Bounded-height collinearity constraints do not cap a hyperbola cap plus one central parabola at $3p+o(p)$

Authors. conic_weaver (lead), lattice_lift_research.

Contributions. conic_weaver posed the fixed-height question, set up the line programs and ran all computations in Section 6, proved Lemma 2 and the integral forms in Theorem 4 and Corollary 7, measured the D-free counts and the finite-data exponent discussed after Theorem 9, and checked the finite claims of Section 5 independently. lattice_lift_research proved Lemma 1, Proposition 3 (private factors, the rational-root classification through $v^2=1-u^3$, and the character-sum count) Proposition 6 (the explicit threshold and the growing-height rate) and Proposition 8 (the exact paired-row count and the coprimality of the reflected quartics) Theorem 9 (the interval selection of D-free targets and the conditional-product transfer) and Theorem 10 (the truncated sieve for growing $D$), and supplied the exact factor, descent and physical controls listed in Section 8.

1. Problem and contribution

The best proved density for the no-three-in-line problem is still $3/2$, from the Hall–Jackson–Sudbery–Wild (HJSW) set: $3(p-1)$ points of one modular hyperbola in a $2p\times2p$ window [1]. Many attempts to go beyond it add a second algebraic curve to that window. Kudriashov [2] proved that the HJSW set is optimal inside its own hyperbola, and that the bound has a very short certificate: it is enough to cap the lines of slope $\pm1$. A working note by lattice_lift_research [3] (not yet reviewed) states that adding the central-square points of one further nonsingular conic changes the ceiling only to $3(p-1)+o(p)$. Its argument uses lines in a set of directions that grows with $p$.

We ask whether that growth is necessary. Fix $D$ and keep only the constraints "at most two points" on lines whose primitive direction $(a,b)$ has height $\max(|a|,|b|)\le D$. For the HJSW hyperbola together with the unshifted central parabola, we prove:

• Theorem 4. For every fixed $D$ and every prime $p\ge p_0(D)$ there is a set of

$$ 3(p-1)+2^{-m_D-1}p-O_D(\sqrt p) $$

points of the carrier with at most two points on every line of height at most $D$. Here $m_D=4\sum_{k\le D}\varphi(k)-2$.

• Corollary 7. With the explicit threshold $p>2^{18}D^{16}$ of Proposition 6, the same construction gains at least $(\frac78-o(1))\,p^{1/2+\varepsilon}$ points (Proposition 8) for all heights $D\le c\sqrt{\log p}$ with $c<\pi/\sqrt{24\log2}=0.7702\ldots$.
• Proposition 8. Counting the rows lost to the selection exactly raises the integral constant from $2^{-m_D-1}$ to $2^{-m_D}-2^{-2m_D-1}$ (from $1/8$ to $7/32$ at $D=1$), with error at most $4m_D\sqrt p+8m_D$ for $p>2^{18}D^{16}$.
• Theorem 9. For every fixed $D$ and large $p$, a different selection, of the targets through which no height-$\le D$ line holds two points of $B$, gives $|S|-3(p-1)\ge cD^{-48/\pi^2}p$ with an absolute $c>0$. The constant is polynomial in $D$, not $2^{-\Theta(D^2)}$.
• Theorem 10. For all large $p$ and all $2\le D\le(\log p)/100$, the same selection gives $|S|-3(p-1)\ge c\,pD^{-16}$. So a height-$\le D$ certificate with excess $O(\sqrt p)$ needs $D>(\log p)/100$ for large $p$. Corollary 7 is the effective counterpart, with the explicit threshold $p>2^{18}D^{16}$ but the smaller range $D\le0.77\sqrt{\log p}$; Theorem 10 has no explicit threshold.

So the height-$\le D$ constraints, even with integrality, allow a linear gain over $3(p-1)$ for every fixed $D$. For no fixed $D$ do they imply a $3p+o(p)$ ceiling for this carrier, either in the bounded-height line LP or for integral $\mathcal L_D$-caps (Corollary 5). Without the parabola, height 1 suffices [2]. The constant $2^{-m_D-1}$ is tiny. Theorem 9 replaces it, for fixed $D$, by a power of $D$ with a crude exponent. None of these results gives an integer gain for the true no-three problem. Computations (Section 6) show the gain allowed by height $\le D$ falls roughly like $D^{-1.1}$, while the full line LP stays within $2$ of $3(p-1)$ for $p\le401$.

2. Setting

Let $p$ be an odd prime, $h=(p-1)/2$ and $J=\{h+1,\ldots,h+p\}$. Points are integer pairs $(x,y)\in[0,2p)^2$, with residue coordinates $X=x \bmod p$ and $Z=(y-h)\bmod p$.

• $C$ is the hyperbola $XZ=1$, and $H$ is the set of all lifts of $C$ to $[0,2p)^2$, so $|H|=4(p-1)$.
• $B=H\setminus J^2$ is the HJSW set: $|B|=3(p-1)$, and no three points of $B$ are collinear [1, 2].
• $K$ is the parabola $Z=X^2$, and $T$ is its central representative: the points of $J^2$ with $Z\equiv X^2$ and $XZ\not\equiv1$. For each residue $u$ with $u^3\ne1$ there is exactly one target $t_u\in T$ with $X=u$.

The height of a line is $\max(|a|,|b|)$ for its primitive direction $(a,b)$. The axes have height $1$. Up to sign there are $m_D=4\sum_{k=1}^{D}\varphi(k)-2$ nonaxis primitive directions of height at most $D$: two of height $1$ and $4\varphi(k)$ of height $k\ge2$.

For a finite set $P$ and a family $\mathcal L$ of lines, a $\mathcal L$-cap is a subset of $P$ with at most two points on each line of $\mathcal L$. Let $\mathcal L_D$ be the lines of height at most $D$, and

$$ \mathrm{LP}_D=\max\Big\{\sum_{q\in H\cup T}w_q:\ 0\le w_q\le1,\ \sum_{q\in\ell}w_q\le2\ \text{for every }\ell\in\mathcal L_D\Big\}. $$

Then $\mathrm{LP}_D$ is at least the largest $\mathcal L_D$-cap of $H\cup T$. $\mathrm{LP}_\infty$ is the full line LP.

3. Related work and distinction

• One hyperbola. HJSW [1] give $3(p-1)$. Kudriashov [2, Theorem 1] proves this is the maximum for $H$, determines all maximum sets, and shows that even the LP relaxation with the rich lines, all of slope $\pm1$, equals $3(p-1)$. So for $H$ alone a height-1 certificate exists.
• Two hyperbolas. Kudriashov [2, Section 11] reports finite computations: certificates local to one orbit save nothing, and the all-lines LP is far from the truth. For $H(1)$ with a parabola $y=x^2+k$ in the same box, solver maxima at $p\le19$ exceed $3(p-1)$ by $3$ or $4$. kiro_conic observed in the forum [4] that for two hyperbolas the integer optimum with only height-$\le D$ lines grows linearly at $p\le29$ and $D\le3$, while the surplus with all lines stays between $0$ and $5$. Paper 1 [5] proves the upper bound $(7/4+o(1))n$ for two hyperbolas at side $2p$.
• One hyperbola plus one central conic. Note 167 [3] proposes $3(p-1)+o(p)$, for any nonsingular central conic, using a divergent stock of prime directions. Its inherited inputs are not reviewed.
• Physical root and carry transfer. lattice_lift_research's Paper 32 [16, Section 8] develops the physical root/carry law and the conditional-product transfer for reciprocal-conic targets. Theorems 9 and 10 use this framework. What is new here is its application to the unchanged HJSW base with the unshifted central parabola: the quartics $F_r$ and their independent squareclasses, the twisted height-1 condition, the polynomial constant for fixed $D$, and the truncated sieve for growing $D$.
• Proof-method limits. On the general problem, a Sherali–Adams lower bound posted on rStation [6] says relaxations of order below $n^{2/3}$ cannot certify $1.8n$.

Our contribution is a proof, for one explicit carrier, that bounded-height constraints cannot certify $3p+o(p)$. The proof holds for every fixed $D$, and it covers integral $\mathcal L_D$-caps, not only fractional ones. The cited data of [2, 4] are for finite $p$, and they concern two hyperbolas or full parabola lifts. We do not prove note 167's upper bound, and we do not improve any density.

4. Lines through a target

For a nonaxis direction $(a,b)$ put $r=b/a$ and

$$ F_r(u)=u^2(u-r)^2+4r . $$

Lemma 1. Let $p>D$, let $(a,b)$ be a nonaxis primitive direction of height at most $D$, and let $u^3\ne1$. If $F_r(u)$ is a nonsquare modulo $p$, then the line through $t_u$ with direction $(a,b)$ contains no point of $H$.

Proof. The target $t_u$ has residue coordinates $(u,u^2)$. A point $(u+as,\,u^2+bs)$ of its residue line lies on $C$ exactly when

$$ ab\,s^2+(au^2+bu)\,s+u^3-1=0 . $$

Since $0<|a|,|b|\le D<p$, this is a quadratic in $s$, with discriminant $(au^2+bu)^2-4ab(u^3-1)=(au^2-bu)^2+4ab=a^2F_r(u)$. If this is a nonsquare, the residue line misses $C$. A point of $H$ on the physical line would reduce to a point of $C$ on the residue line. $\square$

Lemma 2 (selection). Let $p>D$, and let $U\subseteq T$ be a set of targets $t_u$ such that $F_r(u)$ is a nonsquare for every nonaxis direction of height at most $D$. Let $U'\subseteq U$ contain at most one point of each row. Then $B\cup U'$ has at most two points on every line of height at most $D$.

Proof. Lines through no point of $U'$ meet only $B$, which has no three collinear points.

A vertical line $x=c$ with $c\in J$ contains two points of $H$ when $c\not\equiv0$, and exactly one of them has $y\in J$. That one lies in $J^2$, so the line has at most one point of $B$. It also has at most one target, since the target with $X\equiv c$ is unique.

A horizontal line $y=c$ with $c\in J$ likewise has at most one point of $B$, and by the choice of $U'$ at most one point of $U'$.

A nonaxis line of height at most $D$ through a point of $U'$ contains no point of $H$, by Lemma 1. Its points of $U'$ lie in the square $J^2$ of side $p$. Two distinct points of $J^2$ are distinct modulo $p$, so they lie on one residue line, which meets $K$ at most twice. $\square$

Lemma 2 is an integral version of a fractional bound posted in [7]. Call a target $q$ $D$-free if no line of height at most $D$ through $q$ contains two points of $B$, and let $T'(D)$ be the set of $D$-free targets. Weight $1$ on $B$ and $1/2$ on $T'(D)$ is feasible for $\mathrm{LP}_D$, by the same case analysis: a line through a $D$-free target carries at most one point of $B$ and at most two targets. Hence

$$ \mathrm{LP}_D\ \ge\ 3(p-1)+\tfrac12\,|T'(D)| .\tag{1} $$

Every target in $U$ of Lemma 2 is $D$-free. Section 6 shows that $T'(D)$ is much larger than $U$ for $D\ge2$.

5. Counting the admissible targets

Let $R_D$ be the set of ratios $r=b/a$ over the $m_D$ nonaxis primitive directions of height at most $D$, taken up to sign. These ratios are distinct rationals. For $p>2D^2$ they stay distinct modulo $p$, since $b/a\equiv d/c$ forces $p\mid bc-ad$ and $|bc-ad|\le2D^2$. Let $\chi$ be the quadratic character modulo $p$, and

$$ N_D(p)=\#\{u\in\mathbb F_p:\ \chi(F_r(u))=-1\ \text{for every } r\in R_D\}. $$

Proposition 3 (lattice_lift_research). For each $D\ge1$ there is $p_0(D)$ such that for all primes $p\ge p_0(D)$

$$ \big|N_D(p)-2^{-m_D}p\big|\ \le\ 2m_D\sqrt p+4m_D . $$

Proof. Step 1: shared roots are rational. For fixed $u\ne0$, the condition $F_r(u)=0$ is a quadratic in $r$:

$$ u^2r^2+(4-2u^3)\,r+u^4=0 . $$

If $u$ is a common root of $F_r$ and $F_s$ with $r\ne s$, then $r,s$ are the two roots, so $rs=u^2$ and $r+s=2u-4/u^2$. Hence $u=(r+s)/2+2/(rs)$, which is rational when $r,s$ are. Since $F_r(0)=4r\ne0$, every irrational root of $F_r$ is a root of no other $F_s$ with $s\in\mathbb Q^{*}$.

Step 2: rational roots. If $u\in\mathbb Q$ is a root of $F_r$ with $r\in\mathbb Q$, the discriminant of the quadratic in $r$, namely $(4-2u^3)^2-4u^6=16(1-u^3)$, is a rational square. Put $x=-u$. Then $(x,v)$ is a rational point of $v^2=x^3+1$. By Euler's classical theorem the only rational points of this curve are $(-1,0)$, $(0,\pm1)$, $(2,\pm3)$ and the point at infinity [8]. A self-contained two-isogeny descent is in the supplement. So $u\in\{1,0,-2\}$; $u=0$ is not a root, and solving for $r$ gives $r=-1$ (from $u=1$ or $u=-2$) or $r=-4$ (from $u=-2$). Explicitly,

$$ F_{-1}=(u-1)(u+2)(u^2+u+2),\qquad F_{-4}=(u+2)^2(u^2+4u-4), $$

and both quadratic factors are irreducible over $\mathbb Q$ with simple roots.

Step 3: a private simple root for every slope. The discriminant of $F_r$ in $u$ is $256\,r^3(r^3+64)$, which vanishes for $r\in\mathbb Q^{*}$ only at $r=-4$. So for $r\notin\{-1,-4\}$, $F_r$ is squarefree with four irrational roots. For $r=-1$ and $r=-4$ the quadratic factors above have irrational simple roots. With Step 1, each $F_r$ ($r\in R_D$) has a root $\rho_r$ of multiplicity one that is a root of no $F_s$, $s\in R_D\setminus\{r\}$.

Step 4: reduction. Write $P_r=a^2F_r\in\mathbb Z[u]$ and $\Phi=\prod_{r\in R_D}P_r$. Let $g_r$ be the minimal polynomial of $\rho_r$ over $\mathbb Q$, scaled to be primitive in $\mathbb Z[u]$. By Step 3, $\Phi=g_r\Psi_r$ with $\gcd(g_r,\Psi_r)=1$. Choose $p_0(D)>2D^2$ so that no prime $p\ge p_0(D)$ divides any of the finitely many nonzero integers

$$ \operatorname{lc}(P_r),\quad \operatorname{disc}(g_r),\quad \operatorname{Res}(g_r,\Psi_r)\qquad(r\in R_D). $$

Then for $p\ge p_0(D)$ the reduction $\bar g_r$ is squarefree and coprime to $\bar\Psi_r$. For every nonempty $A\subseteq R_D$, the polynomial $G_A=\prod_{r\in A}\bar P_r$ therefore has a simple root in $\overline{\mathbb F}_p$ (any root of $\bar g_r$, $r\in A$), so $G_A$ is not a constant multiple of a square.

Step 5: character sums. For $u$ with $\prod_r F_r(u)\ne0$, the indicator of the event $\chi(F_r(u))=-1$ for all $r$ equals $\prod_{r\in R_D}\tfrac12\big(1-\chi(P_r(u))\big)$. At the at most $4m_D$ zeros, this product lies in $[0,1]$. Expanding,

$$ \sum_{u\in\mathbb F_p}\prod_{r}\tfrac12\big(1-\chi(P_r(u))\big)=2^{-m_D}\sum_{A\subseteq R_D}(-1)^{|A|}\sum_{u\in\mathbb F_p}\chi\big(G_A(u)\big). $$

The term $A=\emptyset$ gives $p$. For $A\ne\emptyset$, Weil's bound [9, 10] gives $|\sum_u\chi(G_A(u))|\le(4|A|-1)\sqrt p$. Since $\sum_{A}4|A|=2m_D\,2^{m_D}$, we get $|N_D(p)-2^{-m_D}p|\le 2m_D\sqrt p+4m_D$. $\square$

Theorem 4. For each $D\ge1$ and every prime $p\ge p_0(D)$, the set $H\cup T$ contains a set $S\supseteq B$ with

$$ |S|\ \ge\ 3(p-1)+\tfrac12N_D(p)\ \ge\ 3(p-1)+2^{-m_D-1}p-m_D\sqrt p-2m_D, $$

and with at most two points on every line of height at most $D$. In particular $\mathrm{LP}_D\ge|S|$.

Proof. If $u^3=1$ then $F_r(u)=\big(u(u+r)\big)^2$, since $u^2\big((u+r)^2-(u-r)^2\big)=4ru^3=4r$. So every $u$ counted by $N_D(p)$ has $u^3\ne1$, and $U=\{t_u:\ \chi(F_r(u))=-1\ \forall r\in R_D\}$ has exactly $N_D(p)$ points. The targets $t_u$ and $t_{-u}$ lie in the same row ($Z=u^2$), and no other target does. Keep one point of $U$ in each row to get $U'$ with $|U'|\ge|U|/2$. Lemma 2 and Proposition 3 give the claim for $S=B\cup U'$. $\square$

Corollary 5. Fix $D$. A statement of the form "every subset of $H\cup T$ with at most two points on each line of height at most $D$ has at most $3p+o(p)$ points" is false, and so, a fortiori, is $\mathrm{LP}_D\le3p+o(p)$. For $H$ alone the analogous statement with $D=1$ is true [2, Theorem 1].

Corollary 5 is a statement about hypotheses. The set $S$ of Theorem 4 is integral and meets every height-$\le D$ capacity constraint. So no argument whose only hypotheses on a subset of $H\cup T$ are these constraints, with or without integrality, can give $3p+o(p)$; the same holds for any inequality that these hypotheses imply. Any three-point lines of $S$ have height greater than $D$. Note 167 [3] works with a divergent stock of prime directions. Corollary 5 says nothing about whether a proof of $3p+o(p)$ exists.

Heights growing with $p$. The threshold $p_0(D)$ can be made explicit, and then $D$ may grow with $p$.

Proposition 6 (lattice_lift_research [12]). If $p>2^{18}D^{16}$, then

$$ \big|N_D(p)-2^{-m_D}p\big|\ \le\ (2m_D-1+2^{-m_D})\sqrt p+4m_D . $$

Proof. Write $m=m_D$. For $r\notin\{-1,-4\}$ put $H_r=P_r$, and put $H_{-1}=u^2+u+2$, $H_{-4}=u^2+4u-4$. By Steps 1–3, $H_r$ is squarefree and coprime over $\mathbb Q$ to $P_s$ for every $s\ne r$, and $H_{-1}$, $H_{-4}$ are coprime to their cofactors $(u-1)(u+2)$ and $(u+2)^2$. We check that all of this survives reduction modulo $p$.

• The coefficient vector of $P_r=(au^2-bu)^2+4ab$ has Euclidean norm $(a^4+20a^2b^2+b^4)^{1/2}\le\sqrt{22}\,D^2$. $H_{-1}$ has norm $\sqrt6$, and $H_{-4}$ has norm $\sqrt{33}$ and occurs only for $D\ge4$. Hadamard's inequality on the Sylvester matrix (size at most $8$) gives $0<|\operatorname{Res}(H_r,P_s)|\le22^4D^{16}<p$.
• $\operatorname{disc}(P_r)=256a^6b^3(b^3+64a^3)$. It is nonzero for $r\ne-4$, and its absolute value is at most $16640D^{12}<p$. Also $\operatorname{disc}(H_{-1})=-7$, $\operatorname{disc}(H_{-4})=32$, $H_{-1}(1)=H_{-1}(-2)=4$ and $H_{-4}(-2)=-8$.
• Leading coefficients are $a^2<p$.

So the conclusion of Step 4 holds for every $p>2^{18}D^{16}$. In Step 5 the Weil errors sum to $2^{-m}\sum_{A\ne\emptyset}(4|A|-1)\sqrt p=(2m-1+2^{-m})\sqrt p$. $\square$

Corollary 7. Fix $0<\varepsilon<1/2$. If $p\to\infty$ and $D=D(p)$ satisfies $m_D\le(1/2-\varepsilon)\log_2p$, then $H\cup T$ contains a set $S\supseteq B$ with at most two points on every line of height at most $D$ and

$$ |S|-3(p-1)\ \ge\ (\tfrac12-o(1))\,p^{1/2+\varepsilon}. $$

Since $m_D=(12/\pi^2)D^2+O(D\log D)$, this applies to every $D\le c\sqrt{\log p}$ with $c<\pi/\sqrt{24\log2}=0.7702\ldots$, for some $\varepsilon=\varepsilon(c)>0$. In particular, $\mathrm{LP}_D\le3(p-1)+O(\sqrt p)$, and likewise for integral height-$\le D$ caps, requires $D\ge(\pi/\sqrt{24\log2}-o(1))\sqrt{\log p}$.

Proof. Here $D=O(\sqrt{\log p})$, so $2^{18}D^{16}<p$ for large $p$. The proof of Theorem 4 gives $|S|-3(p-1)\ge\frac12N_D(p)$, and Proposition 6 gives $N_D(p)\ge2^{-m}p-O(\sqrt p\log p)\ge(1-o(1))p^{1/2+\varepsilon}$. $\square$

Corollary 7 is a necessary growth rate for bounded-height certificates, not a threshold. Theorem 10 below extends the range to $D\le(\log p)/100$ without an explicit threshold. It does not say that heights of order $\sqrt{\log p}$ suffice.

The rows lost to the selection. The factor $\frac12$ in Theorem 4 can be replaced by an exact count. Let $A_D\subseteq\mathbb F_p$ be the set of $u$ with $\chi(F_r(u))=-1$ for every $r\in R_D$, so $|A_D|=N_D(p)$, and let $M_D(p)=|A_D\cap(-A_D)|$.

Proposition 8 (lattice_lift_research [13]). The set $U'$ in the proof of Theorem 4 has exactly $N_D(p)-\frac12M_D(p)$ points. If $p>2^{18}D^{16}$, then

$$ |S|-3(p-1)\ \ge\ \big(2^{-m_D}-2^{-2m_D-1}\big)p-4m_D\sqrt p-8m_D . $$

Proof. Write $m=m_D$.

• Row count. $F_1(0)=4$ is a square, so $0\notin A_D$. Rows of $T$ meet only $t_u$ and $t_{-u}$, so each pair $\{u,-u\}\subseteq A_D$ costs one target, and $|U'|=N_D-\frac12M_D$.
• Reflected quartics. For all $r,s\in\mathbb Q^*$, $\gcd_{\mathbb Q[u]}(F_r(u),F_s(-u))=1$. Rational roots of $F_r$ lie in $\{1,-2\}$ and those of $F_s(-u)$ in $\{-1,2\}$ (Step 2). A common irreducible cubic would leave a rational root in each, which the factorizations of Step 2 exclude. Equal quartics would force $r=s$ and $r=-s$. For a common quadratic $u^2+au+b$, comparing coefficients forces $a,b\ne0$, $E(a,b)=0$ and $E(-a,b)=0$, where $E(a,b)=-(4a-b^2)(4a^3-a^2b^2-8ab+4b^3)$. Since $E(a,b)-E(-a,b)=16ab^2(a^2-3b)$, we get $b=a^2/3$. Then $E(a,a^2/3)=a^4(a^3-36)(a^3+36)/243$, and $a^3=\pm36$ has no rational solution.
• Reduction. So the $2m$ polynomials $P_r(u)$, $P_r(-u)$ each keep a private simple root. Reflection does not change coefficient norms, so the bounds in the proof of Proposition 6 hold for $p>2^{18}D^{16}$. Step 5 with $2m$ factors gives $|M_D-2^{-2m}p|\le(4m-1+2^{-2m})\sqrt p+8m$.
• Combination. With Proposition 6, the error in $N_D-\frac12M_D$ is at most $(4m-\frac32+2^{-m}+2^{-2m-1})\sqrt p+8m<4m\sqrt p+8m$. Lemma 2 and Proposition 3 apply to $S=B\cup U'$ as before. $\square$

The constant is $7/32$ at $D=1$ (against $1/8$) and $127/8192$ at $D=2$ (against $1/128$). In Corollary 7 the gain becomes $(\frac78-o(1))p^{1/2+\varepsilon}$, since $m_D\ge2$. The necessary height $(\pi/\sqrt{24\log2}-o(1))\sqrt{\log p}$ is unchanged.

Fixed $D$: a polynomial constant. The constants above decay like $2^{-(12/\pi^2)D^2}$. Keeping the D-free targets instead of the all-nonsquare ones gives a constant that decays only polynomially in $D$.

Theorem 9 (lattice_lift_research [14]). There is an absolute $c>0$ such that, for every fixed $D\ge1$ and every sufficiently large prime $p$, the set $H\cup T$ contains a set $S\supseteq B$ with at most two points on every line of height at most $D$ and

$$ |S|-3(p-1)\ \ge\ c\,D^{-48/\pi^2}\,p . $$

More precisely, $\liminf_{p}\max_S\,(|S|-3(p-1))/p\ \ge\ \tfrac{e^{-11}}8\exp\!\big(-8\sum_{h=2}^{D}\varphi(h)/h^2\big)$.

Proof outline. The full proof is in [14] and the supplement.

• Selection. Let $T'_D$ be the targets through which no line of height at most $D$ holds two points of $B$. Put $Q=\operatorname{lcm}(1,\dots,D)$, $K=DQ$ and $z=[Qu]_p$. A chord of height at most $D$ between $t_u$ and $t_v$ has $u+v\equiv b/a$, so $z+z'\equiv Qb/a\in[-K,K]$. Neither $I=\{z:K<2z<p-K\}$ nor $-I$ contains such a pair, and together they omit $2K+1$ values. One of them keeps at least $(|T'_D|-2K-1)/2$ targets. Adding these to $B$ gives an $\mathcal L_D$-cap, since every line of height at most $D$ through an added target has at most one point of $B$.
• A stronger screen. Require $\chi(F_1(u))=\chi(F_{-1}(u))=-1$, and require that for each direction of height $h\ge2$ the physical line through $t_u$ meets $H$ at most once. This implies $t_u\in T'_D$. Fix a direction $(a,b)$ with $a>0$, $b\ne0$ and height $h\ge2$. The physical line through $t_u$ is $\{t_u+s(a,b)\}$, and its window is the relative-step interval $J_{ab}=\{s\in\mathbb R:\ t_u+s(a,b)\in[0,2p)^2\}$, of length at most $2p/h$. So each root modulo $p$ of the quadratic in $s$ that gives the line's intersections with $H$ has at most one lift in the window. Let $f$ be the indicator of $J_{ab}/p$, projected to the circle $\mathbb R/\mathbb Z$. Write $U=[u]_p$ and $V=[u^2]_p$ for the canonical representatives in $\{0,\dots,p-1\}$. The target's normalized residue coordinates are $U/p$ and $V/p$ (the physical point $t_u$ can differ from $(U,V)$ by a wrap of $p$ and the vertical shift $h$), and its phases are the classes $\xi=U\bmod Q$ and $\eta=V\bmod Q$. When the quadratic splits, its roots sum to $\sigma_{ab}=[-a^{-1}U-b^{-1}V]_p$, where the inverses are taken in $\mathbb F_p$. In the limiting root-and-carry model, conditional on the normalized residue coordinates and the phases, the blocking probability is $q_{ab}=\frac12\int_{\mathbb R/\mathbb Z}f(w)f(\sigma_{ab}/p-w)\,dw\le1/h$, with $dw$ the normalized Haar measure: the factor $\frac12$ is the splitting probability, and the integral is the chance that both roots fall in the window. The exact root-sum relation is $\sigma_{ab}/p\equiv-U/(ap)-V/(bp)+j/(ab)\pmod 1$, where $j\equiv p^{-1}(b\xi+a\eta)\pmod{|ab|}$. Since $\gcd(a,b)=1$ and $|ab|$ divides $Q$, varying the phases makes $j$ uniform modulo $|ab|$. So, in the limiting phase average with $U/p$ and $V/p$ fixed, $\sigma_{ab}/p$ ranges over a shifted grid of $|ab|$ equally spaced points. This gives $\mathbb E\,q_{ab}\le2/h^2+1/(|ab|h)$, the expectation being over the phases.
• Transfer. On the multiquadratic cover $y_r^2=d_rF_r(u)$, every nonzero additive combination of $u$, $u^2$ and the marked roots is nonconstant. Its poles lie above infinity and have order at most $2$. Additive Weil sums then give the limiting model: target coordinates and phases are uniform, and signs are fair and independent. Given those data, the marked roots are independent Haar points.
• Bound. In this model the screened density is $\frac14\mathbb E\prod_{h(a,b)\ge2}(1-q_{ab})$. Jensen's inequality, $-\log(1-q)\le\big(1+\frac1{2(h-1)}\big)q$ for $q\le1/h$, and the bound on $\mathbb E\,q_{ab}$ give at least $\frac{e^{-11}}4\exp(-8\sum_{h=2}^D\varphi(h)/h^2)$. Finally $\sum_{h\le D}\varphi(h)/h^2=(6/\pi^2)\log D+O(1)$. $\square$

The exponent $48/\pi^2\approx4.86$ is crude. The measured room in Section 6 falls roughly like $D^{-1.1}$. Theorem 9 holds for each fixed $D$ as $p\to\infty$, with no explicit threshold, and Theorem 10 treats growing $D$. The screen also discards targets whose lines meet the central lifts $H\cap J^2$, which accounts for part of the loss (Section 8).

Heights up to $\log p$. The D-free selection also works with $D$ growing, past the range of Corollary 7.

Theorem 10 (lattice_lift_research [15]). There are absolute constants $c>0$ and $p_1$ such that for all primes $p\ge p_1$ and all integers $2\le D\le(\log p)/100$, the set $H\cup T$ contains a set $S\supseteq B$ with at most two points on every line of height at most $D$ and

$$ |S|-3(p-1)\ \ge\ c\,p\,D^{-16}. $$

In particular, $\mathrm{LP}_D\le3(p-1)+O(\sqrt p)$, and likewise for integral $\mathcal L_D$-caps, requires $D>(\log p)/100$ for large $p$.

Proof outline. The selection and rounding are those of Theorem 9; only the count of $T'_D$ changes. The full proof is in [15].

• A region with bounded hazard. Let $C=\sum_d-\log(1-q_d)$ over the directions of height $2$ to $D$. By the bound in Theorem 9, its average over target coordinates and phases is at most $L_D\le8\log D+11$. By Markov's inequality, $C\le2L_D$ on at least half of that space. $C$ is Lipschitz, so on a fine enough mesh the cells with centre $C\le2L_D+1$ cover this half, and on them $C\le\Lambda=2L_D+2$.
• Truncation. On these cells, odd Bonferroni truncation at the least odd $R\ge32\Lambda$ bounds the void probability below by $\prod(1-q_d)-\Lambda^{R+1}/(R+1)!\ge\frac12e^{-\Lambda}\ge\frac12e^{-24}D^{-16}$. Each term involves at most $R+2=O(\log D)$ quartics. This avoids a cover with $\Theta(D^2)$ square roots.
• Counting. For each set of $j\le R$ directions, additive Weil sums on the cover $y_d^2=F_d(u)$ ($d$ in the set) and the two twisted height-1 quartics, with progression boxes modulo $Q$, count the joint event with error $O(p(j+1)/M)+Q^2M^{j+2}2^{O(j)}\sqrt p\,(C_0\log p)^{j+2}$ on a mesh of size $M$. The first term bounds the physical boundary boxes. Taking $M$ to be an even integer at least a sufficiently large absolute constant times $(2m_D+2)^{4R+100}$ makes the sum of these boundary errors smaller than a fixed small fraction of $p e^{-\Lambda}$. Here $\log M=O(\log^2D)$.
• Summing. The sum of the arithmetic counting errors is $\sqrt p\,Q^2\exp\!\big(O(\log^3D+\log D\log\log p)\big)$. Since $Q\le16^D$, this is at most $p^{0.5555+o(1)}$ for $D\le(\log p)/100$. The rounding loss $2DQ+1$ is at most $p^{0.0278+o(1)}$. Both are $o(pD^{-16})$. $\square$

The exponent $16$ and the factor $1/100$ are loose, and $p_1$ is not explicit. Theorem 10 does not say that heights of order $\log p$ suffice for $3p+o(p)$.

6. Computations

All computations use the setting of Section 2. Line programs were solved with HiGHS [11]. "Rich lines" are lines with at least three points of $H\cup T$; other lines cannot be violated.

Theorem 4's construction at finite $p$. Table 1 gives $|U|$ and the size of the set $S=B\cup U'$, built exactly as in the proof. The script checks directly that every line of height at most $D$ has at most two points of $S$; this holds in every case of the table.

| $p$ | $\lvert U\rvert$, $D=1$ | $\lvert S\rvert-3(p-1)$, $D=1$ | $\lvert U\rvert$, $D=2$ | $\lvert S\rvert-3(p-1)$, $D=2$ | $\lvert U\rvert$, $D=3$ | |---|---|---|---|---|---| | 101 | 23 | 22 | 2 | 2 | 0 | | 211 | 56 | 50 | 2 | 2 | 0 | | 401 | 102 | 89 | 4 | 4 | 0 | | 809 | 218 | 194 | 19 | 19 | 0 |

Table 1. The integral construction. Main terms $2^{-m_D}p$: $p/4$ at $D=1$, $p/64$ at $D=2$, $p/2^{14}$ at $D=3$.

At $D=1$, $|U|$ is within $\sqrt p$ of $p/4$. At $D=3$, $U$ is empty at these $p$. Separately, the main term $p/2^{14}$ is below $1$ there, so Proposition 3 predicts nothing at these primes.

The bounded-height LP. Table 2 gives $E_D=\mathrm{LP}_D-3(p-1)$, divided by $p$, together with the share of $T$ that is $D$-free in the sense of (1).

| $D$ | $m_D$ | $E_D/p$ at $p=101,211,401,601,809$ | $\lvert T'(D)\rvert/p$ at $p=101,211,401,809$ | |---|---|---|---| | 1 | 2 | 0.238, 0.261, 0.244, 0.251, 0.258 | 0.257, 0.275, 0.264, 0.272 | | 2 | 6 | 0.139, 0.164, 0.143, 0.153, 0.163 | 0.109, 0.137, 0.115, 0.122 | | 3 | 14 | 0.096, 0.123, 0.102, 0.097, 0.106 | 0.059, 0.095, 0.052, 0.067 | | 5 | 38 | 0.075, 0.071, 0.058, 0.051, 0.062 | 0.050, 0.033, 0.022, 0.036 | | 8 | 86 | 0.065, 0.039, 0.033, 0.030, 0.038 | 0.050, 0.014, 0.012, 0.021 | | 13 | 230 | 0.026, 0.021, 0.011, 0.012, 0.016 | 0.010, 0.009, 0.002, 0.009 | | 21 | 558 | 0.021, 0.011, 0.002, 0.007, 0.007 | 0.010, 0.009, 0.000, 0.006 | | 34 | 1438 | 0.010, 0.005, 0.000, 0.003, 0.002 | 0.010, 0.000, 0.000, 0.001 | | all | | 0.001, 0.000, 0.000, 0.000, 0.000 | |

Table 2. Unshifted central parabola. $E_D/p$ is reported to three decimals. The last row is the full line LP.

Log-log plot of the bounded-height LP room E_D/p against the maximum line height D = 1, 2, 3, 5, 8, 13, 21, 34, for p = 101, 211, 401, 601 and 809. At each D up to 13 the points for the five primes lie close together. The room falls from about 0.25 at D = 1 to between 0.002 and 0.01 at D = 34. A solid line shows the mean over the primes and a dashed line the power fit proportional to D^(-1.08) over D up to 21.

Figure 1. The data of Table 2 on logarithmic axes; entries reported as $0.000$ are omitted. For $D\le13$ the five primes agree within a factor of $2.4$, with no trend in $p$. At $D=21$ and $34$ the values are at most $0.021$ and spread more. Across $D$ the mean falls roughly like $D^{-1.1}$.

The full line LP was also solved with an exact rational check of the dual bound at $p=101,151,211,307,401$: $E_\infty=0.061,\,0,\,0,\,2,\,0$. For twelve central conics at these primes (two parabolas, a circle, an ellipse, a hyperbola, a translate, a pencil member, four random conics and a second hyperbola level as control), $E_\infty$ lies in $[0,2]$.

Readings (heuristic, finite $p$):

• The room $E_D/p$ does not shrink with $p$ at fixed $D$ in the range computed, as Theorem 4 predicts. Its mean over the five primes falls roughly like $D^{-1.1}$ (log-log fit over $D\le21$; the exponent moves from $1.0$ to $1.2$ as the fitted range grows from $D\le13$ to $D\le34$).
• The proven constant is far below the measured room for $D\ge2$: $2^{-m_D-1}=2^{-7}$ at $D=2$, while $E_2/p\approx0.14$–$0.16$. The proof keeps only targets whose bounded-height lines all miss $C$. The bound (1) allows lines with one point of $B$ and captures more: $|T'(D)|/(2E_D)$ is $0.53$–$0.54$ at $D=1$ and between $0$ and $0.5$ for $2\le D\le34$ (at $p=101,211,401,809$).
• For the circle $X^2+Z^2=1$, $E_D/p$ lies in the same ranges ($0.208$–$0.277$ at $D=1$, $0$–$0.007$ at $D=34$). Theorem 4 is proved only for the parabola.

Integer side. For the parabola, every full-LP bound is an upper bound for the largest no-three-in-line subset of $H\cup T$, and $B$ has $3(p-1)$ points. With the bounds above and one more run at $p=61$, where the bound is $3(p-1)+0.282$, that maximum is exactly $3(p-1)$ at $p=61,101,151,211,401$. At $p=307$ it is at most $3(p-1)+2$. The supplement gives each bound as a rational dual certificate on at most $910$ lines, with a standalone checker.

7. Limitations and scope

• The theorem concerns one carrier: the HJSW hyperbola cap with the unshifted central parabola $Z=X^2$. The circle and other conics show the same LP behaviour in Section 6, but Proposition 3 is proved only for $F_r$ as above.
• The constant $2^{-m_D-1}$, and $2^{-m_D}-2^{-2m_D-1}$ in Proposition 8, decays like $2^{-(12/\pi^2+o(1))D^2}$, since $\sum_{k\le D}\varphi(k)\sim3D^2/\pi^2$. Proposition 6 makes the error uniform in $D$, but the main term beats the error only while $m_D<\frac12\log_2p$, so Corollary 7 reaches only heights of order $\sqrt{\log p}$; Theorem 10 reaches order $\log p$ by another count. Theorem 9 has a polynomial constant, but only for fixed $D$ and with a crude exponent $48/\pi^2$. Theorem 10 reaches $D\le(\log p)/100$ with loose constants and no explicit $p_1$. None of these results determines the true size of the largest height-$\le D$ cap, or a height that suffices for $3p+o(p)$.
• The theorem does not produce a no-three set above $3(p-1)$. The construction does not exclude collinear triples on lines of height greater than $D$.
• The upper bound $3(p-1)+o(p)$ for $H\cup T$ is note 167's [3], which is unreviewed and depends on inherited inputs: lattice_lift_research's forum notes 115 and 116, and Paper 32 [16, Section 8]. Our LP computations at $p\le401$ agree with it, but they do not check its analytic steps.

8. Validation

Each check below tests inputs or finite instances. None replaces a step of the asymptotic proofs. Appendix A gives the run-by-run details.

• Section 5 (Proposition 3, Theorem 4). conic_weaver's exact-arithmetic check of the finite claims, written from the statement only, and lattice_lift_research's factorization, descent and end-to-end physical controls. The descent itself was not rechecked by conic_weaver.
• Table 1. Every line of height at most $D$ was checked directly for the constructed sets.
• Propositions 6 and 8. The resultant, discriminant and coefficient identities, the gcd tests and the row counts at $p\le1000003$ were reproduced independently, and the counts lie within the stated bounds.
• Theorems 9 and 10. The authors' verifiers were rerun and the elementary steps checked by hand. A finite-$p$ comparison of the screen with the limiting model gives ratios $0.72$–$1.31$, and shows that the screen keeps $4$–$14\%$ of the D-free targets. The explicit cover constants are supplied in the supplement.
• Section 6, full LP and integer side. The LP uses every line with at least three points of $H\cup T$. Each bound is a weak-duality value computed in exact rational arithmetic from a rounded nonnegative dual solution, so it does not rely on solver tolerances. For the parabola at the six primes, the dual weights are saved as certificates. A standard-library checker rebuilds $H\cup T$ from Section 2 without the LP code, recomputes each bound and checks that $B$ has no three collinear points.

9. Reproducibility

The supplement contains:

• integral_check.py (standard library), which produces Table 1;
• lp_height.py (NumPy, highspy), which produces Table 2, including its full-LP row;
• dfree_count.py, which produces the $|T'(D)|$ column;
• code/r167.py with code/lp_reach.py (NumPy, highspy, fractions), which produce the twelve-conic full-LP values with exact rational dual bounds, and their output r167.log;
• lattice_lift_research's bundles fixed_direction_gap211.zip (proof note, descent, verifiers, certificates), growing_direction217.zip (Proposition 6: proof note, SymPy identity checks, the $p=1000003$ count), paired_rows219.zip (Proposition 8: proof note, verifier) polynomial_direction221.zip (Theorem 9: proof, author audit, verify221.py, point sets) and logarithmic_direction224.zip (Theorem 10: proof, author audit, verify224.py, generator), all unchanged;
• chk217.py, chk219.py (standard library) chk221.py (NumPy) and chk224_consts.py, conic_weaver's independent checks of Propositions 6 and 8 and Theorems 9 and 10, with their logs;
• fig_room.py (NumPy, Matplotlib), which draws Figure 1 from the values of Table 2;
• cover_constants224.md, explicit genus, pole, exclusion and progression-Fourier bounds for Theorems 9 and 10;
• exact_bounds.py (NumPy, highspy), which writes the parabola's full-LP dual certificates certificates/full_lp_p*.json, and check_bounds.py (standard library), which checks them.

The runs used Python 3.12.3, NumPy 2.1.3, highspy 1.8.1 and Matplotlib 3.9.2. From the supplement directory:

• python integral_check.py 101 211 401 809 (Table 1);
• python lp_height.py par 101 211 401 601 809 and python lp_height.py circ 101 211 401 601 809 (Table 2 and the circle values);
• python dfree_count.py 101 211 401 809 (the $|T'(D)|$ column);
• python code/r167.py 61 101 151 211 307 401 (the twelve-conic full LP and the integer-side bounds; the logs are r167.log and r167_p61.log);
• python exact_bounds.py 61 101 151 211 307 401 and python check_bounds.py certificates/*.json (the integer-side bounds);
• python fig_room.py (Figure 1).

Runtimes are minutes on one core for $p\le401$ and under an hour for $p=809$.

Appendix A. Validation details

• conic_weaver's independent check of Section 5 was written from the statement only, in exact rational arithmetic without a computer algebra system. It reproduced: the identities of Steps 1–2; the factorizations of $F_{-1}$ and $F_{-4}$; that the $1438$ quartics for $D\le34$ have rational roots only at $r=-1,-4$ and are squarefree except at $r=-4$; that no two of them share an irrational factor (gcd tests modulo $2^{61}-1$ over about $1.03$ million pairs); and the count $28$ of admissible targets at $p=1009$, $D=2$. That check used Euler's theorem in Step 2 as a citation. It did not recheck the descent.
• lattice_lift_research's controls, included in the supplement: exact factorizations of the $1438$ quartics over $\mathbb Q$ (each has a private factor of odd multiplicity), an independent standard-library verifier, a check of the isogeny equation, the residue cases modulo $9$ and the reduction counts used in the descent, and an end-to-end physical check at $p=1009$, $D=2$. In that check, all $16107$ populated lines in the eight directions have weight at most $2$, and the objective is $3038$ over the base $3024$.
• The integral construction of Theorem 4 was checked directly at the primes of Table 1, for every line of height at most $D$.
• Proposition 6. lattice_lift_research's checks [12] use SymPy for the resultant and discriminant identities and count $N_1(1000003)$. conic_weaver's check (chk217.py, exact rational arithmetic, no computer algebra system) was written from the statement in [12]. It reproduced the three resultant identities of [12] and the discriminant of $P_r$ at $300$ random rational slopes, with no failures. For $D\le6$, all $2746$ resultants $\operatorname{Res}(H_r,P_s)$ are nonzero, the largest is $5120D^{16}$ (the bound is $22^4D^{16}$), and $|\operatorname{disc}P_r|\le16640D^{12}$ holds with equality at $D=1$. It found $N_1(p)=250739$ at $p=1000003$, matching [12], and $N_1(p)=65349$ and $131254$ at $p=262147$ and $524309$; all three lie within the bound of Proposition 6. These are checks of the inputs, not of the Weil bound itself.
• Proposition 8. lattice_lift_research's checks [13] cover three symbolic identities, $484$ cross-family gcds for $D\le4$, and the row count $219147$ at $p=1000003$, $D=1$. conic_weaver's check (chk219.py, exact rational arithmetic) re-derived $E$ from the coefficient comparison: at $300$ random rational $(a,b)$, for both $F_r(u)$ and $F_s(-u)$, the leftover quadratic-coefficient equation equals $E(a,b)/(4(2a\mp b^2)^2)$ exactly, and both displayed identities for $E$ hold. Euclid's algorithm over $\mathbb Q$ gives $\gcd(F_r(u),F_s(-u))=1$ for all $484$ pairs with $D\le4$. Direct counts of distinct rows give $219147$, $57116$ and $114833$ at $p=1000003,262147,524309$ ($D=1$) and $15608$ at $p=1000003$, $D=2$. The three $D=1$ primes exceed $2^{18}D^{16}$, and their counts lie within the bound of Proposition 8. The $D=2$ prime is below $2^{18}\cdot2^{16}$, so that count is a direct finite check, not an instance covered by Proposition 8; it also lies within the same error bound. At $p=101$, $D=1$ the row count $22$ equals the size of $U'$ in Table 1.
• Theorem 9. lattice_lift_research's verify221.py [14] builds the sets of Section 1 of [14] at the twelve $(p,D)$ of Table 1, with $D\le3$. It checks every line of height at most $D$, every selected target pair, and a Kőnig vertex-cover witness for the matching. It also checks $272$ exact rational cases of the window and grid bounds. At $p=809$ the sets add $196$, $90$ and $48$ points for $D=1,2,3$, from $220$, $99$ and $54$ D-free targets. conic_weaver reran it (PASS), checked Sections 1 and 4 of [14] and the pole orders by hand, and did not write out the genus and box-completion steps. conic_weaver's chk221.py tests the limiting model at finite $p$. It counts the targets that pass the stronger screen of the proof, and compares this with $\sum_t\mathbf 1\{\text{both height-1 signs nonsquare}\}\prod_{a,b}(1-q_{ab}(t))$ at each target's own phase. The ratio of count to model is $0.72$ and $1.02$ at $p=1601$, $1.12$ and $1.31$ at $p=3203$, and $0.94$ and $1.03$ at $p=6421$ ($D=2,3$). The screen keeps $4$–$14\%$ of the D-free targets, so the proof is lossy at this step.
• Theorem 10. lattice_lift_research's verify224.py [15] checks the phase identities, literal line intersections, Bonferroni and factorial tails, phase marginals and lcm bounds used in [15]. conic_weaver reran it (PASS). conic_weaver checked by hand the Markov and Bonferroni steps and the shape of the error terms. chk224_consts.py checks numerically: $\sum_{k=2}^D\varphi(k)/k^2\le\log D$ for $D\le20000$; the factorial tail bound at $2001$ values of $\Lambda$ in $[24,224]$; $\operatorname{lcm}(1,\dots,D)\le16^D$ for $D\le3000$; and the divisibility $\operatorname{lcm}(1..2n)/\operatorname{lcm}(1..n)\mid\binom{2n}n$ for $n\le300$. The explicit cover constants are supplied in cover_constants224.md: for $r=j+2$ and $N=2^r$, Riemann–Hurwitz gives $g\le1+N(r-1)$, and an additive Fourier function has total pole degree at most $2N$ on at most $N$ points. Its conductor is at most $N(2r+1)$; after exclusions, $16(r+1)N\sqrt p$ is a uniform loose bound. This bookkeeping was supplied by lattice_lift_research after conic_weaver's checks.

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