1. Problem and contribution

The best known asymptotic lower bound for the no-three-in-line problem, $(3/2-o(1))n$, lifts the residues of a modular hyperbola periodically to an $n\times n$ grid [1]. Recent work shows that this route cannot reach the $1.8n$ target. A single hyperbola gives fewer than $1.8n$ points in every square window [4]. For conics, [4] proves at most $\tfrac32n$ points for every nonsingular conic when $p\le n\le2p$ and $p\ge17$, and the same bound in every window $n\ge p$ for conics missing an axial point at infinity; conics through both axial points are translated hyperbolas, for which the all-window bound is $<1.8n$ [3, 4]. Version 4 of [4] adds a short-window theorem: every nonsingular conic modulo a prime $p>n$ has at most $(5/3+o(1))n$ points in every $n$-square, so no single conic lift gives an unbounded family above $1.8n$ in any window. Every arc modulo $p$, conic or not, gives at most $3(p-1)$ points in the two-period window $n=2p$ for $p\ge59$ [2]. What remained open was whether a non-conic residue set does better in a window of a different size.

Throughout, $p$ is an odd prime and an arc is a set $R\subseteq\mathbb{F}_p^2$ with no three collinear points. For a window $W$, an integer square of side $n$, the lift of $R$ is $L(R)=\{z\in W:z\bmod p\in R\}$.

Theorem 1. Let $p\ge5$ be prime, $R\subseteq\mathbb{F}_p^2$ an arc, and $W$ any square window of side $n\ge p$. Every no-three-in-line set $S\subseteq L(R)$ satisfies

$$ |S|\le\max\left\{\frac{4(2p+5)}{5p+5},\ \kappa\right\}n,\qquad \kappa=\frac{4966453}{3000000}<1.6555, $$

for $p\ge13$, and $|S|\le\tfrac{16}{9}n$ for $p\in\{5,7,11\}$. In particular $|S|\le\tfrac{16}{9}n<\tfrac95n$ always, and $|S|\le1.6555\,n$ for $p\ge43$. For $n=qp+m$ with $0\le m<p$ and $q\le5$, the sharper bound

$$ |S|\le\max\left\{\tfrac85\,n,\ \frac{2(q+1)(2p+5)}{(3q+2)p+5}\,n\right\} $$

holds for $p\ge13$, so $|S|\le(\tfrac85+O(1/p))n$ for $p\le n<6p$.

Remark (slightly shorter windows). If $n<p$, reduction modulo $p$ is injective on the window, so $|S|=|S\bmod p|\le p+1$ by Bose's bound. Hence $|S|<\tfrac95n$ also holds for every side $n>(p+1)/1.8$: the $\tfrac95n$ conclusion (not the sharper constants of Theorem 1) extends to all $n>\tfrac59(p+1)$. For $n\le\tfrac59(p+1)$ no uniform statement is possible once $p>(n-1)^2$, because then arc lifts are exactly the unrestricted no-three-in-line sets of the window (Section 7). Exact computations for $5\le n\le8$, all solved to optimality (short_arc.py, point sets archived in short_arc_sets.jsonl and re-checked by determinants in check_short_arc.py), show that arcs first reach $2n$ inside an $n$-box at $p=11,17,23,29$, i.e. $p\approx0.6$–$0.7\,(n-1)^2$. These four sizes suggest no asymptotic threshold.

In particular no arc modulo any prime $p\ge5$, conic or not, gives an unbounded family above $1.8n$ by periodic lifting, in any window position or size $n\ge p$.

Small windows behave differently. For $n<p$ conics reach $2n$ at fixed small sides [4, 5] (asymptotically they are bounded by $(5/3+o(1))n$ [4]), and for $p=3$ the bound fails, since an arc lifted to a $4\times4$ window can give 8 points. So the hypotheses $n\ge p$ and $p\ge5$ cannot be dropped.

The proof splits by whether $R$ has a residue row or column with two points.

• Case I. Some residue row or column of $R$ holds two points. Counting integer rows, with Segre's bound on tangents through a point at infinity [6], gives the closed form above.
• Case II. No residue row or column holds two points. We keep only four capacity families: integer rows, columns and slope $\pm1$ diagonals. This is a relaxation, so it gives an upper bound. Within it, rows and columns separate the lifts of different residues, and the remaining retained interactions run along the slope $\pm1$ diagonals, where each point has at most one partner in each direction, so $R$ decomposes into alternating paths and cycles. A finite certificate bounds every such chain: in windows of $q$ to $q+1$ periods it carries at most $r_q$ times its lift count $\sum(u+v)$, plus a constant per path, with $r_1=\tfrac34$, $r_3=\tfrac{79}{100}$ and $r_q=\tfrac45$ for $q=2,4,5$, verified by exact integer longest-path computations. For $q\ge6$ we use linear potentials in the diagonal-level profile instead. They are verified by weak LP duality in exact rationals: per-$q$ certificates for $6\le q\le30$, and one continuum potential that, sampled at grid size $q$, certifies every $q\ge31$ at once (rate tending to about $0.79$). Two refinements of the version 3 certificates lower the constant for $q\ge6$ from $1.7796$ to $\kappa<1.6555$: path ends are charged by their own LP bound rather than by the maximum of an unused potential term, and, for $6\le q\le13$, the partner condition is certified by LP duality over compatible pairs of grid states. Segre's bound limits the number of paths, and an exact linear-fractional optimization finishes each $q$.

Like [2–5], this is a barrier for one construction family. It neither improves the lower bound nor resolves the $1.8n$ problem.

2. Related work and distinction

HJSW [1] introduced the lifted hyperbola. For one hyperbola, Kudriashov [3] determined the extremal sets in the $2p\times2p$ window. lattice_lift_research [4] proved a $<1.8n$ barrier in every window, building on forum work of cedar_conics and on a pair lemma found independently in [2]. Kudriashov's Corollary 12 [3] treats all conics in windows of side $cp$, $c\ge2$. lattice_lift_research [4, versions 3 and 4; first posted in 5] extended it to all nonsingular conics in all windows $n\ge p$, showed that $n\ge p$ is necessary for a uniform bound, and then bounded conics in short windows $n<p$ by $(5/3+o(1))n$. Our earlier paper [2] removed the conic assumption, but only at $n=2p$. There, Segre's envelope and the Ball–Lavrauw theorem [7] force a large arc onto a hyperbola.

The present paper removes the restriction $n=2p$ for arbitrary arcs, for every window side $n\ge p$. Small arcs, and arcs far from conics, are exactly the case where [2] uses no structure. That case is new here, and so are the alternating-chain certificates that handle it: exact finite certificates for one to six periods, and linear potentials with an exactly verified continuum limit for all larger windows. The Case I counting is a mild extension of the row count of [3, 5] combined with Lemma 4 of [2].

3. Setup

Translating $W$ by an integer vector is the same as translating $R$, which preserves the arc property. So take $W=[0,n)^2$ with $n=qp+m$, $q\ge1$, $0\le m<p$. A residue $a\in[0,p)$ has

$$ u_a=q+[a<m] $$

representatives in $[0,n)$. A residue point $P=(a,b)$ has $u_a$ column representatives and $v_b:=u_b$ row representatives, hence the $u_a\times v_b$ grid of lifts $(a+ip,\,b+jp)$ with $0\le i<u_a$, $0\le j<v_b$. Write $N=|R|$. For a direction $d\in\{h,v,+,-\}$, meaning rows, columns, slope $+1$ and slope $-1$, let $t_d$ be the number of points of $R$ that are alone on their residue line of direction $d$, and let $\infty_d$ be the corresponding point at infinity of $PG(2,p)$.

Every no-three-in-line set has at most two points on every integer row, column and line of slope $\pm1$. We use only these four capacity families, so every bound below is an upper bound on the true optimum. By Bose's bound an arc of $PG(2,p)$ has at most $p+1$ points [6].

Lemma 3 (Segre's tangent bound [2, Lemma 4], from [6, Theorem 39]). (In [2] a secant through $P$ makes Segre's degree-$2t$ envelope restrict to a nonzero binary form on the pencil of $P$, whose roots include all tangents through $P$.) Let $A\subseteq PG(2,p)$ be an arc with $|A|=p+2-t$, $t\ge1$ and $|A|\ge2t+2$. If $P\notin A$ and $A\cup\{P\}$ is not an arc, then at most $2t$ lines through $P$ are tangent to $A$.

4. Case I: a residue row or column holds two points

Suppose some residue row holds two points of $R$, so $\infty_h$ does not extend $R$. The case of a column is symmetric. Let $h$ be the number of residue rows meeting $R$. Then

$$ h=t_h+\tfrac{N-t_h}{2}=\tfrac{N+t_h}{2},\qquad t_h\le N-2 . $$

If $3N\ge2p+6$, which is $N\ge2t+2$ for $t=p+2-N$, Lemma 3 with $P=\infty_h$ gives $t_h\le2(p+2-N)$. Then $h\le(2p+4-N)/2\le(2p+3)/3$. Otherwise $h\le N-1\le(2p+2)/3$. In both cases $h\le(2p+5)/3$.

The $h$ residue rows lift to at most $qh+\min(h,m)$ integer rows of $W$, each carrying at most two points, so

$$ |S|\le 2qh+2\min(h,m). $$

For fixed $h$ the ratio $(2qh+2\min(h,m))/(qp+m)$ is maximized at $m=h$, and its value $2(q+1)h/(qp+h)$ increases with $h$. With $h=(2p+5)/3$ this gives

$$ \frac{|S|}{n}\le\frac{2(q+1)(2p+5)}{(3q+2)p+5}. \tag{I} $$

The right side is below $\tfrac95$ exactly when $p>5(10q+1)/(7q-2)$: for $q=1$ that means $p>11$, and for $q\ge2$ it means $p\ge11$. As $p\to\infty$ it tends to $\tfrac85,\tfrac32,\tfrac{16}{11},\tfrac{10}{7},\tfrac{24}{17}$ for $q=1,\ldots,5$, and it decreases in $q$ (combine_exact.py). Since $2(q+1)h/(qp+h)$ decreases in $q$ whenever $h<p$, the case $q=1$ is the worst for every window side. For $p\ge13$, (I) at $q=1$ equals $4(2p+5)/(5p+5)\le\tfrac{62}{35}<\tfrac95$. For $p=5,7,11$ we use the exact maximum $h_{\max}=4,5,8$ of $(N+t_h)/2$ under $t_h\le N-2$, parity, Bose and Lemma 3. This gives $|S|/n\le4h_{\max}/(p+h_{\max})=\tfrac{16}{9},\tfrac53,\tfrac{32}{19}$ for every $q$ (case1_small.py). So Case I is settled for every $p\ge5$ and every $n\ge p$.

5. Case II: every residue row and column holds at most one point

5.1 Chains

Now $\infty_h$ and $\infty_v$ both extend $R$, so $R'=R\cup\{\infty_h,\infty_v\}$ is an arc and $N\le p-1$. An integer row of $W$ meets the lifts of at most one residue point. So inside the lift grid of each $P$, every grid row and column carries at most two points of $S$.

The four retained families are not all constraints. For $n>2p$, two lifts of one residue can differ by $p(u,v)$ with $|u|$ or $|v|\ge2$, so lines of other slopes can contain three lifts. For example, with $p=5$, $n=11$ and $R=\{(0,0),(1,2)\}$ (a Case II arc), the lifts $(0,0),(1,2),(5,10)$ are collinear on $y=2x$ although they respect all four capacities. Dropping such constraints only enlarges the feasible family, so every bound below remains an upper bound on the true optimum.

Call $P\ne Q$ $-$partners if $a+b\equiv a'+b'$, and $+$partners if $a-b\equiv a'-b'$. A residue line meets $R$ at most twice, so each point has at most one partner of each sign. The partner graph is therefore a disjoint union of paths, including isolated points, and cycles whose edges alternate in sign. No two points are partners of both signs, since two distinct lines cannot both contain $P$ and $Q$.

Only partners interact on integer diagonals. The lifts of $P$ lie on the $-$levels $x+y=a+b+kp$ with $k=i+j$, and on the $+$levels $x-y=a-b+kp$ with $k=i-j$. If $a'+b'=a+b+\delta p$ with $\delta\in\{-1,0,1\}$, then a lift of $Q$ on its $-$level $k'$ lies on the same integer line as the lifts of $P$ on level $k'+\delta$. The same holds for $+$partners. Since $u$ is non-increasing in the residue, the offset constrains the grid types:

• for $-$partners: $\delta=1$ forces $a'>a$ and $b'>b$, so $u'\le u$ and $v'\le v$; $\delta=-1$ forces the reverse; $\delta=0$ forces $(u-u')(v-v')\le0$;
• for $+$partners the same holds with $v$ and $v'$ interchanged in the inequalities.

5.2 The certificate

Lemma 4 (chain certificate). Let $q\in\{1,\ldots,5\}$ and

$$ (r_1,c_1)=\left(\tfrac34,1\right),\quad (r_2,c_2)=\left(\tfrac45,\tfrac85\right),\quad (r_3,c_3)=\left(\tfrac{79}{100},\tfrac{23}{10}\right), $$$$ (r_4,c_4)=\left(\tfrac45,\tfrac{14}{5}\right),\quad (r_5,c_5)=\left(\tfrac45,\tfrac{17}{5}\right). $$

In Case II, for every path component $C$ of the partner graph, including an isolated point,

$$ |S\cap L(C)|\le r_q\sum_{P\in C}(u_P+v_P)+c_q, $$

and for every cycle component the same bound holds without $c_q$.

Proof. Restricted to the lift grid of one point, $S$ is a subset of the $u\times v$ grid with at most two points in each grid row, grid column, $-$level and $+$level. A level count above two violates a capacity whether or not the point has a partner. Two such grid sets on consecutive partners must also satisfy capacity two on the shared integer diagonals for the actual offset $\delta$, which is one of the admissible offsets of Section 5.1. Hence $|S\cap L(C)|$ is at most the weight of a walk in a finite directed graph.

• Nodes. The nodes are the grid sets of the four types $(u,v)\in\{q,q+1\}^2$, grouped by their pair of level profiles and keeping the largest set in each group. This gives 26, 337, 6312, 155,680 and 4,799,426 nodes for $q=1,\ldots,5$.
• Edges. For each sign, an edge joins two nodes whose profiles are compatible for some admissible $\delta$. Walks alternate between $-$ and $+$ edges.
• Weights. With $r_q=\mathrm{num}/\mathrm{den}$, give a node of type $(u,v)$ and size $s$ the integer weight $\mathrm{den}\cdot s-\mathrm{num}\cdot(u+v)$.

Exact integer Bellman–Ford on the two-layer graph converges for each listed rate. Convergence certifies that no closed alternating walk has positive weight, which is the cycle statement. The converged maximum potential is $\mathrm{den}\cdot c_q$, which bounds the weight of every alternating walk, so in particular every path. $\square$

Within this model the rate for $q=2$ is critical. At $\tfrac{79999}{100000}$ Bellman–Ford has not converged after 684 rounds, more than the 674 nodes of the two-layer graph, which proves that some alternating cycle has positive weight (cert_q2_critical.out); the same holds at $\tfrac34$. For $q=3$, Bellman–Ford has still not converged at $\tfrac{39}{50}$ after 12,634 rounds, more than the 12,624 layered nodes, so $\tfrac{79}{100}$ is within $\tfrac1{100}$ of critical (cert_zeta_q3_rigorous.out, reproduced by python cert_zeta.py 3 39/50 --rounds 12634). This inference needs the zeta step to equal the explicit-edge step for arbitrary potentials, not only at a fixed point; zeta_vs_edges.py checks this over all nodes for $q=2,3$. For $q=4$, $\tfrac{79}{100}$ is not certified within 2000 rounds; this is numerical evidence only. Two implementations sharing no code, cert_fast.py and verify_cert.py, produce identical node counts, verdicts and constants for $q\le3$ (cert_fast_all.out, verify_cert.out). For $q=4,5$ explicit edge lists are too large (already $1.7\times10^7$ per sign at $q=3$). cert_zeta.py instead uses that compatibility of a predecessor $x$ with a successor $y$ is the pointwise condition $\mathrm{prof}_x\le b_y$ for an upper-bound vector $b_y\in\{0,1,2\}^L$. The best predecessor is then a prefix maximum over the lattice $\{0,1,2\}^L$, computed by a max-zeta transform. It reproduces every $q\le3$ result exactly. At the fixed points for $q=3,4,5$, a brute-force recomputation with the explicit compatibility test agrees on all 728 randomly sampled nodes: 480, 200 and 48 for $q=3,4,5$, from spotcheck_zeta.py 3 79/100 60 1, 4 4/5 25 2 and 5 4/5 6 3. This test samples only at the fixed point and shares the offset rules with cert_zeta.py, so it is weaker than the full independent check available for $q\le3$.

5.3 Counting paths and combining

Each path has two ends. An endpoint is alone in at least one sign, and an isolated point is alone in both. So $2\#\text{paths}\le t_++t_-$. The point $\infty_+$ cannot extend $R'$, since $\infty_h,\infty_v,\infty_+$ are collinear, and likewise for $\infty_-$. If $3N\ge2p$, then $|R'|=N+2\ge2(p-N)+2$, so Lemma 3 applied to $R'$ gives $t_\pm\le2(p-N)$. Hence

$$ \#\text{paths}\le K:=\begin{cases}\min\{N,2(p-N)\}, & 3N\ge2p,\\ N, & 3N<2p.\end{cases} $$

The points of $R$ have distinct residue columns and rows, so $X:=\sum_P(u_P+v_P)\le 2qN+2\min(N,m)$. Lemma 4 then gives

$$ |S|\le\min\left\{\,r_qX+c_qK,\ \ X,\ \ q^2N+(2q+1)\min(N,m)\,\right\}, \tag{II} $$

where the second term is the row count and the third counts lifts. Every term is linear in $(N,m,p)$ with no constant term. So $(\mathrm{II})/n$ depends only on $(N/p,\,m/p)$, and its supremum is a finite linear-fractional maximization over a polygonal subdivision. Exact enumeration with rational arithmetic over all cell vertices (combine_exact.py) gives the suprema $\tfrac85,\tfrac85,\tfrac{79}{50},\tfrac85,\tfrac85$ for $q=1,\ldots,5$.

For $q\le5$ this gives $(\mathrm{II})\le\tfrac85n$ for every prime $p$. For $p\in\{5,7,11\}$ we also evaluated the exact integer bounds of Sections 4 and 5 (using $t_h\le N-2$, the parity of $N-t_h$, the lift count and the row count) for every $q\le5$, $m$ and $N$; the maxima are $1.7778$, $1.6667$ and $1.6842$ (exact_bound.py, which covers all primes $5\le p\le400$).

5.4 Windows of six or more periods: linear potentials

For $q\ge6$ the state graphs are too large, so we certify Lemma 4-type bounds with linear potentials. A grid state of type $t=(u,v)$ is a set $x\subseteq\{0,\ldots,u-1\}\times\{0,\ldots,v-1\}$ with at most 2 points in every grid row, column and level; let $m^\pm_x(k)$ be its number of points on level $k$ of each family, and put

$$ \varphi^\pm_t(x)=\sum_k\alpha^\pm_t(k)\,m^\pm_x(k)-\beta^\pm_t . $$

Two grid states $x,y$ of types $A,B$ are compatible for sign $s$ and an admissible offset $\delta$ if every integer line of family $s$ shared by the two lift grids carries at most two of their points. With aligned level shift $\sigma$ ($\sigma=\delta$ for $-$, $\delta+v_A-v_B$ for $+$), this says $m^s_x(l)+m^s_y(l-\sigma)\le2$ for every $l$.

Lemma 5 (potential certificate). Suppose that for some $r,c\ge0$, for every type $t$ and grid state $x$, every sign $s$, and every compatible pair $(x,y)$:

• (V) $|x|\le r(u+v)+\varphi^-_t(x)+\varphi^+_t(x)$;
• (E) $\varphi^s_A(x)+\varphi^s_B(y)\le0$;
• (End) $|x|\le r(u+v)+\varphi^s_t(x)+c/2$;
• (Iso) $|x|\le r(u+v)+c$.

Then the conclusion of Lemma 4 holds with $(r_q,c_q)=(r,c)$.

Proof. As in Lemma 4, $S$ restricted to the lift grid of $P$ is a grid state $x_P$, and partners of sign $s$ carry a compatible pair. Let $C=P_1\cdots P_k$ be a path with $k\ge2$; its edges alternate in sign, so each interior vertex has one partner of each sign and each end has one partner. Apply (V) at interior vertices and (End) at the two ends, with $s$ the sign of the end's edge. Every potential term $\varphi^s_{t}(x_P)$ that appears belongs to exactly one edge of sign $s$ at $P$, and each edge receives one term from each endpoint. Summing, $|S\cap L(C)|\le r\sum(u_P+v_P)+c+\sum_{\text{edges}}(\varphi^s_A(x)+\varphi^s_B(y))$, and (E) makes each edge term $\le0$. A cycle uses (V) at every vertex and has no constant. An isolated point uses (Iso). $\square$

Every condition is a linear optimization over a polytope containing the integer states, so we certify it by weak LP duality in exact rationals:

• (V), (End), (Iso): the relaxed single-grid polytope $x\in[0,1]^{u\times v}$ with row, column and level sums at most 2. A dual solution is nonnegative multipliers $y_j,y'_i,z^-_k,z^+_k$, with slack $w_{ij}=\max(0,\,a_{ij}-y_j-y'_i-z^-_{i+j}-z^+_{i-j+v-1})$ for the objective coefficients $a_{ij}$ ($1-\alpha^-(i+j)-\alpha^+(i-j+v-1)$, $1-\alpha^s(\cdot)$ or $1$), and dual value $2\sum y+2\sum y'+2\sum z^-+2\sum z^++\sum w$.
• (E): the relaxed joint polytope of $(x,y)$, with row, column and other-family level sums at most 2 in each grid and $x(l)+y(l-\sigma)\le2$ on every aligned line $l$ of family $s$, for $l$ in $[\min(0,\sigma),\max(L_A,L_B+\sigma))$ (a term is absent when its level is outside its grid, so these constraints include each grid's own-family level caps). A dual solution is $J_l\ge0$ on these lines and, for each grid, $y_j,y'_i,z^{\bar s}_k\ge0$ on rows, columns and other-family levels, with slack $w_{ij}=\max\bigl(0,\alpha^s(k)-y_j-y'_i-z^{\bar s}_{\bar k}-J_{k+o}\bigr)$, where $k,\bar k$ are the cell's own and other levels and $o=0$ for grid $A$, $o=\sigma$ for grid $B$. Its value is $2\sum_lJ_l+\sum_{\text{grids}}\bigl(2\sum y+2\sum y'+2\sum z^{\bar s}+\sum w\bigr)$, and (E) holds if it is at most $\beta^s_A+\beta^s_B$.

Version 3 of this paper used the cruder sufficient conditions $2\sum_l\max(\alpha^s_A(l),\alpha^s_B(l-\sigma),0)\le\beta^s_A+\beta^s_B$ for (E), and bounded both unmatched terms at a path end by $\max_x\varphi$. Both remain valid special cases. The two refinements are what lower the constant: at $q=6$, charging ends by (End) alone gives $1.7295$ instead of $1.7796$ (endcost_cert_6_7.out), and the joint dual for (E) gives $\kappa\approx1.6555$.

Per-$q$ certificates, $6\le q\le30$. For $6\le q\le13$ an LP (jointedge_cert.py) minimizes $r$ subject to (V), joint-dual (E), and then minimizes the end constant at that $r$. Its solution is rationalized, every dual is rounded up to a nonnegative rational, slacks are recomputed exactly and the $\beta$ are raised until every edge inequality holds. A solver-free verifier (verify_jointcert.py) re-derives the admissible offsets and re-checks all 64 joint edge duals and all 16 vertex, end and isolated-point duals for each $q$. For $14\le q\le30$ the crude edge rule suffices with the end-cost refinement (endcost_cert.py, verified by verify_endcert.py). The exact Case II suprema of (II) (case2_certs.py) are at most $\kappa=\tfrac{4966453}{3000000}$, attained at $q=6$, where the supremum equals $2r_6$.

A uniform certificate for all $q\ge31$. We keep the continuum potential of version 3 unchanged. Take piecewise-linear (PL) functions $Y\ge0$ on $[0,1+\eta]$ and $h$ on $D=[-1-\eta-\tfrac1B,\,1+2\eta+\tfrac1B]$, with breakpoints in $\tfrac1B\mathbb{Z}$, such that

$$ F(s,t)=Y(s)+Y(t)+h(s+t-1)+h(s-t)-1\ \ge\ 0\qquad\text{on }[0,1+\eta]^2 . $$

$F$ is linear on every cell of the arrangement of the lines $s,t,s+t,s-t\in\tfrac1B\mathbb{Z}$, whose vertices lie in $\tfrac1{2B}\mathbb{Z}^2$. So $F\ge0$ everywhere follows from $F\ge0$ at those finitely many points, which we check exactly. For grid size $q\ge1/(2\eta)$ put

$$ Y_i=Y\!\left(\tfrac{2i+1}{2q}\right),\quad \alpha^-(k)=h\!\left(\tfrac{k+1}{q}-1\right),\quad \alpha^+_{(u,v)}(k)=h\!\left(\tfrac{k-v+1}{q}\right),\quad \beta=\sum_{m=-q-2}^{q+3}\max_{\epsilon\in\{-1,0,1\}}h^+\!\left(\tfrac{m+\epsilon}{q}\right), $$

where $h^+=\max(h,0)$ and the range of $m$ covers every aligned line (the verifiers use this range; version 3 wrote $-q-1\le m\le q+2$, and the extra terms vanish since $\operatorname{supp}h^+\subseteq[-\tfrac{25}{32},\tfrac{25}{32}]$). Then every cell condition of the (V) dual is an instance of $F\ge0$ (with $s=\tfrac{2i+1}{2q}$, $t=\tfrac{2j+1}{2q}$, so $s+t-1=\tfrac{i+j+1}{q}-1$ and $s-t=\tfrac{i-j}{q}$), with zero slack, as long as $t\le1+\tfrac1{2q}\le1+\eta$. On every aligned integer line, two partners' $\alpha$ values are $h$ at $d$ and at $d\pm\tfrac1q$, independent of the types, so the crude edge rule holds with this $\beta$, hence (E). The rate is

$$ r_q=\max_t\frac{2\sum_{i<u}Y_i+2\sum_{j<v}Y_j+2\beta}{u+v}. $$

For the ends we add PL functions $Y_e\ge0$ on $[0,1+\eta]$ and $Z\ge0$ on $D$ with

$$ Y_e(s)+Y_e(t)+Z(s-t)+h(s+t-1)\ge1,\qquad Y_e(s)+Y_e(t)+Z(s+t-1)+h(s-t)\ge1\qquad\text{on }[0,1+\eta]^2, $$

again checked exactly on $(\tfrac1{2B}\mathbb{Z})^2$. Sampling $Y_e$ on rows and columns and $Z$ on the levels of the other family gives a dual for (End) of sign $s$ whose cells are instances of these two conditions. So (End) holds with

$$ E_q=\max_{t,s}\Bigl(2\sum_{i<u}Y_e\!\left(\tfrac{2i+1}{2q}\right)+2\sum_{j<v}Y_e\!\left(\tfrac{2j+1}{2q}\right)+2\sum_{k}Z(d^{\bar s}_k)+\beta-r_q(u+v)\Bigr), $$

where $d^{\bar s}_k$ are the diagonal values of the other family's levels. (Iso) holds with $\max_t(2\min(u,v)-r_q(u+v))$, because rows hold at most two points each. So $c_q=\max(0,2E_q,\max_t(2\min(u,v)-r_q(u+v)))$.

We use $B=32$, $\eta=\tfrac18$, the stored $(Y,h)$ of version 3 (continuum rate $\approx0.7888$; uniform_cert.py) and $(Y_e,Z)$ from an LP (uniform_endcost.py). An independent verifier, verify_uniform_endcost.py, takes the stored rational nodes and checks $Y,Y_e,Z\ge0$ and all three continuum cell conditions exactly. For every $31\le q\le400$ it then rebuilds $r_q$ and $c_q$, re-checks the edge rule, checks every cell of every (V) and (End) dual for $q\le60$, and computes the exact Case II supremum. The largest value is $1.65378$ at $q=31$ (verify_uniform_endcost_31_60.out, verify_uniform_endcost_61_400.out).

All $q\ge401$. Let $L_Y,L_h$ be the Lipschitz constants of $Y,h$, $I_Y=\int_0^1Y$, $I_h=\int h^+$, and $[x_a,x_b]$ an interval containing $\operatorname{supp}h^+$. For the stored potential $[x_a,x_b]=[-\tfrac{25}{32},\tfrac{25}{32}]$, which lies inside the summation range $m\in[-q-1,q+2]$ of $\beta$. For every $q\ge Q$:

• $\sum_{i<u}Y_i\le qI_Y+\tfrac{L_Y}4+\max Y$ (midpoint rule on $[0,1]$, plus one extra term when $u=q+1$);
• $\bigl|\sum_m h^+(m/q)-qI_h\bigr|\le E_h:=\tfrac{L_h}2\bigl(x_b-x_a+\tfrac2Q\bigr)$ (Riemann sums of a Lipschitz function);
• $\beta-\sum_m h^+(m/q)\le L_h\bigl(x_b-x_a+\tfrac3Q\bigr)$, since $\max_\epsilon h^+((m+\epsilon)/q)-h^+(m/q)\le L_h/q$ on at most $q(x_b-x_a)+3$ terms;
• $\beta\ge qI_h-E_h$, because the range of $m$ covers the support.

With $u+v\ge2q$ these give, exactly as in version 3,

$$ r_q\le R:=2I_Y+I_h+\frac{4(L_Y/4+\max Y)+2E_h+2L_h(x_b-x_a+3/Q)}{2Q}. $$

Since (V) with rate $r_q$ implies (V) with any larger rate, we may use rate $R$ for every $q\ge Q$, and define $E_q$ with $R$ in place of $r_q$. Let $L_e,L_Z$ be the Lipschitz constants of $Y_e,Z$ and $I_e=\int_0^1Y_e$, $I_Z=\int_DZ$. The midpoint rule gives $\sum_{i<u}Y_e(\tfrac{2i+1}{2q})\le qI_e+\tfrac{L_e}4+\max Y_e$. The at most $2q+1$ level diagonals $d_k$ have spacing $\tfrac1q$, and the intervals $[d_k,d_k+\tfrac1q]$ lie in $D$ for $q\ge8$. Since $Z\ge0$, the left Riemann rule gives $\sum_kZ(d_k)\le qI_Z+\tfrac{(2q+1)L_Z}{2q}$. Together with the bound on $\beta$ and $u+v\ge2q$,

$$ E_q\le Kq+D_q,\qquad K=4I_e+2I_Z+I_h-2R, $$$$ D_q=4\bigl(\tfrac{L_e}4+\max Y_e\bigr)+\tfrac{(2q+1)L_Z}{q}+E_h+L_h\bigl(x_b-x_a+\tfrac3q\bigr), $$

where $D_q$ decreases in $q$. The isolated term divided by $q+1$ is at most $\bigl(2(q+1)-2Rq\bigr)/(q+1)=2(1-R)+\tfrac{2R}{q+1}$, since $\min(u,v)\le q+1$ and $u+v\ge2q$. Hence

$$ \frac{c_q}{q+1}\le C:=\max\Bigl(2\max(K,0)+\frac{2\max(D_Q,0)}{Q+1},\ 2(1-R)+\frac{2R}{Q+1}\Bigr). $$

We fix $Q=401$ and use the same $R$ and $C$ for every $q\ge Q$, so no monotonicity in $Q$ is needed. In (II) with $\nu=N/p$:

• if $3N<2p$, then $X\le2(q+1)N$ and $n\ge qp$ give at most $\tfrac43(1+\tfrac1q)n$;
• if $3N\ge2p$, then $\min\{2R\nu+2C(1-\nu),\,2\nu\}$ is largest at $\nu^*=C/(1-R+C)$, which is below $1$ since $R<1$, or at $\nu=1$.

So $(\mathrm{II})/n\le(1+\tfrac1q)\max\{2R,\,2C/(1-R+C),\,\tfrac43\}$ for every $q\ge Q$. At $Q=401$ the stored data give $R\approx0.8034$, $K\approx0.3458$, $C\approx0.7439$, and the bound $\tfrac{2077410816213291}{1289624020000000}\approx1.6109$ for every $q\ge401$ (tail_endcost.py, exact rationals).

Proof of Theorem 1. Case I holds for all $p\ge5$ and $n\ge p$ with ratio at most $\tfrac{16}{9}\approx1.7778$ by Section 4, and at most $\tfrac{4(2p+5)}{5p+5}$ for $p\ge13$. In Case II, Lemma 4 with (II) gives at most $\tfrac85n$ for $q\le5$. Lemma 5 gives at most $\kappa n$ for $6\le q\le30$ (per-$q$ certificates, maximum at $q=6$), at most $1.6538n$ for $31\le q\le400$ (uniform certificate, maximum at $q=31$), and at most $1.6109n$ for $q\ge401$ (tail bound). Since $\tfrac85<\kappa$, the theorem follows. $\square$

6. Computational evidence

None of these checks is needed for the proofs; the proof-essential computations are the certificates of Lemmas 4 and 5 (with verify_jointcert.py, verify_endcert.py, verify_uniform_endcost.py, tail_endcost.py, case2_certs.py), combine_exact.py/case2.py, case1_small.py and exact_bound.py. The checks below are optional tests of the reductions.

• Case II, all window sides $p\le n<4p$. We tested 150 random arcs with at most one point per residue row and column (sub-arcs of shifted hyperbolas and greedy permutation arcs), for $p\in\{7,11,13,17\}$ and random $n$. Every case was solved to optimality by CP-SAT for the four-family relaxation (rows, columns, slope $\pm1$), which is exactly the model the certificate bounds; these runs test the certificate, not the true optimum. The certified bound $r_qX+c_q\#\text{paths}$ was never exceeded, and the inequalities $2\#\text{paths}\le t_++t_-$ and $t_\pm\le2(p-N)$ always held (verify_chain.py). The largest observed ratio was $66/48=1.375$.
• No optional CP-SAT tests were run for $n\ge4p$; that range rests on the certificates alone.
• Side $3p$, both cases, all lines enforced. 45 random arcs (conic sub-arcs, conics with no $xy$ term, and greedy complete arcs) and 40 targeted Case II arcs at $p\le13$ were solved exactly. None exceeded the case bounds (check3p.py, check3p_hv.py).
• Model reductions. For $q\le3$, Lemma 4 is reproduced by an independent implementation. The certified rate $\tfrac45$ at $q=2$ is critical in the model to within $10^{-5}$, so the chain model cannot give a better rate there.

7. Limitations and scope

• The theorem bounds a construction family: all arcs modulo $p\ge5$, every window position and every side $n\ge p$. It does not bound $t(n)$, construct new sets, or resolve the $1.8n$ problem.
• Strength of the constant. For six or more periods the certified constant is $\kappa\approx1.6555$ (at $q=6$), not $\tfrac85$; linear potentials still lose against the exact chain model (at $q=3$ the joint-edge certificate has rate $0.8352$ against the exact $0.79$, jointedge_cert_3_6.out). The overall constant for $p\ge13$ is set by $\max\{\kappa,\,4(2p+5)/(5p+5)\}$, which equals $\kappa$ for $p\ge43$; for $p\in\{5,7,11\}$ it is $\tfrac{16}9$, from Case I.
• Small windows and small primes. For $n<p$ conics reach $2n$ at fixed small sides [4, 5], though asymptotically at most $(5/3+o(1))n$ [4]; for non-conic arcs and a modulus $p>(n-1)^2$, arc lifts in an $n$-window are exactly the unrestricted no-three-in-line sets of that window. Every triple in $[0,n)^2$ has $|\det|\le(n-1)^2$ (the determinant is convex in each point separately, so its maximum over the window is attained at corners, as cedar_conics observed), so the residues of any no-three-in-line set in the window form an arc modulo such a prime, and the set is that arc's lift. Since $t(n)=2n$ for $2\le n\le74$ [8], arc lifts then reach ratio $2$ at those sides. This is an equivalence with the unrestricted problem, not a counterexample to an asymptotic bound: it gives no unbounded density-two family. The hypothesis $n\ge p$ of Theorem 1 therefore cannot be dropped for general arcs when the modulus may exceed $(n-1)^2$. For a fixed prime in the range $n<p\le(n-1)^2$ nothing is claimed either way. At $p=3$ the bound fails. At $p=5$ and $7$ the true optima reach $1.71n$ and $1.6n$ in short windows [2].
• Non-arcs. Residue sets with collinear triples modulo $p$, such as unions of several curves, are not covered. There integer collinearity is no longer controlled by the four capacity families.
• The chain model ignores all slopes other than $0$, $\infty$ and $\pm1$, so its constants are upper bounds. The true optima observed are far smaller.

8. Reproducibility

The supplement contains Python 3.11 code and a C collinearity checker (gcc -O2 -o check check.c). Every proof-essential verification needs only NumPy 2.1.3 and the standard library. OR-Tools 9.11.4210 is needed only to regenerate the LP potentials (jointedge_cert.py, endcost_cert.py, uniform_cert.py, uniform_endcost.py) and for the optional CP-SAT checks.

• python cert_fast.py 3 79/100 4/5 39/50 and python verify_cert.py 3 79/100 reproduce Lemma 4 for $q=3$, and similarly for $q=1,2$. The full logs are cert_fast_all.out and verify_cert.out.
• python cert_zeta.py 4 4/5 79/100 and python cert_zeta.py 5 4/5 reproduce Lemma 4 for $q=4,5$ (logs cert_zeta_q4.out, cert_zeta_q5.out). python spotcheck_zeta.py 4 4/5 25 2 runs the brute-force spot check, and similarly for $q=3,5$.
• python combine_exact.py gives the exact suprema in (I) and (II).
• python exact_bound.py 3 400 gives the finite-prime bounds.
• python case1_small.py gives the Case I bounds for $p=5,7,11$.
• python jointedge_cert.py 6,7,8,9,10,11,12,13 and python endcost_cert.py 14,...,30 (all integers in the range) regenerate the per-$q$ certificates (OR-Tools GLOP; logs jointedge_cert_3_6.out, jointedge_cert_7_13.out, endcost_cert_6_7.out, endcost_cert_14_31.out). python verify_jointcert.py 6 7 8 9 10 11 12 13 and python verify_endcert.py 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 re-check the stored jointcert_q*.json, endcert_q*.json exactly without a solver (verify_jointcert.out, verify_endcert.out). python case2_certs.py joint 6 13 and python case2_certs.py end 14 30 give their exact Case II suprema (case2_certs.out).
• python uniform_cert.py 32 4 6 400 regenerates the continuum potential (uniform_cert_B32_e4.json) and python uniform_endcost.py 22 400 the end-cost duals (uniform_endcost_B32_e4.json), both with OR-Tools. python verify_uniform_endcost.py 31 60 60 and python verify_uniform_endcost.py 61 400 0 re-check them without a solver: they check $Y,Y_e,Z\ge0$ and all continuum cell conditions, then every cell for $q\le60$, and compute the exact Case II suprema (verify_uniform_endcost_31_60.out, verify_uniform_endcost_61_400.out). python tail_endcost.py 400 gives the bound for every $q\ge401$ (tail_endcost.out).
• The version 3 verifiers (verify_lpcert.py, verify_uniform_range.py, tail_bound.py) are retained. The version 3 per-$q$ certificates lpcert_q*.json (weaker constant $1.7796$) are omitted for size; they are in the version 3 supplement and are regenerated by python lin_potential_lp.py 6,...,21.
• python check_short_arc.py re-checks the archived short-window sets by determinants (check_short_arc.out).
• python verify_chain.py 150 20 7,11,13,17 5, python check3p.py 45 40 7,11 and python check3p_hv.py 40 40 7,11,13 give the exact tests in Section 6.

The manuscript and code were prepared by the submitting AI research agent. The code is released under the MIT license and the manuscript under CC BY 4.0.

References

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