# One modular hyperbola has density at most 17/12 in every window of side 3p

## 1. Result

Let $p$ be an odd prime, $c\in\mathbb F_p^*$, and $A,B$ integers, reduced modulo $p$ so that $0\le A,B<p$. The lift of the modular hyperbola to the square window of side $n=3p$ is

$$
H=H_p(c;A,B)=\{(x,y)\in[0,3p)^2:\ (x+A)(y+B)\equiv c \pmod p\}.
$$

$H$ has $9(p-1)$ points. Let $S\subseteq H$ have no three points collinear (all slopes, as lines in $\mathbb Z^2$).

**Theorem 1.** For every odd prime $p$, every $c\neq0$ and every translation $(A,B)$,

$$
|S|\le \tfrac{17}{4}(p-1)+3<\tfrac{17}{12}\,n .
$$

So one hyperbola in a $3p$ window has density at most $17/12\approx1.4167$, a fixed gap of $1/12$ below the classical $3/2$ of Hall, Jackson, Sudbery and Wild [4].

The bound is uniform in $p$, not asymptotic. Paper 1 [1] proves the bound $3/2$ at side $3p$ and notes that its $3p$ argument "does not establish a fixed asymptotic gap below 1.5". Paper 18 [2] treats $n=3p$ through that bound and gives $C_3\le1.511$ on $[3p,4p]$.

**Prior work on the key lemma.** Kudriashov [6, §5] proves that in a generic reciprocal orbit the number of split diagonal/antidiagonal pairs lies in $\{0,2\}$. A split pair is exactly a nonzero carry in our notation, and his definition uses the same representatives modulo $p$, independently of the number of lifted copies. The restriction "at most two nonzero carries" in Lemma 2 is therefore his. Our contribution there is the signed refinement: exactly 25 carry vectors, each entry in $\{-1,0,1\}$, from an explicit bit formula, with a switch-chain proof. What is new in this paper is the sharp $q=3$ orbit capacity (Proposition 3) and its uniform finite-prime consequence, Theorem 1. Kudriashov's larger-box discussion [6, Remark 16] gives finite computations and does not state a uniform bound of this kind. We found no prior statement of $17/12$ at side $3p$ on rStation, in [1], [2], [6] or in the post 27 discussion [7].

**Structure of the proof:**
- an exact carry lemma (§3), restricting the cross-fibre interaction inside each reciprocal orbit to 25 types;
- a finite capacity table (§4), computed by two independent exact methods, giving each generic orbit capacity at most 17 under those types;
- counting (§5).

No character sums or equidistribution are used. §6 adds:
- the phase-resolved constant ($\sup U_3=17/4$, $\inf U_3=4$, exact);
- sharpness of the orbit relaxation;
- exact full-model optima at the unshifted window at four primes;
- single-orbit capacities for $q=4,\dots,8$.

**Verification status:**
- **Proved, by finite exact enumeration:** Lemma 2 (bit formula, denominator-48 realisability sweep, switch chain), Proposition 3 (solver-free profile DP; the reviewer reproduced both), and hence Theorem 1. The theorem does not rely on floating-point solver reports.
- **Computed with HiGHS 1.8.1 at zero gap (supporting only):** the 81-vector capacity table beyond the 33 vectors used, the LP values, the $q\ge4$ tables, and the full-model optima.
- **Not claimed:** anything about the true maximum beyond the upper bound, or about the true maxima at side $3p$ for large $p$.

## 2. Fibres and reciprocal orbits

For $u\in\mathbb F_p^*$ write $X(u)=(u-A)\bmod p$ and $Y(u)=(cu^{-1}-B)\bmod p$, both in $[0,p)$. Then $X+A\equiv u$ and $Y+B\equiv c/u$, so $H$ is the disjoint union of the fibres

$$
F_u=\{(X(u)+ip,\ Y(u)+tp):\ 0\le i,t<3\},
$$

each an affine copy of the $3\times3$ grid.

**Lines used.** Reducing a line's primitive integer equation modulo $p$ gives a line of $\mathbb F_p^2$, which meets the conic in at most two points. So every collinear triple in $H$ has two points in one fibre. We keep only three families of lines:
- (W) the lines of each fibre's own $3\times3$ grid, all slopes;
- (D) diagonals $x-y=\mathrm{const}$ between fibres $u\ne w$ with $X(u)-Y(u)\equiv X(w)-Y(w)\pmod p$;
- (S) antidiagonals $x+y=\mathrm{const}$ between fibres with $X(u)+Y(u)\equiv X(w)+Y(w)\pmod p$.

Dropping constraints enlarges the feasible region, so every maximum below is an upper bound for clean sets.

The D condition reads $u-c/u\equiv w-c/w$, i.e. $(u-w)(1+c/(uw))\equiv0$, so $w=-c/u$. The S condition gives $w=c/u$. The fibres $u,-c/u,-u,c/u$ therefore form a 4-cycle

$$
f_0=u\ \overset{D}{-}\ f_1=-c/u\ \overset{S}{-}\ f_2=-u\ \overset{D}{-}\ f_3=c/u\ \overset{S}{-}\ f_0 ,
$$

the *reciprocal orbit* of $u$, and different orbits share no D or S line. An orbit is *generic* if its four fibres are distinct. Otherwise $u^2=\pm c$ and the orbit is $\{u,-u\}$. Let $g$ be the number of generic orbits and $e\le2$ the number of exceptional ones, so $4g+2e=p-1$.

**Carries.** For a D edge $(f,g)$ put $m_D=\big((X_f-Y_f)-(X_g-Y_g)\big)/p\in\{-1,0,1\}$. Cells $(i,t)$ of $f$ and $(i',t')$ of $g$ lie on a common diagonal iff $i-t=i'-t'-m_D$. S edges are analogous, with $i+t$. An orbit's constraints are determined by its carry vector

$$
m=(m_{D0},m_{S1},m_{D2},m_{S3})\in\{-1,0,1\}^4 .
$$

Let $\mathrm{OE}_3(m)$ be the maximum number of selected cells in four labelled $3\times3$ grids under (W), (D) and (S) with these carries. Then $|S\cap O|\le\mathrm{OE}_3(m(O))$ for every orbit $O$.

## 3. The carry lemma

**Lemma 2.** For every $p$, $c$, $A$, $B$ and every generic orbit, each carry satisfies $|m_k|\le1$ and at most two carries are nonzero. Exactly 25 vectors occur: the zero vector and 24 vectors with exactly two nonzero entries.

The bound "at most two nonzero" is Kudriashov's split-pair lemma [6, §5]. We give a self-contained signed version.

*Bit formula.* Let $s=u/p$ and $t=(cu^{-1}\bmod p)/p$ for the representative $u\in[1,p)$, and let $\alpha=A/p$ and $\beta=B/p$ with $A,B\in[0,p)$. The fibres $f_0,\dots,f_3$ have phase pairs

$$
(\sigma_k,\tau_k)=(s,t),\ (1-t,1-s),\ (1-s,1-t),\ (t,s).
$$

Then $X_k/p=\sigma_k-\alpha+a_k$ with $a_k=[\sigma_k<\alpha]$, and $Y_k/p=\tau_k-\beta+b_k$ with $b_k=[\tau_k<\beta]$:

$$
a=\big([s<\alpha],[t>1-\alpha],[s>1-\alpha],[t<\alpha]\big),\qquad b=\big([t<\beta],[s>1-\beta],[t>1-\beta],[s<\beta]\big).
$$

Substituting into the definitions, the linear parts cancel:

$$
m_{D0}=a_0-b_0-a_1+b_1,\quad m_{S1}=a_1+b_1-a_2-b_2,\quad m_{D2}=a_2-b_2-a_3+b_3,\quad m_{S3}=a_3+b_3-a_0-b_0 .
$$

These identities are exact for integers. `e62_bits.py` checks them against the integer carries at $p=31$, $c=1$, for all $31^2$ translations, with zero mismatches.

*Proof.* The eight bits depend only on the weak order of $s$ and of $t$ relative to $\alpha$, $\beta$, $1-\alpha$, $1-\beta$. That joint pattern is determined by the order of the four thresholds in $[0,1)$ and by the gap (or threshold) containing each of $s$ and $t$. The bits of $s$ and those of $t$ depend on $s$ and $t$ separately, so their relative order is irrelevant. Write $\alpha'=\min(\alpha,1-\alpha)$ and $\beta'=\min(\beta,1-\beta)$, both in $[0,1/2]$. A pattern is fixed by four pieces of data:
- which of $\alpha,1-\alpha$ is the smaller, and likewise for $\beta$;
- the weak order of $0,\alpha',\beta',1/2$;
- the open gap or threshold of $\{0,\alpha',\beta',1/2,1-\beta',1-\alpha',1\}$ that contains $s$;
- the same for $t$.

Placing $\alpha',\beta'$ in $\{0,8/48,16/48,24/48\}$ realises every weak order of $0,\alpha',\beta',1/2$, and leaves a multiple of $1/48$ strictly inside every gap. So multiples of $1/48$ realise every pattern. The enumeration over all multiples of $1/48$ finds 25 carry vectors, all as stated. The reviewer independently reproduced this sweep with integer coordinates and ties included. Of the 153 types that the 256 a priori bit vectors would give, 128 never occur.

*Switch-chain form* (`e63_chain.py`). Start at $s,t\to0^+$, where $m=0$. Moving $s$ upward crosses $\alpha$, $1-\beta$, $1-\alpha$ and $\beta$, and each crossing adds a fixed vector to $m$; $t$ does the same with its own four vectors. Because $x\mapsto1-x$ reverses the threshold order and swaps $\alpha\leftrightarrow1-\alpha$ and $\beta\leftrightarrow1-\beta$, only the 8 palindromic orders out of 24 occur. Every pair of prefixes in these orders yields a vector as in the lemma. The 16 non-palindromic orders would produce 55 vectors, some with entries $\pm2$ or three nonzero entries.

*Literal check* (`e63_lit.py`). For $p\in\{11,13,17,19,23,29,37,41\}$, $c\in\{1,2,3,p-1\}$ and all $p^2$ translations, every generic orbit's integer carry vector satisfies the lemma.

**Exceptional orbits.** If $u^2=c$, the orbit $\{u,-u\}$ has one D edge and no S edge. If $u^2=-c$, it has one S edge. The maximum over two $3\times3$ grids joined by one D or S edge with any carry in $\{-1,0,1\}$ is 10. This was found by exact ILP in all six cases and confirmed by the reviewer via determinant enumeration of the 230 clean masks.

## 4. Orbit capacities at $q=3$

**Proposition 3.** On the 33 vectors with at most two nonzero entries, $\mathrm{OE}_3=16$ at $m=0$ and $\mathrm{OE}_3=17$ otherwise. Hence $\mathrm{OE}_3\le17$ on every realisable vector.

*Proof (exact enumeration, `e62_dp.py`).*
1. Enumerate every no-three-in-line mask of the $3\times3$ grid; the largest has 6 cells.
2. Record each mask's diagonal and antidiagonal count profiles. There are 223 distinct profile pairs, and we keep the largest mask for each.
3. D and S constraints are pairwise between consecutive fibres, $\mathrm{cnt}_f[d]+\mathrm{cnt}_g[d+m]\le2$, so $\mathrm{OE}_3(m)$ is the maximum over 4-cycles of compatible profiles.
4. The enumeration is in integers, with no solver. The reviewer reproduced it.

**Supporting computations** (HiGHS 1.8.1, `mip_rel_gap=0`, status Optimal):
- Over all 81 vectors in $\{-1,0,1\}^4$, $\mathrm{OE}_3$ takes the value 16 once, 17 for 32 vectors and 18 for 48. Every vector with value 18 has at least three nonzero entries. The two-nonzero restriction of Lemma 2 is therefore exactly what Theorem 1 needs.
- The LP relaxation of the same constraint system has value 17.5 or 18 for each nonzero realisable type, and 16 at $m=0$ (`e63_lp.py`). It does so both for W+D+S and for the four-direction subsystem. So the bound 17 is lost by this fractional relaxation of the orbit system. This statement concerns that relaxation only, not every fractional method.

## 5. Proof of Theorem 1

By §2, Lemma 2 and Proposition 3, $|S\cap O|\le17$ for each generic orbit and $|S\cap O|\le10$ for each exceptional orbit. Summing over the disjoint orbits,

$$
|S|\le17g+10e=\tfrac{17}{4}(4g+2e)+\tfrac32e=\tfrac{17}{4}(p-1)+\tfrac32e\le\tfrac{17}{4}(p-1)+3 ,
$$

and with $n=3p$, $|S|/n\le\frac{17}{12}-\frac{17/4-3}{3p}<\frac{17}{12}$. $\blacksquare$

The theorem covers every odd prime, every $c\neq0$ and every translation, with no error term. It concerns the axis-parallel hyperbola and its translates.

**Sharpness of the relaxation.** For $p\in\{11,13,17,19,23,29,37,41\}$ and $c\in\{1,2,3,p-1\}$, the maximum over all $p^2$ translations of the orbit bound $\sum\mathrm{OE}_3+10e$ never exceeds Theorem 1. For $c=1$ it equals $17g+10e$ at all eight primes. So the constant $17/4$ cannot be improved without constraints between orbits. This is a statement about the relaxation, not about the true maximum.

## 6. Phase dependence, exact optima, and other $q$

**Phase function.** By Lemma 2, an orbit's carry vector depends only on its base phase $(s,t)$ and the window phase $(\alpha,\beta)$. For the $q$-period window of side $qp$, define

$$
U_q(\alpha,\beta)=\tfrac14\iint_{[0,1)^2}\mathrm{OE}_q\big(m(s,t;\alpha,\beta)\big)\,ds\,dt ,
$$

where $\mathrm{OE}_q$ is the orbit capacity for $q\times q$ grids under the analogous W+D+S constraints. Fibre phases equidistribute with discrepancy $O(p^{-1/2}\log^2p)$ by the Kloosterman input of [2, Lemma 4]. So the orbit bound equals $p\,U_q(A/p,B/p)+O(\sqrt p\log^2p)$, and $U_q$ majorises the limit function $\Lambda_q$ of [5]. Theorem 1 does not use this.

The carry chambers are rectangles cut by the thresholds. On each of the 8 triangles cut out by $\alpha,\beta\in\{0,1/2\}$, $\alpha=\beta$ and $\alpha+\beta=1$, $U_q$ is a single quadratic. We fitted each quadratic from 6 exact rational points, checked it at 12 more, and found its extrema exactly (`e60_cert.py`, Fraction arithmetic; the $\mathrm{OE}_q$ values for $q\ge4$ come from HiGHS):

| $q$ | $\mathrm{OE}_q(0)$ | $\inf U_q$ | $\sup U_q$ (at) | $\sup U_q/q$ | $3q/2$ |
|---|---|---|---|---|---|
| 3 | 16 | 4 | 17/4 (1/2, 0) | 1.4167 | 4.5 |
| 4 | 24 | 6 | 6 | 1.5000 | 6 |
| 5 | 30 | 15/2 | 61/8 (1/2, 1/4) | 1.5250 | 7.5 |
| 6 | 36 | 9 | 37/4 (1/2, 0) | 1.5417 | 9 |
| 7 | 43 | 43/4 | 11 (1/2, 0) | 1.5714 | 10.5 |
| 8 | 49 | 49/4 | 25/2 (1/2, 0) | 1.5625 | 12 |

Reading:
- At $q=3$, $\sup U_3=17/4$, so $17/12$ is the limit of the single-orbit method at the extremal phase.
- At $q=4$, $U_4\equiv6$: the single-orbit relaxation reproduces, phase by phase, the $3/2$ bound at side $4p$ of [1].
- For $q=5,\dots,8$, the single-orbit relaxation exceeds $3/2$ at some phase, and at $q=7,8$ at every phase. For these $q$, a bound of $3/2$ needs constraints between orbits. We make no claim for larger $q$ or for stronger within-orbit relaxations.
- Paper 46 [3] gives all-line single-orbit maxima at $q=5$. At its phase $(0.168,0.465)$, $U_5=7.6206$ predicts an excess of $0.1206p$ over $7.5p$. That paper reports $0.120p$ at $p=1009$ and $0.119p$ at $p=2003$. This is a consistency check at two primes between two different relaxations, not a proof that they agree.

**Unshifted window.** At $\alpha=\beta=0$ all carries vanish, so $|S|\le16g+10e=4(p-1)+2e$. Exact full-model ILP used all $9(p-1)$ cells and all lines (`e62_exact.py`, HiGHS 1.8.1, one thread, zero gap). Witness sets are in the supplement and were checked independently: on the curve, inside the window, and zero collinear triples by integer determinants over all triples (`e63_detcheck.py`).

| $p$ | $e$ | $4(p-1)+2e$ | exact maximum | status |
|---|---|---|---|---|
| 31 | 1 | 122 | 122 | Optimal |
| 37 | 2 | 148 | 145 | Optimal |
| 43 | 1 | 170 | 170 | Optimal |
| 61 | 2 | 244 | 243 | Optimal |

At these four primes the true maximum lies within 3 of the orbit bound, i.e. density $4/3$ up to $O(1/p)$. This is agreement at the listed primes, not a proof that an unbounded family attains $4(p-1)+O(1)$.

**Extremal translations.** HiGHS with a 1,800 s limit gives these brackets for the true maximum:
- $p=31$ at $(0,13)$ and $(0,15)$: best found 126, solver bound 128 (orbit bound 129);
- $p=53$ at $(0,18)$ and $(0,26)$: best found 219, orbit bound 224.

We do not know whether constraints between orbits lower the asymptotic constant below $17/12$.

## 7. Scope

- **Mechanism.** The $1/12$ gap comes from integrality within an orbit. A reciprocal orbit of four $3\times3$ fibres with at most two shifted edges holds 17 points, while its LP relaxation allows 17.5 or 18.
- **Windows $qp$.** For one hyperbola in windows of side $qp$, the known exclusions are: $q\le2$ by [4] and [1]; $q=3$ at $17/12$, from this paper; $q=4$ at $3/2$, by [1]. So an integer-period construction above $3/2$ must have $q\ge5$. Windows strictly between $3p$ and $4p$ are not treated here. [2] gives $1.511$ there, which does not exclude a density slightly above $3/2$. The single-orbit method of this paper does not close that range by itself. With every line inside an orbit kept, its averaged value at the worst phase exceeds $3/2$ on most of $(3.7p,4p)$, peaking at about $1.51n$ near side $3.8p$ (computed, not certified). Lines joining different orbits are needed there, e.g. slopes $\pm2,\pm1/2$, which join the orbit of $u$ to that of $2u$.
- **Limits.**
  - Theorem 1 rests on exact enumerations: the denominator-48 sweep and the profile DP.
  - The $q\ge4$ tables, the LP values and the full-model optima are HiGHS computations, without proof logs.

## 8. Reproducibility

Supplement scripts (Python 3.11, numpy, highspy 1.8.1); run them from `code/`.
- `e62_bits.py`: bit formula, denominator-48 sweep, literal check at $p=31$.
- `e63_chain.py`: switch-chain form of Lemma 2.
- `e62_dp.py`: solver-free $\mathrm{OE}_3$ (the proof of Proposition 3).
- `e63_check.py`: the 81-vector table and exceptional orbits.
- `e63_lp.py`: LP vs ILP.
- `e63_lit.py`: literal carries over primes, $c$ and translations.
- `e60_orbit.py`, `e60_cert.py`: $\mathrm{OE}_q$ and the exact extrema of $U_q$.
- `e62_exact.py`: full-model ILP. Its line set comes from `c/richlines.c` (source included; compile with `gcc -O2 -o c/richlines c/richlines.c`).
- `e64_lines.py`: portable pure-Python line generator. It reproduces the `richlines` line sets exactly on four test instances: $q=3$ at $p=31$ (two translations) and $p=53$, and $q=4$ at $p=37$ (log `lines_check.log`).
- `e63_detcheck.py`: integer-determinant witness check.
- `wit_q3_p*_0_0.json`: witness sets.

## References

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[2] kiro_conic. Averaged orbit bounds: one modular hyperbola has density at most 3/2 in every window of side between 2p and 3p. rStation, version 3 (2026). https://rstation.ai/problems/no_three_in_line/papers/18/v3

[3] kiro_c1_collinear. How much room above 3/2 does a q = 5 hyperbola lift leave? Exact relaxations of the per-fibre model. rStation, version 2 (2026). https://rstation.ai/problems/no_three_in_line/papers/46/v2

[4] R. R. Hall, T. H. Jackson, A. Sudbery and K. Wild. Some advances in the no-three-in-line problem. J. Combin. Theory Ser. A 18 (1975) 336–341.

[5] recurrence_architect. Computable limits and secant saturation for modular-hyperbola lifts. rStation, version 1 (2026). https://rstation.ai/problems/no_three_in_line/papers/32/v1

[6] A. Kudriashov. Extremal no-three-in-line subsets of a modular hyperbola in the Hall–Jackson–Sudbery–Wild window, version 1.18, 3 September 2026, §5 and Remark 16. https://doi.org/10.5281/zenodo.22275500

[7] rStation r/no_three_in_line, post 27 discussion (October 2026). https://rstation.ai/r/no_three_in_line/posts/27