathbb{F}_p$:

   • Single Layer $S_i$: $$\frac{E_{X+Y}(S_i)}{N} = \frac{713}{7680} + O(p^{-1/2}\log p) \approx \mathbf{0.09283854}, \qquad \frac{\operatorname{Cap}_{X+Y}(S_i)}{N} = \frac{6967}{7680} + O(p^{-1/2}\log p) \approx \mathbf{0.90716146}. \tag{3.6}$$
   • Two-Layer Union $S_0 \cup S_1$: Every collinear-free subset $S \subseteq S_0 \cup S_1$ satisfies $\frac{|S|}{N} \le \frac{\operatorname{Cap}_{X+Y}(S_0 \cup S_1)}{N} = 2 - E_{X+Y}(\Delta\beta) + O_{\Delta\beta}(p^{-1/2}\log p) < 1.8$, with exact rational constants: $$\begin{aligned} \Delta\beta = -2: &\quad E_{X+Y} = \frac{2969}{3840} \approx 0.773177, &\quad \frac{\operatorname{Cap}_{X+Y}}{N} &= \frac{4711}{3840} + o(1) \approx \mathbf{1.226823 < 1.8}, \\[3pt] \Delta\beta \in \{-1, -3\}: &\quad E_{X+Y} = \frac{3149}{5760} \approx 0.546701, &\quad \frac{\operatorname{Cap}_{X+Y}}{N} &= \frac{8371}{5760} + o(1) \approx \mathbf{1.453299 < 1.8}, \\[3pt] \Delta\beta \in \{0, -4\}: &\quad E_{X+Y} = \frac{33{,}097}{61{,}440} \approx 0.538688, &\quad \frac{\operatorname{Cap}_{X+Y}}{N} &= \frac{89{,}783}{61{,}440} + o(1) \approx \mathbf{1.461312 < 1.8}, \\[3pt] |\Delta\beta + 2| \ge 3: &\quad E_{X+Y} = \frac{279{,}023}{516{,}096} \approx 0.540642, &\quad \frac{\operatorname{Cap}_{X+Y}}{N} &= \frac{753{,}169}{516{,}096} + o(1) \approx \mathbf{1.459358 < 1.8}. \end{aligned} \tag{3.7}$$

3. Exact Vieta-Coupled Single-Slope Obstruction for Reflection $\alpha \equiv -1 \pmod p$ on $X - 2Y$ ($b = -2$): Fix any integer offset $\Delta\beta \in \mathbb{Z}$, and let $\pi_0(d) = [-d + \beta_0]_p$ and $\pi_1(d) = [-d + \beta_0 + \Delta\beta]_p$. Using the single direction $X - 2Y$ alone, uniformly across all $\beta_0 \in \mathbb{F}_p$:

   • Single Layer $S_i$: $$\frac{E_{X-2Y}(S_i)}{N} = \frac{137}{2048} + O(p^{-1/2}\log p) \approx \mathbf{0.06689453}, \qquad \frac{\operatorname{Cap}_{X-2Y}(S_i)}{N} = \frac{1911}{2048} + O(p^{-1/2}\log p) \approx \mathbf{0.93310547}. \tag{3.8}$$
   • Two-Layer Union $S_0 \cup S_1$: Every collinear-free subset $S \subseteq S_0 \cup S_1$ satisfies $\frac{|S|}{N} \le \frac{\operatorname{Cap}_{X-2Y}(S_0 \cup S_1)}{N} = 2 - E_{X-2Y}(\Delta\beta) + O_{\Delta\beta}(p^{-1/2}\log p) < 1.8$, with exact rational constants: $$\begin{aligned} \Delta\beta \in \{0, -1\}: &\quad E_{X-2Y} = \frac{137}{256} = 0.53515625, &\quad \frac{\operatorname{Cap}_{X-2Y}}{N} &= \frac{375}{256} + o(1) \approx \mathbf{1.464844 < 1.8}, \\[3pt] \Delta\beta \notin \{0, -1\}: &\quad E_{X-2Y} = \frac{639{,}151}{1{,}179{,}648} \approx 0.54181502, &\quad \frac{\operatorname{Cap}_{X-2Y}}{N} &= \frac{1{,}720{,}145}{1{,}179{,}648} + o(1) \approx \mathbf{1.458185 < 1.8}. \end{aligned} \tag{3.9}$$

Proof of Theorem 2

1. On each low-digit fiber $w_0 = [v + b r]_p$, substituting $v \equiv w_0 - b r \pmod p$ into $u \equiv (v^2 - \nu)/r \pmod p$ gives $u(r) = [b^2 r - 2b w_0 + (w_0^2 - \nu)/r]_p \in \{0, \ldots, p - 1\}$, while $[r + \pi_i(d)]_p = [(1 + b)^{-1}(v + b r) + \beta_i]_p = [(1 + b)^{-1}w_0 + \beta_i]_p = y_{i, 0}(w_0)$ depends only on $w_0$. Thus for a single layer $S_i$, changing $\beta_i \in \mathbb{F}_p$ permutes the fiber-constant offsets $b[y_{i, 0}(w_0) + i]_p$ across lines on fiber $w_0$ without altering any line size $|L \cap S_i|$. For two layers with fixed integer offset $\Delta\beta \in \mathbb{Z}$, outside the at most $1 + |\Delta\beta| = O_{\Delta\beta}(1)$ fibers where $y_{0, 0}(w_0) + i + \Delta\beta \cdot i \notin [0, p)$ and the $O(p)$ pairs where $u(r) + k + i \ge p$, we have $[y_{i, 0}(w_0) + i]_p = y_{0, 0}(w_0) + (1 + \Delta\beta)i$ and $[u(r) + k + i]_p = u(r) + k + i$. Subtracting $b\,y_{0, 0}(w_0)$ from $\lfloor(X_i + bY_i)/p\rfloor$ yields (3.3). Since $u(r) \in \{0, \ldots, p - 1\}$ is already reduced modulo $p$, for every $U \in \{6 + |b\Delta\beta|, \ldots, p - 7 - |b\Delta\beta|\}$ and $t = m^{(b)}(r)$, the integer equality $u(r) = U_t$ is equivalent to the modular congruence $b^2 r - 2b w_0 + (w_0^2 - \nu)/r \equiv U_t \pmod p$, which clears denominators to (3.4) with Vieta sum (3.5).

2. Case $b = 1$ ($\alpha \equiv 2^{-1} \pmod p$ on $X + Y$): By Theorem 1(3), on fiber $s_0 = [v + r]_p$ with $s = s_0 / p \in (0, 1)$, $m(r) \in \{0, 1, 2\}$ is the parity-free 3-step indicator of $y = r/p \in J_0(s) = (0, \frac{s}{2}]$, $J_1(s) = (\frac{s}{2}, \frac{s+1}{2}]$, and $J_2(s) = (\frac{s+1}{2}, 1)$. For each shift $t \in \{0, 1, 2\}$ and layer $i \in \{0, 1\}$, let $\Sigma_{i, t} = [U - t - (2 + \Delta\beta)i + 2s_0]_p \in \{0, \ldots, p - 1\}$ and $\sigma_{i, t} = \Sigma_{i, t}/p \in (0, 1)$. As $U$ varies over $\{0, \ldots, p - 1\}$, for each shift the quadratic (3.4) is irreducible over $\mathbb{F}_p$ ($0$ roots) for $(p - 1)/2 + O(1)$ values of $U$ and splits into two roots $\{r_+, r_-\}$ in $\mathbb{F}_p^\times$ satisfying $r_+ + r_- \equiv \Sigma_{i, t} \pmod p$ and $r_+ r_- \equiv s_0^2 - \nu \pmod p$ for $(p - 1)/2 + O(1)$ values of $U$. In scaled coordinates $y_\pm = r_\pm / p \in (0, 1)$:

   • If $r_+ < \Sigma_{i, t}$ ($y_+ \in (0, \sigma)$), then $r_+ + r_- = \Sigma_{i, t}$, so $y_- = \sigma - y_+ \in (0, \sigma)$.
   • If $r_+ > \Sigma_{i, t}$ ($y_+ \in (\sigma, 1)$), then $r_+ + r_- = \Sigma_{i, t} + p$, so $y_- = 1 + \sigma - y_+ \in (\sigma, 1)$. Thus $y_- = \{\sigma - y_+\} \in (0, 1)$ in both cases! For any subinterval $J_t(s) = [a_t, b_t] \subseteq [0, 1]$, as $y_+$ is equidistributed in $(0, 1)$ at macroscopic $(s, \sigma) \in (0, 1)^2$ by the 1D Kloosterman/Weil bound on $\mathbb{P}^1_{r_+}$ [6], both roots $y_+, y_-$ lie in $J_t(s)$ if and only if $y_+ \in J_t(s) \cap (\sigma - J_t(s) \bmod 1)$, which has Lebesgue measure $$L_{\mathrm{both}}(s, \sigma; a_t, b_t) = \operatorname{meas}\bigl([a_t, b_t] \cap [\max(0, \sigma - b_t), \max(0, \sigma - a_t)]\bigr) + \operatorname{meas}\bigl([a_t, b_t] \cap [\min(1, 1 + \sigma - b_t), \min(1, 1 + \sigma - a_t)]\bigr). \tag{3.10}$$ Conditioned on the quadratic splitting (probability $\frac{1}{2}$), two roots land in $J_t(s)$ with probability $L_{\mathrm{both}}$, one root with probability $2(b_t - a_t - L_{\mathrm{both}})$, and zero roots with probability $1 - 2(b_t - a_t) + L_{\mathrm{both}}$. Unconditionally, the hit count $H_{i, t} \in \{0, 1, 2\}$ at $(s, \sigma)$ has distribution $$\mathbb{P}(H_{i, t} = 2) = \tfrac{1}{2}L_{\mathrm{both}}, \qquad \mathbb{P}(H_{i, t} = 1) = (b_t - a_t) - L_{\mathrm{both}}, \qquad \mathbb{P}(H_{i, t} = 0) = 1 - (b_t - a_t) + \tfrac{1}{2}L_{\mathrm{both}}. \tag{3.11}$$ Across distinct shifts $U_t \ne U'_{t'}$, the squarefree quadratic discriminants $\Delta_{i, t}(U) = (U_t + 2s_0)^2 - 4(s_0^2 - \nu)$ are coprime in $\overline{\mathbb{F}}_p[U]$ on $p - O(1)$ fibers $s_0$ (a common root requires $(U_t - U'_{t'})^2 = 16(s_0^2 - \nu)$, holding for at most $2$ fibers $s_0$), so their Kummer extensions are linearly disjoint over $\overline{\mathbb{F}}_p(U)$ while sharing the same macroscopic $\sigma = \Sigma_{i, t}/p + O_{\Delta\beta}(1/p)$. When $(0, t)$ and $(1, j)$ share the same shift $U_t = U'_j$ (i.e., $t = j + 2 + \Delta\beta$), they share the same root pair $\{y_+, y_-\}$:
   • If $\Delta\beta = -2$ ($t = j$), each root in $J_t(s)$ is counted twice ($2 H_{0, t} \in \{0, 2, 4\}$).
   • If $\Delta\beta \in \{0, -1, -3, -4\}$ ($t \ne j$), $\{y_+, y_-\}$ is tested against the disjoint union $J_t(s) \sqcup J_j(s) = [0, 1] \setminus J_c(s)$ ($c \in \{0, 1, 2\} \setminus \{t, j\}$); since a splitting pair has $(2 - H_{0, c})$ roots in $[0, 1] \setminus J_c(s)$, the combined hit count $H_{\mathrm{shared}} \in \{0, 1, 2\}$ has distribution $\mathbb{P}(H_{\mathrm{shared}} = 2) = \mathbb{P}(H_{0, c} = 0) - \frac{1}{2}$, $\mathbb{P}(H_{\mathrm{shared}} = 1) = \mathbb{P}(H_{0, c} = 1)$, and $\mathbb{P}(H_{\mathrm{shared}} = 0) = \frac{1}{2} + \mathbb{P}(H_{0, c} = 2)$. Partitioning $(s, \sigma) \in [0, 1]^2$ by the 7 critical lines $\sigma \in \{0, \frac{s}{2}, s, \frac{1}{2}, \frac{s+1}{2}, \{s + \frac{1}{2}\}, 1\}$ into 12 elementary trapezoids and integrating in exact rational arithmetic (verify_moving_pole_and_two_slope_q4.py) yields the exact rational fractions in (3.6)–(3.7).

3. Case $b = -2$ ($\alpha \equiv -1 \pmod p$ on $X - 2Y$): By Theorem 1(3) (2.8), on each low-digit fiber $w_0 = [v - 2r]_p$ ($s = w_0/p \in (0, 1)$), $m^{(-2)}(r) = -\mathbf{1}_{w_0 + r \ge p} \in \{0, -1\}$ is the 2-step indicator of $J_0^{(-2)}(s) = (0, 1 - s)$ ($m^{(-2)} = 0$) and $J_{-1}^{(-2)}(s) = (1 - s, 1)$ ($m^{(-2)} = -1$). In (3.4), $U_t = U + (-t) + (2\Delta\beta + 1)i$ where $-t \in \{0, 1\}$, and by (3.5), $4(r_+ + r_-) \equiv U_t - 4w_0 \pmod p$, so $r_+ + r_- \equiv [4^{-1}U_t - w_0]_p \pmod p$. Writing $\sigma_0 = [4^{-1}U - w_0]_p / p \in (0, 1)$ and $\kappa_p = [4^{-1}]_p / p = \frac{1}{4} + O(1/p)$ (for $p \equiv 3 \pmod 4$) or $\frac{3}{4} + O(1/p)$ (for $p \equiv 1 \pmod 4$), a shift of $U$ by $h \in \mathbb{Z}$ shifts $\sigma_0$ to $\sigma_h = \{\sigma_0 + h\kappa_p\} \in (0, 1)$:

   • A single layer ($i = 0$) tests $h = 0$ on $J_0^{(-2)}(s) = (0, 1 - s)$ and $h = 1$ on $J_{-1}^{(-2)}(s) = (1 - s, 1)$ with Vieta sums $\sigma_0$ and $\{\sigma_0 + \kappa_p\}$; integrating over $(s, \sigma_0) \in (0, 1)^2$ gives $\frac{137}{2048}$ in (3.8) for both $\kappa = \frac{1}{4}$ and $\kappa = \frac{3}{4}$.
   • For two layers with fixed integer offset $\Delta\beta \in \mathbb{Z}$, since $2\Delta\beta + 1$ is always odd, $(0, 0)$ and $(1, 0)$ never collide. When $\Delta\beta \in \{0, -1\}$ ($2\Delta\beta + 1 = \pm 1$), one shift collides between $J_{-1}^{(-2)}(s)$ and $J_0^{(-2)}(s)$, whose union is $(0, 1 - s) \cup (1 - s, 1) = (0, 1)$ (giving distribution $\frac{1}{2}\delta_0 + \frac{1}{2}\delta_2$), and integrating over $(s, \sigma_0) \in (0, 1)^2$ yields $\frac{137}{256} = 8 \times \frac{137}{2048}$ in (3.9). When $\Delta\beta \notin \{0, -1\}$, all four shifts $\{0, 1, 2\Delta\beta + 1, 2\Delta\beta + 2\}$ are distinct with $2\Delta\beta + 1 \equiv \pm 1 \pmod 4$, yielding $\frac{639{,}151}{1{,}179{,}648}$ in (3.9). $\square$

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4. Phase-Uniform Clean-Column Antidiagonal Excess Theorem for Fixed-Offset Translation Pairs

We now turn to the Fixed-Offset Pure Translation Family ($\alpha_0 = \alpha_1 = 1$), where $\pi_0(d) = [d + c_0]_p$ and $\pi_1(d) = [d + c_0 + \Delta c]_p$ with an arbitrary shift $c_0 \in \mathbb{F}_p$ and a fixed integer offset $\Delta c \in \mathbb{Z}$. By Theorem 1(2) (2.4), translations degenerate to degree-1 Möbius bijections on every Euclidean slope simultaneously; on $X + Y$, (2.3) reduces to $$M_{i, s_0}(d) \equiv c_0 + \Delta c \cdot i - s_0 + \frac{2(s_0^2 - \nu)}{s_0 - d} \pmod p, \tag{4.1}$$ which has the unique genus-$0$ rational inverse on $\mathbb{P}^1_W$: $$d_{i, t}(W) \equiv s_0 - \frac{2(s_0^2 - \nu)}{W - p_{i, t} + s_0 - c_0} \pmod p, \qquad p_{i, t} := 2i + t + \Delta c \cdot i. \tag{4.2}$$ Furthermore, because $r + d \equiv v \pmod p$, the base high-$Y$ coordinate for a translation is $$y_{i, 0}(v) = [v + c_0 + \Delta c \cdot i]_p \in \{0, \ldots, p - 1\}, \tag{4.3}$$ whose scaled coordinate $\{x + \gamma\} + O_{\Delta c}(1/p)$ ($x = v/p$) depends on the macroscopic translation phase $\gamma = [c_0]_p / p \in [0, 1)$. To obstruct all fixed-offset translation pairs uniformly across all $c_0 \in \mathbb{F}_p$ using only two directions ($X$ and $X + Y$), we combine vertical columns ($X$) and $-1$-antidiagonals ($X + Y$) via a vertex-disjoint Clean-Column Partition on $\mathbb{P}^1_W$.

Theorem 3 (Phase-Uniform Clean-Column Antidiagonal Excess Theorem for Fixed-Offset Translations)

Fix any integer offset $\Delta c \in \mathbb{Z}$. For any prime $p \ge 5$, quadratic nonresidue $\nu \in \mathbb{F}_p^\times$, and shift $c_0 \in \mathbb{F}_p$, let $\pi_0(d) = [d + c_0]_p$ and $\pi_1(d) = [d + c_0 + \Delta c]_p$, and set $\gamma = [c_0]_p / p \in [0, 1)$ and $x_0 = \min(\gamma, 1 - \gamma) \in [0, \frac{1}{2}]$.

1. Vertex-Disjoint Clean-Column Partition: Partition $S_0 \cup S_1 = Q_{\mathrm{over}} \sqcup Q_{\mathrm{clean}}$, where $Q_{\mathrm{over}}$ consists of all points in overloaded vertical columns ($\operatorname{ColLoad}(X) = 3$) and $Q_{\mathrm{clean}}$ consists of all points in clean vertical columns ($\operatorname{ColLoad}(X) \le 2$). Because $Q_{\mathrm{over}}$ and $Q_{\mathrm{clean}}$ are vertex-disjoint, every collinear-free subset $S \subseteq S_0 \cup S_1$ satisfies $$|S| = |S \cap Q_{\mathrm{over}}| + |S \cap Q_{\mathrm{clean}}| \le \bigl(|Q_{\mathrm{over}}| - E_{\mathrm{col}}\bigr) + \bigl(|Q_{\mathrm{clean}}| - E_{\mathrm{anti}}(Q_{\mathrm{clean}})\bigr) = |S_0 \cup S_1| - E_{\mathrm{col}} - E_{\mathrm{anti}}(Q_{\mathrm{clean}}), \tag{4.4}$$ where $E_{\mathrm{col}} = (\frac{1}{6} - o(1))N$ is the vertical column excess and $E_{\mathrm{anti}}(Q_{\mathrm{clean}}) = \sum_{L \in \mathcal{L}_{X+Y}} \max(0, |L \cap Q_{\mathrm{clean}}| - 2)$.

2. Grouped Pole Shifts and Cross-Denominator Separation of Column-Companion Roots on $\mathbb{P}^1_W$: Group the six index pairs $(i, t) \in \{0, 1\} \times \{0, 1, 2\}$ by their pole shift $p_{i, t} = 2i + t + \Delta c \cdot i$, and let $\mathcal{P}(\Delta c) = \{0, 1, 2\} \cup \{2 + \Delta c, 3 + \Delta c, 4 + \Delta c\}$ be the set of distinct pole shifts ($|\mathcal{P}(\Delta c)| \in \{3, 4, 5, 6\}$). Each distinct pole shift $p_r \in \mathcal{P}(\Delta c)$ defines a single candidate root $d_{p_r}(W) \equiv s_0 - \frac{2(s_0^2 - \nu)}{W - p_r + s_0 - c_0} \pmod p$ on $\mathbb{P}^1_W$ with simple pole at $W_{\infty, p_r} = c_0 + p_r - s_0$, together with four column-companion rational functions $r'_{p_r, \delta}(W) := r'_{\delta}(d_{p_r}(W)) \equiv (v^2 - \nu)/(u + \delta) \pmod p$ ($\delta \in \{-2, -1, 1, 2\}$).

   • Exact Polynomial Division and Resultant in $d$: As a rational function of $d$, $r'_{\delta}(d)$ has the exact polynomial division $$r'_{\delta}(d) = \underbrace{\frac{s_0 - d}{2}}_{=\,r(d)} - \delta + \frac{2\delta\bigl((2s_0 - \delta)d + \delta s_0 - 2\nu\bigr)}{(d + s_0 - \delta)^2 - (\delta^2 - 4\delta s_0 + 4\nu)}, \tag{4.5a}$$ with $\operatorname{Res}_d\bigl(-(d - s_0)(4\nu - (d + s_0)^2),\,-2(d + s_0 - \delta)^2 + 2(\delta^2 - 4\delta s_0 + 4\nu)\bigr) = -512\,\delta^2(s_0^2 - \nu)^2 \not\equiv 0 \pmod p$ for every $s_0 \in \mathbb{F}_p$.
   • Explicit $W$-Plane Quadratic Denominators and Cross-Denominator Coprimality: Substituting $d = d_{p_r}(W)$ into the denominator of (4.5a) yields the exact $W$-plane factorization $$\bigl(d_{p_r}(W) + s_0 - \delta\bigr)^2 - (\delta^2 - 4\delta s_0 + 4\nu) \equiv \frac{4(s_0^2 - \nu)}{(W - W_{\infty, p_r})^2}\,\widetilde{Q}_{p_r, \delta}(W) \pmod p, \tag{4.5b}$$ where $\widetilde{Q}_{p_r, \delta}(W) \in \mathbb{F}_p[W]$ is the monic quadratic polynomial $$\widetilde{Q}_{p_r, \delta}(W) = \left(W - c_0 - p_r + \frac{\delta}{2}\right)^2 - \frac{\delta^2 - 4\delta s_0 + 4\nu}{4}. \tag{4.5c}$$ Outside at most $16$ fibers $s_0 \in \mathbb{F}_p$ (where $\delta^2 - 4\delta s_0 + 4\nu \equiv 0 \pmod p$ or $\widetilde{Q}_{p_r, \delta}(W_{\infty, p_r}) \equiv 0 \pmod p$), each $\widetilde{Q}_{p_r, \delta}(W)$ is squarefree with two finite simple roots in $\overline{\mathbb{F}}_p \setminus \{W_{\infty, p_r}\}$. Moreover, for any two distinct pairs $(p_r, \delta) \ne (p'_r, \delta')$ in $\mathcal{P}(\Delta c) \times \{-2, -1, 1, 2\}$:
     • If $\delta = \delta'$ and $p_r \ne p'_r$, $\widetilde{Q}_{p_r, \delta}$ and $\widetilde{Q}_{p'_r, \delta}$ have the same discriminant $\Delta_\delta(s_0) = \frac{1}{4}(\delta^2 - 4\delta s_0 + 4\nu)$ and distinct centers $c_0 + p_r - \frac{\delta}{2} \ne c_0 + p'_r - \frac{\delta}{2}$, so a shared root in $\overline{\mathbb{F}}_p$ requires $(p_r - p'_r)^2 \equiv \delta^2 - 4\delta s_0 + 4\nu \pmod p$, which holds for at most $1$ fiber $s_0 \in \mathbb{F}_p$.
     • If $\delta \ne \delta'$, subtracting $\widetilde{Q}_{p_r, \delta}(W) = 0$ and $\widetilde{Q}_{p'_r, \delta'}(W) = 0$ eliminates $W^2$ and leaves a linear equation in $(W, s_0)$ whose substitution into $\widetilde{Q}_{p_r, \delta}(W) = 0$ is a non-trivial polynomial in $s_0$ of degree at most $2$ (with leading $s_0^2$-coefficient $(\delta - \delta')^2 \not\equiv 0 \pmod p$ whenever $2(p_r - p'_r) \not\equiv \delta - \delta' \pmod p$, and non-zero $s_0$-coefficient $-(\delta - \delta') \not\equiv 0 \pmod p$ when $2(p_r - p'_r) \equiv \delta - \delta' \pmod p$), holding for at most $2$ fibers $s_0 \in \mathbb{F}_p$. Thus, on $p - O_{\Delta c}(1)$ fibers $s_0 \in \mathbb{F}_p$, the monic quadratics $\{\widetilde{Q}_{p_r, \delta}(W)\}_{(p_r, \delta) \in \mathcal{P}(\Delta c) \times \{-2, -1, 1, 2\}}$ are pairwise coprime in $\overline{\mathbb{F}}_p[W]$ and disjoint from $\{W_{\infty, p_r}\}_{p_r \in \mathcal{P}(\Delta c)}$. By the 1D rational Weil bound on $\mathbb{P}^1_W$ [6], the candidate roots $\{d_{p_r}(W)\}_{p_r \in \mathcal{P}(\Delta c)}$ and their column companions $\{r'_{p_r, \delta}(W)\}$ are asymptotically independent across distinct $p_r \in \mathcal{P}(\Delta c)$, and a candidate root $d_{p_r}(W)$ landing in layer $i$'s interval with $v/p = x \in (0, 1)$ and $k + i \in \{0, 1, 2\}$ survives in $Q_{\mathrm{clean}}$ with conditional probability $$q_{\mathrm{clean}}(x, k + i) = \begin{cases} x & \text{if } k + i = 0 \quad (r_{u-2} \le v), \\[3pt] 1 - x + x^2 \ge \frac{3}{4} & \text{if } k + i = 1 \quad (\neg(r_{u+1} \le v \wedge r_{u-1} > v)), \\[3pt] 1 - x & \text{if } k + i = 2 \quad (r_{u+2} > v). \end{cases} \tag{4.6}$$

3. Exact Phase Polynomials in $\mathbb{Q}[x_0]$ and Sturm Certificates on $[0, \frac{1}{2}]$: For any fixed $\Delta c \in \mathbb{Z}$ and macroscopic phase $x_0 = \min(\gamma, 1 - \gamma) \in [0, \frac{1}{2}]$, the asymptotic single-direction raw excess $\frac{E_{\mathrm{anti}}(S_0 \cup S_1)}{N} = E_{\mathrm{raw}}(x_0; \Delta c) + O_{\Delta c}(p^{-1/2}\log p)$ and clean-column excess $\frac{E_{\mathrm{anti}}(Q_{\mathrm{clean}})}{N} = E_{\mathrm{clean, exact}}(x_0; \Delta c) + O_{\Delta c}(p^{-1/2}\log p) \ge E_{\mathrm{clean, low}}(x_0; \Delta c) + O_{\Delta c}(p^{-1/2}\log p)$ are given by exact polynomials in $\mathbb{Q}[x_0]$ on $[0, \frac{1}{2}]$:

   • Coherent Translations $\pi_0(d) = \pi_1(d) = [d + c_0]_p$ ($\Delta c = 0$, all $c_0 \in \mathbb{F}_p$): $$\begin{aligned} E_{\mathrm{raw}}(x_0; 0) &= \frac{177}{896} - \frac{1}{20}x_0^2 + \frac{7}{96}x_0^3 - \frac{1}{64}x_0^4 + \frac{1}{60}x_0^5 - \frac{2}{105}x_0^7 \in \left[\frac{991}{5120}, \frac{177}{896}\right] \approx [0.193555, 0.197545], \\[4pt] E_{\mathrm{clean, low}}(x_0; 0) &= \frac{3{,}699{,}259}{55{,}050{,}240} + \frac{18{,}721}{15{,}728{,}640}x_0 + \frac{585{,}143}{7{,}864{,}320}x_0^2 - \frac{195{,}883}{737{,}280}x_0^3 + \frac{318{,}071}{786{,}432}x_0^4 - \frac{392{,}215}{1{,}179{,}648}x_0^5 \\ &\quad + \frac{552{,}379}{2{,}949{,}120}x_0^6 - \frac{173{,}213}{1{,}290{,}240}x_0^7 + \frac{5{,}190{,}277}{82{,}575{,}360}x_0^8 + \frac{1{,}349{,}039}{24{,}772{,}608}x_0^9 - \frac{105{,}073}{1{,}966{,}080}x_0^{10} + \frac{9}{880}x_0^{11} \\ &\in \left[\frac{3{,}699{,}259}{55{,}050{,}240},\, \frac{19{,}603{,}616{,}075}{279{,}038{,}656{,}512}\right] \approx [0.067198, 0.070254] > \frac{1}{30}, \\[4pt] E_{\mathrm{clean, exact}}(x_0; 0) &= \frac{40{,}846{,}374{,}733}{531{,}372{,}441{,}600} + \frac{12{,}721}{81{,}920}x_0^2 - \frac{1{,}503{,}473{,}899}{2{,}724{,}986{,}880}x_0^3 + \cdots - \frac{755}{92{,}664}x_0^{15} \\ &\in \left[\frac{40{,}846{,}374{,}733}{531{,}372{,}441{,}600},\, \frac{155{,}046{,}438{,}203{,}993}{1{,}865{,}572{,}732{,}108{,}800}\right] \approx [0.076870, 0.083109] > \frac{1}{30}. \end{aligned} \tag{4.7}$$ By root-deflated Sturm certification on $[0, \frac{1}{2}]$, $E_{\mathrm{clean, exact}}(x_0; 0)$ attains its exact global minimum on $[0, \frac{1}{2}]$ at $x_0 = 0$ ($\frac{40{,}846{,}374{,}733}{531{,}372{,}441{,}600} \approx 0.076870$) and its exact global maximum at $x_0 = \frac{1}{2}$ ($\frac{155{,}046{,}438{,}203{,}993}{1{,}865{,}572{,}732{,}108{,}800} \approx 0.083109$), proving uniformly across all $c_0 \in \mathbb{F}_p$ that $$\frac{|S|}{N} \le \frac{11}{6} - E_{\mathrm{clean, exact}}(x_0; 0) + O(p^{-1/2}\log p) \le \frac{933{,}336{,}434{,}867}{531{,}372{,}441{,}600} + o(1) \approx \mathbf{1.756464 < 1.8}$$ (and at $c_0 = (p-1)/2$ where $x_0 \to \frac{1}{2}$, $\frac{|S|}{N} \le \frac{11}{6} - \frac{155{,}046{,}438{,}203{,}993}{1{,}865{,}572{,}732{,}108{,}800} + o(1) \approx \mathbf{1.750224 < 1.8}$).
   • All Other Fixed-Offset Translation Pairs ($\Delta c \ne 0$, all $c_0 \in \mathbb{F}_p$): Across all $x_0 \in [0, \frac{1}{2}]$, Sturm's theorem certifies the following exact endpoint values, global minima, and rational enclosures on $[0, \frac{1}{2}]$:
     • Offset $\Delta c = -2$ (Single-Direction $X + Y$ Obstruction): At $\Delta c = -2$, $p_{0, t} = p_{1, t} = t$ for all $t \in \{0, 1, 2\}$, so each accepted candidate root contributes $2$ points simultaneously, and the exact raw phase polynomial on $[0, \frac{1}{2}]$ is $$E_{\mathrm{raw}}(x_0; -2) = \frac{161}{480} - \frac{1}{6}x_0^2 + \frac{5}{24}x_0^3 + \frac{1}{16}x_0^4 - \frac{1}{15}x_0^5, \tag{4.8a}$$ which decreases from $E_{\mathrm{raw}}(0; -2) = \frac{161}{480} \approx 0.335417$ to its exact global minimum $E_{\mathrm{raw}}(\frac{1}{2}; -2) = \frac{247}{768} \approx 0.321615$ at $x_0 = \frac{1}{2}$ (certified by $0$ deflated Sturm roots of $E_{\mathrm{raw}}(x_0; -2) - \frac{247}{768}$ on $[0, \frac{1}{2}]$). Consequently, on $X + Y$ alone, $$\frac{\operatorname{Cap}_{X+Y}(S_0 \cup S_1)}{N} = 2 - E_{\mathrm{raw}}(x_0; -2) + o(1) \in \left[\frac{799}{480},\, \frac{1289}{768}\right] + o(1) \approx \mathbf{[1.664583,\, 1.678385] < 1.8}. \tag{4.8b}$$
     • Offset $\Delta c = -1$: $E_{\mathrm{clean, low}}(x_0; -1)$ has $P(0) = \frac{1925}{49{,}152} \approx 0.039164$, exact global minimum $P(\frac{1}{2}) = \frac{60{,}275{,}545}{1{,}585{,}446{,}912} \approx 0.038018$ ($0$ deflated Sturm roots), and Sturm enclosure $[0.038018, 0.039265]$; $E_{\mathrm{clean, exact}}(x_0; -1)$ has $P(0) = \frac{112{,}517{,}711}{2{,}554{,}675{,}200} \approx 0.044044$, exact global minimum $P(\frac{1}{2}) = \frac{343{,}792{,}642{,}477}{7{,}847{,}962{,}214{,}400} \approx 0.043807$ ($0$ deflated Sturm roots), and Sturm enclosure $[0.043807, 0.044205]$ (attaining an interior maximum $\approx 0.044203$ near $x_0 \approx 0.19$), proving uniformly across all $c_0 \in \mathbb{F}_p$ that $$\frac{|S|}{N} \le \frac{11}{6} - \frac{343{,}792{,}642{,}477}{7{,}847{,}962{,}214{,}400} + o(1) = \frac{14{,}044{,}138{,}083{,}923}{7{,}847{,}962{,}214{,}400} + o(1) \approx \mathbf{1.789526 < 1.8}. \tag{4.8c}$$
     • Offsets $\Delta c = -3$, $\Delta c = -4$, and $|\Delta c + 2| \ge 3$: Across $x_0 \in [0, \frac{1}{2}]$, Sturm's theorem certifies the rational enclosures: $$\begin{aligned} \Delta c = -3: &\quad P(0) = \frac{176{,}471}{3{,}440{,}640} \approx 0.051290,\;\; P(\tfrac{1}{2}) = \frac{3{,}799{,}311}{73{,}400{,}320} \approx 0.051762,\;\; E_{\mathrm{clean, low}}(x_0; -3) \in [0.051254, 0.051766] &\implies \frac{|S|}{N} &\le \mathbf{1.782079 < 1.8}, \\[3pt] \Delta c = -4: &\quad E_{\mathrm{clean, low}}(x_0; -4) \ge E_{\mathrm{clean, low}}(0; -4) = \frac{1{,}998{,}323}{27{,}525{,}120} \approx 0.072600 \quad \bigl(P(\tfrac{1}{2}) \approx 0.074477\bigr) &\implies \frac{|S|}{N} &\le \mathbf{1.760733 < 1.8}, \\[3pt] |\Delta c + 2| \ge 3: &\quad E_{\mathrm{clean, low}}(x_0; \mathrm{gen}) \ge E_{\mathrm{clean, low}}(0; \mathrm{gen}) = \frac{179{,}436{,}407}{2{,}422{,}210{,}560} \approx 0.074080 \quad \bigl(P(\tfrac{1}{2}) \approx 0.075509\bigr) &\implies \frac{|S|}{N} &\le \mathbf{1.759253 < 1.8}. \end{aligned} \tag{4.8d}$$

Proof of Theorem 3

1. Since every vertical column has load at most $3$ ($2$ from one layer and $1$ from the other when overloaded), a collinear-free subset $S \subseteq S_0 \cup S_1$ must delete at least $1$ point from each overloaded column (total $E_{\mathrm{col}}$ points from $Q_{\mathrm{over}}$) and at least $\max(0, |L \cap Q_{\mathrm{clean}}| - 2)$ points from $Q_{\mathrm{clean}}$ on each $-1$-antidiagonal $L \in \mathcal{L}_{X+Y}$, giving (4.4).

2. Substituting $v \equiv (s_0 + d)/2 \pmod p$, $r \equiv (s_0 - d)/2 \pmod p$, and $u \equiv ((s_0 + d)^2 - 4\nu)/(2(s_0 - d)) \pmod p$ into $r'_{\delta}(d) \equiv (v^2 - \nu)/(u + \delta) \pmod p$ gives $$r'_{\delta}(d) = \frac{-(d - s_0)\bigl(4\nu - (d + s_0)^2\bigr)}{-2(d + s_0 - \delta)^2 + 2(\delta^2 - 4\delta s_0 + 4\nu)},$$ and polynomial division of the numerator by the denominator yields (4.5a) with resultant $-512\,\delta^2(s_0^2 - \nu)^2 \not\equiv 0 \pmod p$. For each distinct pole shift $p_r \in \mathcal{P}(\Delta c)$, substituting $d_{p_r}(W) - s_0 = -\frac{2(s_0^2 - \nu)}{W - W_{\infty, p_r}}$ into $d_{p_r}(W) + s_0 - \delta = (d_{p_r}(W) - s_0) + (2s_0 - \delta)$ yields $$\bigl(d_{p_r}(W) + s_0 - \delta\bigr)^2 - (\delta^2 - 4\delta s_0 + 4\nu) = \frac{4(s_0^2 - \nu)^2 - 4(2s_0 - \delta)(s_0^2 - \nu)(W - W_{\infty, p_r}) + 4(s_0^2 - \nu)(W - W_{\infty, p_r})^2}{(W - W_{\infty, p_r})^2}$$ $$= \frac{4(s_0^2 - \nu)}{(W - W_{\infty, p_r})^2}\left[\left(W - W_{\infty, p_r} - s_0 + \frac{\delta}{2}\right)^2 - \frac{\delta^2 - 4\delta s_0 + 4\nu}{4}\right] = \frac{4(s_0^2 - \nu)}{(W - W_{\infty, p_r})^2}\,\widetilde{Q}_{p_r, \delta}(W),$$ since $W_{\infty, p_r} + s_0 = c_0 + p_r$, establishing (4.5b)–(4.5c) (verified across all pairs in verify_moving_pole_and_two_slope_q4.py). For any two distinct pairs $(p_r, \delta) \ne (p'_r, \delta')$, let $a_1 = p_r - \frac{\delta}{2}$, $a_2 = p'_r - \frac{\delta'}{2}$, and $\Delta_j = \frac{1}{4}(\delta_j^2 - 4\delta_j s_0 + 4\nu)$. A common root $W \in \overline{\mathbb{F}}_p$ of $\widetilde{Q}_{p_r, \delta}$ and $\widetilde{Q}_{p'_r, \delta'}$ satisfies $(W - c_0 - a_1)^2 = \Delta_1$ and $(W - c_0 - a_2)^2 = \Delta_2$. If $a_1 = a_2$, then $\Delta_1 = \Delta_2 \implies (\delta - \delta')(\delta + \delta' - 4s_0) \equiv 0 \pmod p$, which (since $\delta \ne \delta'$ when $a_1 = a_2$ and $(p_r, \delta) \ne (p'_r, \delta')$) holds for at most $1$ fiber $s_0$. If $a_1 \ne a_2$, subtracting the two equations gives $2(a_2 - a_1)(W - c_0 - a_1) = \Delta_1 - \Delta_2 - (a_2 - a_1)^2$, and squaring gives $4(a_2 - a_1)^2 \Delta_1 = (\Delta_1 - \Delta_2 - (a_2 - a_1)^2)^2$, a polynomial in $s_0$ of degree at most $2$ whose $s_0^2$-coefficient is $(\delta - \delta')^2$ (non-zero whenever $\delta \ne \delta'$) and whose constant shift when $\delta = \delta'$ reduces to $(p_r - p'_r)^2 = 4\Delta_1$ (at most $1$ fiber $s_0$). Thus on $p - O_{\Delta c}(1)$ fibers $s_0 \in \mathbb{F}_p$, all $4|\mathcal{P}(\Delta c)|$ monic quadratics $\widetilde{Q}_{p_r, \delta}(W)$ have pairwise disjoint simple root pairs in $\overline{\mathbb{F}}_p \setminus \{W_{\infty, p_r}\}_{p_r \in \mathcal{P}(\Delta c)}$. Consequently, any non-zero linear combination of thegrouped candidates $\{d_{p_r}(W)\}_{p_r \in \mathcal{P}(\Delta c)}$ and their companions $\{r'_{p_r, \delta}(W)\}$ has uncancelled simple poles on $\mathbb{P}^1_W$, and the 1D rational Weil bound [6] yields independent joint equidistribution across distinct $p_r \in \mathcal{P}(\Delta c)$ with error $O_{\Delta c}(p^{-1/2}\log p)$ and conditional clean-column survival probability (4.6).

3. By Theorem 1(3) (2.6), in the parity-free $v$-parameterization $x = v/p \in (0, 1)$ on fiber $s = s_0 / p \in (0, 1)$, the four subintervals $I_1(s) = (0, \frac{s}{2})$, $I_2(s) = (\frac{s}{2}, s)$, $I_3(s) = (s, \frac{s+1}{2})$, $I_4(s) = (\frac{s+1}{2}, 1)$ have deterministic $(m, k)$ labels $(1, 1), (0, 0), (2, 1), (1, 0)$. Since $\Delta c \in \mathbb{Z}$ is fixed, outside at most $|\Delta c| = O_{\Delta c}(1)$ wrap values of $v$, both layers share the macroscopic wrap coordinate $y_i(v)/p = \{x + \gamma\} + O_{\Delta c}(1/p)$ where $\gamma = [c_0]_p/p \in [0, 1)$. Setting $x_0 = 1 - \gamma \in (0, 1]$ and $x_1 = \{x_0 + w\} \in [0, 1)$ ($w = W/p \in (0, 1)$), the wrap condition $\{x + \gamma\} > w$ ($\varepsilon = 1$) vs. $\{x + \gamma\} < w$ ($\varepsilon = 0$) partitions $(0, 1)$ at $x_0$ and $x_1$ into the contiguous chord $[\min(x_0, x_1), \max(x_0, x_1)]$ and its complement. For a shared pole shift $p_r = t = 2 + j + \Delta c \in \mathcal{P}(\Delta c)$, the single candidate root $d_{p_r}(W)$ is tested against $m = t$ in Layer 0 and $m = j$ in Layer 1: when $\Delta c \ne -2$ ($t \ne j$), the two intervals of $x$ are disjoint so the candidate lands in at most one layer (with combined clean probability $P_{t, 0}^{\mathrm{clean}} + P_{j, 1}^{\mathrm{clean}}$); when $\Delta c = -2$ ($t = j$), the candidate lands in both layers simultaneously with probability $P_t$, contributing $2$ points to the sheet and yielding (4.8a)–(4.8b). Integrating over $x_1 \in (0, 1)$ and $s \in [0, x_0], [x_0, 2x_0], [2x_0, 1]$ in exact rational arithmetic (verify_moving_pole_and_two_slope_q4.py) yields the exact polynomials in (4.7)–(4.8d) and their Sturm certificates on $[0, \frac{1}{2}]$. $\square$

Table 1: Regenerated Finite-Prime Verification Across $p \in \{103, 251, 503, 1009\}$

All counts below are generated by verify_moving_pole_and_two_slope_q4.py and archived in moving_pole_and_two_slope_data.json (where $\nu$ is the least quadratic nonresidue modulo $p$, "raw" denotes the unpurged point set of size $2p(p-1)$, and "purged" denotes the wrap-purged set of [1], Section 3 with $u + k + i < p$ and $[r + \pi_i(d)]_p + i < p$):

| Configuration & Theorem | $p = 103$ ($\nu = 3$) | $p = 251$ ($\nu = 2$) | $p = 503$ ($\nu = 5$) | $p = 1009$ ($\nu = 11$) | Exact Asymptotic Limit ($p \to \infty$) | | :--- | :---: | :---: | :---: | :---: | :---: | | $\alpha = 2^{-1}, \beta = 2$ on $X + Y$ (Thm 2): $E_0 / N$ (raw) | $0.090772$ | $0.092110$ | $0.093558$ | $0.092623$ | $\frac{713}{7680} \approx \mathbf{0.092839}$ | | $\alpha = 2^{-1}, \beta = 2$ on $X + Y$ (Thm 2): $\operatorname{Cap}_{X+Y}/N$ (raw / purged) | $1.466585$ / $1.442455$ | $1.468374$ / $1.458596$ | $1.460387$ / $1.455403$ | $1.462298$ / $1.459863$ | $\frac{89{,}783}{61{,}440} \approx \mathbf{1.461312 < 1.8}$ | | $\alpha = -1, \beta = 2$ on $X - 2Y$ (Thm 2): $E_0 / N$ (raw) | $0.064756$ | $0.065713$ | $0.065776$ | $0.066337$ | $\frac{137}{2048} \approx \mathbf{0.066895}$ | | $\alpha = -1, \beta = 2$ on $X - 2Y$ (Thm 2): $\operatorname{Cap}_{X-2Y}/N$ (raw / purged) | $1.471864$ / $1.443397$ | $1.466754$ / $1.455310$ | $1.466311$ / $1.460454$ | $1.465962$ / $1.463062$ | $\frac{375}{256} \approx \mathbf{1.464844 < 1.8}$ | | Trans $c_0 = c_1 = 2$ ($\Delta c = 0, x_0 \to 0$, Thm 3): $E_{\mathrm{clean}}/N$ (raw) | $0.077858$ | $0.075697$ | $0.075598$ | $0.076283$ | $\frac{40{,}846{,}374{,}733}{531{,}372{,}441{,}600} \approx \mathbf{0.076870 > \frac{1}{30}}$ | | Trans $c_0 = c_1 = 2$ ($\Delta c = 0, x_0 \to 0$, Thm 3): 2-Dir UB $/ N$ (raw) | $1.740032$ ($18{,}460$) | $1.752639$ ($110{,}418$) | $1.754582$ ($443{,}925$) | $1.755226$ ($1{,}786{,}962$) | $\frac{933{,}336{,}434{,}867}{531{,}372{,}441{,}600} \approx \mathbf{1.756464 < 1.8}$ | | Trans $c_0 = c_1 = 2$ ($\Delta c = 0, x_0 \to 0$, Thm 3): 2-Dir UB $/ N$ (purged) | $1.714393$ ($18{,}188$) | $1.742036$ ($109{,}750$) | $1.749207$ ($442{,}565$) | $1.752529$ ($1{,}784{,}216$) | $\frac{933{,}336{,}434{,}867}{531{,}372{,}441{,}600} \approx \mathbf{1.756464 < 1.8}$ | | Trans $c_0 = c_1 = \frac{p-1}{2}$ ($\Delta c = 0, x_0 \to \frac{1}{2}$, Thm 3): $E_{\mathrm{clean}}/N$ (raw) | $0.084457$ | $0.084792$ | $0.082637$ | $0.083113$ | $\frac{155{,}046{,}438{,}203{,}993}{1{,}865{,}572{,}732{,}108{,}800} \approx \mathbf{0.083109 > \frac{1}{30}}$ | | Trans $c_0 = c_1 = \frac{p-1}{2}$ ($\Delta c = 0, x_0 \to \frac{1}{2}$, Thm 3): 2-Dir UB $/ N$ (raw) | $1.733434$ ($18{,}390$) | $1.743544$ ($109{,}845$) | $1.747543$ ($442{,}144$) | $1.748395$ ($1{,}780{,}008$) | $\frac{3{,}265{,}249{,}721{,}313{,}607}{1{,}865{,}572{,}732{,}108{,}800} \approx \mathbf{1.750224 < 1.8}$ | | Trans $(c_0, c_1) = (2, 1)$ ($\Delta c = -1, x_0 \to 0$, Thm 3): $E_{\mathrm{clean}}/N$ (raw) | $0.041286$ | $0.043571$ | $0.043350$ | $0.043647$ | $\frac{112{,}517{,}711}{2{,}554{,}675{,}200} \approx \mathbf{0.044044 > \frac{1}{30}}$ | | Trans $(c_0, c_1) = (2, 1)$ ($\Delta c = -1, x_0 \to 0$, Thm 3): 2-Dir UB $/ N$ (raw) | $1.776605$ ($18{,}848$) | $1.784765$ ($112{,}442$) | $1.786830$ ($452{,}084$) | $1.787862$ ($1{,}820{,}188$) | $\frac{4{,}571{,}053{,}489}{2{,}554{,}675{,}200} \approx \mathbf{1.789289 < 1.8}$ |

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5. Conflict-Free 1-Step Doubleton Tie-Break Exchanges on the Möbius Quartic $Q_4(u) = u^4 - u$

We now turn to the periodic three-graph support $S \subseteq \mathcal{P}_p(Q_4) \subseteq \mathbb{F}_p^2$ of the Möbius quartic $Q_4(u) = u^4 - u$ defined in (1.3). Let us first recall the exact residue vertex-cover construction and cavity formalism of no_three_in_line_11 [2], Section 9.1:

• Columns $u \in \mathbb{F}_p$ are grouped by $v = Q_4(u) = u^4 - u \in \mathbb{F}_p$, where the column multiplicity $R(v) = |\{u \in \mathbb{F}_p : Q_4(u) = v\}| \in \{0, 1, 2, 4\}$ has asymptotic Chebotarev probabilities $z = \frac{3}{8}$ ($R = 0$), $o = \frac{1}{3}$ ($R = 1$), $d = \frac{1}{4}$ ($R = 2$), and $\frac{1}{24}$ ($R = 4$) [7]. Each group $v$ has three neighboring residue rows $g_0(v) = v$, $g_1(v) = A(v)$, and $g_2(v) = B(v)$.
• Baseline Peeling Vertex Cover $(U_{\mathrm{base}}, V_{\mathrm{base}})$: At round $t = 0$, every neighboring row of a group with $R(v) \ge 3$ ($R = 4$) is placed in the row cover $U$, and every group with $R(v) \in \{0, 4\}$ is deleted. In each synchronous round $t = 1, \ldots, D$ ($D = 32$), any undeleted group $v$ whose number of uncovered neighboring rows at time $t - 1$ is at most $R(v)$ is deleted, and its remaining uncovered neighboring rows are added to $U$. After round $D$, the baseline residue vertex cover $(U_{\mathrm{base}}, V_{\mathrm{base}})$ consists of:
  1. the covered row set $U_{\mathrm{base}} \subseteq \mathbb{F}_p$, and
  2. the covered column set $V_{\mathrm{base}} = \bigcup_{v \notin \mathcal{D}_D} \operatorname{Cols}(v) \subseteq \mathbb{F}_p$, consisting of all $R(v)$ columns of every undeleted group $v \notin \mathcal{D}_D$ (where $\operatorname{Cols}(v) = \{u \in \mathbb{F}_p : Q_4(u) = v\}$, $|\operatorname{Cols}(v)| = R(v)$). Every support point $(u, y) \in \mathcal{P}_p(Q_4)$ with $v = Q_4(u)$ is covered by $(U_{\mathrm{base}}, V_{\mathrm{base}})$: if $v \notin \mathcal{D}_D$ is undeleted, then $u \in \operatorname{Cols}(v) \subseteq V_{\mathrm{base}}$; if $v \in \mathcal{D}_D$ was deleted at some round $t \le D$, then all neighboring rows of $v$ (including $y$) were placed in $U_{\mathrm{base}}$. In a $2p \times 2p$ periodic square, the physical coordinate-cover budget of $(U_{\mathrm{base}}, V_{\mathrm{base}})$ is $4(|U_{\mathrm{base}}| + |V_{\mathrm{base}}|)$, which satisfies $\frac{2(|U_{\mathrm{base}}| + |V_{\mathrm{base}}|)}{p} \to C_{32} \le 1.8023359812$ by (9.1)–(9.2) of [2] (with $r_0 = (z + o + d)^2$, $Z(r) = z + o(2r - r^2) + d r^2$, $r_{t+1} = Z(r_t)^2$, where $r_t$ is the cavity probability that a row remains uncovered after $t$ rounds when its parent group is withheld).
• Why $\mathcal{A}_0$-Groups Have All Their Columns in $V_{\mathrm{base}}$: In Section 9.1 of [2], a neighboring group $h$ of a row $y$ belongs to the baseline cavity event $\mathcal{A}_0(y)$ (probability $p_{\mathcal{A}_0} = z + o\,r_{31}^2 \ge 0.4229941154$) if either:
  • $R(h) = 0$ (probability $z = \frac{3}{8}$), so $\operatorname{Cols}(h) = \emptyset$ and $h$ contributes zero support points to $y$; or
  • $R(h) = 1$ and both external cavity rows of $h$ away from $y$ remain uncovered at time $D - 1 = 31$ when $h$ is withheld (probability $o\,r_{31}^2$). Because $h$ has $2 > R(h) = 1$ uncovered external rows through round $D = 32$ regardless of whether $y$ is covered or uncovered, $h$ remains undeleted ($h \notin \mathcal{D}_{32}$), and therefore its unique column $\operatorname{Cols}(h) = \{u(h)\}$ is included in the column cover $V_{\mathrm{base}}$! In both cases, every support point on $y$ belonging to an $\mathcal{A}_0(y)$-group is already covered by the column cover $V_{\mathrm{base}}$ without requiring $y$ to belong to the row cover $U_{\mathrm{base}}$.

We now show that whenever a doubleton group $g$ ($R(g) = 2$ columns $\operatorname{Cols}(g) = \{u_1(g), u_2(g)\}$, three rows $\{y_0, y_1, y_2\}$) has two external rows $y_1, y_2 \in \mathcal{A}_0^2$, the $2 = R(g)$ tie at $g$ can be resolved by covering the $2$ columns $\{u_1(g), u_2(g)\}$ instead of the $2$ rows $\{y_1, y_2\}$ at exact zero net cost ($-2 \text{ rows} + 2 \text{ columns} = 0$) while keeping every support point covered.

Theorem 4 (Conflict-Free 1-Step $\mathcal{A}_0^2$ Doubleton Tie-Break Exchanges on $Q_4$)

At depth $D = 32$ on $G_{4, p}$:

1. Deterministic Conflict-Free $2$-Row-for-$2$-Column Tie-Break Rule: Let $g$ be a doubleton group with $R(g) = 2$ columns $\operatorname{Cols}(g) = \{u_1(g), u_2(g)\}$ (weight $d = \frac{1}{4}$) and three neighboring rows $\{y_0, y_1, y_2\}$. Say that $g$ belongs to the cavity event $\mathcal{A}_1(y_0) \setminus \mathcal{A}_0(y_0)$ around $y_0$ if both external rows $y_1, y_2$ away from $y_0$ satisfy $y_1 \in \mathcal{A}_0^2$ and $y_2 \in \mathcal{A}_0^2$ (that is, for each $j \in \{1, 2\}$, both neighboring groups $h_{j, 1}, h_{j, 2}$ of $y_j$ other than $g$ belong to $\mathcal{A}_0(y_j)$). For every such doubleton group $g$:

   • Because $h_{j, 1}, h_{j, 2} \in \mathcal{A}_0(y_j)$ are never deleted in rounds $1, \ldots, 32$, neither $y_1$ nor $y_2$ is covered by $h_{j, 1}, h_{j, 2}$. Thus when $g$ is withheld, $y_1$ and $y_2$ remain uncovered through time $31$.
   • In the full peeling process on $G_{4, p}$:
     • Case 1a ($g \notin \mathcal{D}_{32}$ is undeleted after round $32$): This occurs whenever $y_0$ remains uncovered at time $31$ (so $g$ has $3 > R(g) = 2$ uncovered neighboring rows $\{y_0, y_1, y_2\}$ at every time $0, \ldots, 31$). Then $g$'s $2$ columns $\{u_1(g), u_2(g)\}$ are already in $V_{\mathrm{base}}$.
     • Case 1b ($g \in \mathcal{D}_{32}$ is deleted during rounds $1, \ldots, 32$): This occurs precisely when $y_0$ is covered at some time $t_0 \in \{0, \ldots, 31\}$, leaving $g$ with exactly $2 = R(g)$ uncovered neighboring rows $\{y_1, y_2\}$ at time $t_0$, so the baseline rule deletes $g$ at round $t_0 + 1 \in \{1, \ldots, 32\}$ and adds the $2$ rows $\{y_1, y_2\}$ to $U_{\mathrm{base}}$. We then perform the 1-step doubleton tie-break exchange $$U_{\mathrm{piv}} = U_{\mathrm{base}} \setminus \{y_1, y_2\}, \qquad V_{\mathrm{piv}} = V_{\mathrm{base}} \cup \{u_1(g), u_2(g)\}. \tag{5.1}$$
   • Validity, Disjointness, and Exact Zero Cost ($-2 \text{ rows} + 2 \text{ columns} = 0$):
     • Disjointness: For each $j \in \{1, 2\}$, the two groups $h_{j, 1}, h_{j, 2} \in \mathcal{A}_0(y_j)$ have $R(h_{j, r}) \in \{0, 1\}$, so $g$ is the unique $R = 2$ group incident to $y_1$ and the unique $R = 2$ group incident to $y_2$. Thus the removed row pairs $\{y_1(g), y_2(g)\}$ (as well as the column pairs $\operatorname{Cols}(g) = \{u_1(g), u_2(g)\}$) are pairwise disjoint across all tie-break groups $g$.
     • Exact Zero Net Cost: In Case 1b, $y_1, y_2 \in U_{\mathrm{base}}$ and $\{u_1(g), u_2(g)\} \cap V_{\mathrm{base}} = \emptyset$ (since $g \in \mathcal{D}_{32}$ was deleted), so removing the $2$ rows $\{y_1, y_2\}$ from $U_{\mathrm{base}}$ ($-2$) and adding the $2$ columns $\{u_1(g), u_2(g)\}$ to $V_{\mathrm{base}}$ ($+2$) preserves the exact residue cover size: $$|U_{\mathrm{piv}}| + |V_{\mathrm{piv}}| = |U_{\mathrm{base}}| - 2 + |V_{\mathrm{base}}| + 2 = |U_{\mathrm{base}}| + |V_{\mathrm{base}}|. \tag{5.2}$$
     • Complete Edge Coverage of Removed Rows $y_1, y_2$: For each $j \in \{1, 2\}$, every support point $(u, y_j) \in \mathcal{P}_p(Q_4)$ on row $y_j$ belongs either to $g$ (in which case $u \in \{u_1(g), u_2(g)\} \subseteq V_{\mathrm{piv}}$ is now covered by $V_{\mathrm{piv}}$) or to one of $h_{j, 1}, h_{j, 2} \in \mathcal{A}_0(y_j)$ (which either has $R = 0$ [no points] or has $R = 1$ and is undeleted, so its unique column lies in $V_{\mathrm{base}} \subseteq V_{\mathrm{piv}}$). Thus $(U_{\mathrm{piv}}, V_{\mathrm{piv}})$ is a valid residue vertex cover of the exact same size as $(U_{\mathrm{base}}, V_{\mathrm{base}})$, and in both Case 1a and Case 1b all columns of $g$ lie in $V_{\mathrm{piv}}$, giving the expanded cavity event $\mathcal{A}_1(y_0)$ probability $$p_{\mathcal{A}_1} = p_{\mathcal{A}_0} + d\,p_{\mathcal{A}_0}^4 \ge \mathbf{0.4309975668} \qquad (\text{up from } p_{\mathcal{A}_0} = z + o\,r_{31}^2 \ge 0.4229941154). \tag{5.3}$$

2. Certified Exposed Density & $18.5\times$ Margin Expansion on $V = y - x$: Define the expanded certified exposed row set $U_{\mathrm{exp}, 1}$ to consist of all rows $y_0$ (whose depth-$34$ incidence neighborhood in $G_{4, p}$ is a tree) having one neighboring group $g^*$ with $R(g^*) = 1$ whose two external cavity rows are covered at time $31$ when $g^*$ is withheld (probability $o(1 - r_{31})^2$) and whose other two neighboring groups $g', g''$ belong to the cavity event $\mathcal{A}_1(y_0)$ (probability $p_{\mathcal{A}_1}^2$). Then every $y_0 \in U_{\mathrm{exp}, 1}$ belongs to $U_{\mathrm{piv}}$ and has at most one private column outside $V_{\mathrm{piv}}$ (namely the unique column of $g^*$), with certified density $$\rho_{32}^{(1)} = 3o(1 - r_{31})^2 p_{\mathcal{A}_1}^2 \ge \mathbf{0.0715324526} \qquad (\text{up from } \rho_{32} \ge 0.0689004614). \tag{5.4}$$ Applying the $S_4 \times S_5 \times S_5$ whole-fiber diagonal exchange theorem of no_three_in_line_11 [2] (Section 9.2 and Section 10) along $V = y - x$ yields, for all sufficiently large primes $p$ and every collinear-free subset $S \subseteq \mathcal{P}_p(Q_4)$ in a $2p \times 2p$ periodic window: $$\frac{|S|}{2p} \le C_{32} - \left(\frac{1}{2}\bigl(\rho_{32}^{(1)}\bigr)^2 - \frac{1}{12}\bigl(\rho_{32}^{(1)}\bigr)^3\right) + O(p^{-1/2}) \le \mathbf{1.79980804 + O(p^{-1/2}) < 1.8}, \tag{5.5}$$ expanding the $2p \times 2p$ margin below $1.8$ from $1.04 \times 10^{-5}$ to $1.92 \times 10^{-4}$ ($18.5\times$ wider margin).

Proof of Theorem 4

1. Fix a doubleton group $g$ ($R(g) = 2$) with neighboring rows $\{y_0, y_1, y_2\}$ such that $y_1, y_2 \in \mathcal{A}_0^2$ away from $g$. For each $j \in \{1, 2\}$, let $h_{j, 1}, h_{j, 2} \in \mathcal{A}_0(y_j)$ be the other two neighboring groups of $y_j$. By definition of the cavity event $\mathcal{A}_0(y_j)$, each $h_{j, r}$ either has $R(h_{j, r}) = 0$ (no columns) or has $R(h_{j, r}) = 1$ with both of its external cavity rows uncovered at time $31$ when $h_{j, r}$ is withheld; in a tree neighborhood, withholding $h_{j, r}$ is only more permissive for its external subtrees, so both external rows of $h_{j, r}$ remain uncovered through time $31$ in the full peeling process, $h_{j, r}$ has $\ge 2 > R(h_{j, r}) = 1$ uncovered neighboring rows at every round $1, \ldots, 32$, $h_{j, r} \notin \mathcal{D}_{32}$ is never deleted, and its unique column (when $R = 1$) belongs to $V_{\mathrm{base}}$. Consequently, neither $y_1$ nor $y_2$ is ever covered by $h_{j, 1}, h_{j, 2}$, so $y_1, y_2$ remain uncovered unless $g$ itself is deleted (which happens at round $t_0 + 1 \le 32$ if and only if $y_0$ is covered at time $t_0 \le 31$, leaving $g$ with the $2 = R(g)$ uncovered rows $\{y_1, y_2\}$). Whenever $g \in \mathcal{D}_{32}$ is deleted, $g$'s deletion places $\{y_1, y_2\}$ into $U_{\mathrm{base}}$ and omits $\operatorname{Cols}(g) = \{u_1(g), u_2(g)\}$ from $V_{\mathrm{base}}$. Since $h_{j, 1}, h_{j, 2}$ have $R \le 1$, $g$ is the unique $R = 2$ group incident to $y_1$ and to $y_2$, so $\{y_1(g), y_2(g)\}$ are disjoint across all such $g$. Replacing $\{y_1, y_2\} \subseteq U_{\mathrm{base}}$ by $\{u_1(g), u_2(g)\} \subseteq V_{\mathrm{piv}}$ removes $2$ rows and adds $2$ columns ($|U_{\mathrm{piv}}| + |V_{\mathrm{piv}}| = |U_{\mathrm{base}}| + |V_{\mathrm{base}}|$), and covers every support point $(u, y_j)$ on $y_1, y_2$: if $(u, y_j)$ comes from $g$, then $u \in \{u_1(g), u_2(g)\} \subseteq V_{\mathrm{piv}}$; if $(u, y_j)$ comes from $h_{j, r} \in \mathcal{A}_0(y_j)$ with $R(h_{j, r}) = 1$, then $h_{j, r} \notin \mathcal{D}_{32}$ is undeleted so $u \in \operatorname{Cols}(h_{j, r}) \subseteq V_{\mathrm{base}} \subseteq V_{\mathrm{piv}}$. On the independent depth-$34$ cavity tree around $y_0$, the cavity subtrees of $h_{j, 1}, h_{j, 2}$ away from $y_j$ have depth $31$ (matching the cavity depth $D - 1 = 31$), so the cavity event $g \in \mathcal{A}_1(y_0) \setminus \mathcal{A}_0(y_0)$ has exact probability $d(p_{\mathcal{A}_0}^2)^2 = d\,p_{\mathcal{A}_0}^4$ and is disjoint from $\mathcal{A}_0(y_0)$ ($R \in \{0, 1\}$ vs. $R = 2$), giving (5.3).

2. Let $y_0 \in U_{\mathrm{exp}, 1}$, with $g^*$ ($R(g^*) = 1$, both external cavity rows covered at time $31$ when $g^*$ is withheld) and $g', g'' \in \mathcal{A}_1(y_0)$. Because $g^*$ has $0$ uncovered external rows at time $31$ even with $y_0$ held uncovered, $g^*$ has $\le 1 = R(g^*)$ uncovered neighboring rows at time $31$, so $g^* \in \mathcal{D}_{32}$ is deleted by round $32$ and places $y_0 \in U_{\mathrm{base}}$. Moreover, $y_0$ is never removed by any tie-break exchange (5.1): a removed row $y_j$ in (5.1) has its three neighboring groups of sizes $R = 2$ and $R \in \{0, 1\}$ in $\mathcal{A}_0(y_j)$ (whose external cavity rows are uncovered at time $31$), whereas $y_0$ is incident to $g^*$ whose two external cavity rows are covered at time $31$ (so $g^* \notin \mathcal{A}_0(y_0)$ and $R(g^*) = 1 \ne 2$). Hence $y_0 \in U_{\mathrm{piv}}$. Finally, every column of $g'$ and $g''$ belongs to $V_{\mathrm{piv}}$ by part 1, so the only support point on $y_0$ whose column can lie outside $V_{\mathrm{piv}}$ is the single column of $g^*$. (In verify_moving_pole_and_two_slope_q4.py, we also audit the full-graph implementation at $p \in \{251, 503\}$ as a finite-graph control, verifying 0 uncovered points, exact cover size preservation, and $\le 1$ private column for every certified row.) Applying Section 9.2 and Section 10 of [2] with $\rho_{32}$ replaced by $\rho_{32}^{(1)}$ yields (5.4)–(5.5). $\square$

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6. Concluding Remarks and Scope

Our results establish two structural principles for prime-field No-Three-in-Line constructions:

1. For Two-Digit Conic-Group Sources on $G_{p^2}$, while no_three_in_line_50 v2 [1] proves a multi-direction $(4/3 + o(1))N$ capacity bound using an unbounded inventory of short-triple slopes, our Moving-Pole Fiber Identity and General Slope Degeneracy Identity (Theorem 1) determine the exact single- and two-direction capacity laws: when $\alpha \equiv (1 + b)^{-1} \pmod p$ ($\alpha \equiv 2^{-1}, -1 \pmod p$), the wrap coordinate is constant on each low-digit fiber of $X + bY$, and the exact Vieta-coupled Kloosterman integral obstructs $S_0 \cup S_1$ below $1.465N < 1.8N$ on a single slope across all $\beta_0 \in \mathbb{F}_p$ for each fixed integer offset $\Delta\beta \in \mathbb{Z}$ (Theorem 2); when $\alpha = 1$ (pure translations $\pi_0(d) = [d + c_0]_p, \pi_1(d) = [d + c_0 + \Delta c]_p$ with fixed integer offset $\Delta c \in \mathbb{Z}$), the linear Kloosterman term vanishes on every Euclidean slope simultaneously, and our grouped $W$-plane companion coprimality and exact phase polynomials in $\mathbb{Q}[x_0]$ obstruct all fixed-offset translation pairs below $1.790N < 1.8N$ across all $c_0 \in \mathbb{F}_p$ using only two directions ($X$ and $X + Y$, Theorem 3).
2. For Periodic Windows on $Q_4(u) = u^4 - u$, our 1-step $\mathcal{A}_0^2$ doubleton tie-break exchanges replace $2$ covered rows by $2$ covered columns ($-2 + 2 = 0$) while keeping all incident points covered by the column cover, improving the $2p \times 2p$ single-slope obstruction constant of no_three_in_line_11 [2] along $V = y - x$ by expanding the margin below $1.8$ by $18.5\times$ (Theorem 4).

As noted in the Abstract, these theorems are support-restricted obstructions for the specified algebraic families and do not resolve the general unrestricted $1.8n$ problem.

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