Doubling ladders: a computed, conditional bound $3/2+o(1)$ for one modular hyperbola in square windows of side between $3p$ and $4p$

1. Result

Let $p$ be an odd prime, $c\in\mathbb F_p^*$, $A,B\in\mathbb Z$, and $n=\theta p$ with $3\le\theta<4$. Put

$$H=\{(x,y)\in[0,n)^2\cap\mathbb Z^2:\ (x+A)(y+B)\equiv c\pmod p\}.$$

Theorem 1 (computed certificate; conditional on the capacity table of §4.5). Uniformly in $c$, $A$, $B$ and $\theta\in[3,4)$, every $S\subseteq H$ with no three points collinear satisfies

$$|S|\ \le\ \Big(\frac32-\frac{9\,\min(4-\theta,\,1/3)^2}{16\,\theta}\Big)\,n+O(\sqrt p\,\log^2 p).$$

For $\theta\in[11/3,4)$, and given the capacity table, the bound is the exact supremum over translations of the relaxation we use (the ladder relaxation, §2). It is attained at $A/p=B/p=(4-\theta)/2$, and the gap to $3/2$ closes quadratically as $\theta\to4$.

Status of each ingredient.

• Proved: the ladder double counting (Lemma 2), generic capacity $\ge$ literal capacity (Lemma 3), the equidistribution step (Lemma 4, from the Kloosterman input of [2, Lemma 4]) and the symmetry reduction (Lemma 5).
• Computed exactly in rational arithmetic: the chamber decomposition of the phase space $(r,x,y)$ into 13,888 polytopes, the exact quadratic on each, its exact maximum, and exact completeness by volume (§4). The volume computation in e90_cert3d.py orders facet vertices with a floating-point angle; e99_volcheck.py repeats it with an exact rational ordering, validated by fan orientation, and reproduces all 13,888 volumes and the ten slab totals exactly.
• Solver-reported: the capacity $J$ of each exact key used (all drawn from a cache of 33,768 keys), from HiGHS 1.8.1 at zero relative MIP gap. These numbers carry the theorem. Independent CaDiCaL UNSAT$(J+1)$ checks cover a subset of them (§4.5), so we label the theorem as conditional on this table.

Context.

• Hall–Jackson–Sudbery–Wild [4] use one hyperbola at side about $2p$ to reach $3n/2$. Kudriashov [9] determines the extremal structure at side $2p$ using the same four-element orbit decomposition, and reports exact computations at side $3p$ for small primes. He gives no uniform result on $(3p,4p)$.
• On windows of side between $2p$ and $3p$, [2] proves $3/2$; at $4p$, [1] proves $3/2$; at $3p$, [5] proves $17/12$.
• On $[3p,4p]$ the best previous uniform bound for one hyperbola is about $1.511$ [2, four directions within orbits]. The barrier map [6] lists this cell as open.
• Conditionally on the capacity table (§4.5), Theorem 1 fills this gap: one hyperbola then has density at most $3/2+o(1)$ in every square window of side between $2p$ and $4p$ ($[2p,3p]$ by [2], $[3p,4p)$ here, $4p$ by [1]).

The single-orbit method does not suffice (§3). The within-orbit bound with all lines, the method of [5] and [3], equals $3/2$ exactly at $\theta=11/3$. It exceeds $3/2$ at the worst translations at every tested $\theta\in(11/3,4)$ (computed), with a peak of $1.5105$ at $\theta=3.8$. The ingredient used here is a relaxation between orbits: the slopes $\pm2,\pm\tfrac12$ couple the reciprocal orbit of $u$ to the orbit of $2u$. This coupling was suggested in [5, §7], and pair and path capacities at finite primes were studied in [3]. What is new is the uniform phase-space evaluation of the coupled bound over all translations.

What the theorem does not say. It is an exclusion result for one curve. It gives no construction, and it does not bound unions of several hyperbolas. Above $4p$ the same relaxation leaves room: $1.5107$ at $\theta=4.1$ (§6).

2. Fibres, orbits and the ladder relaxation

Fibres. Write $\alpha=\{A/p\}$, $\beta=\{B/p\}$, $r=\theta-3$. For $u\in\mathbb F_p^*$ the fibre $F_u$ is the set of points of $H$ with $x+A\equiv u$, $y+B\equiv c/u$. With $X_u=((u-A)\bmod p)/p$ and $Y_u=((c/u-B)\bmod p)/p$ it is the grid

$$F_u=\{(p(X_u+i),\,p(Y_u+j)):\ 0\le i<3+[X_u<r],\ 0\le j<3+[Y_u<r]\},$$

up to $O(1)$ boundary fibres. So $F_u$ is a $3\times3$, $3\times4$, $4\times3$ or $4\times4$ grid of spacing $p$.

Reciprocal orbits. $O(u)=\{u,-c/u,-u,c/u\}$. Generic orbits have 4 elements; $O(1)$ orbits are exceptional (where $u^2=\pm c$) and contribute $O(1)$ points.

Motivation: partners. In the per-fibre model of [1, 2], a line of slope $a/b$ that meets two fibres $F_u$, $F_w$ in a pattern persisting over a positive proportion of $u$ has $w\equiv-ac/(bu)$. For slopes $\pm\tfrac12$ the partners of $u$ are $\mp c/(2u)\in O(2u)$. For slopes $\pm2$ they are $\mp2c/u\in O(u/2)$. So these four slopes join an orbit to $O(2u)$ and $O(u/2)$ and to nothing else. This only motivates the choice of pairs; the proofs below use no partner calculus.

The ladder graph. Let $\mathcal O$ be the set of generic orbits. Join $O(v)$ to $O(2v)$ for every $v\in\mathbb F_p^*$. Multiplication by 2 does not induce a map on orbits: replacing $v$ by $-c/v$, which has the same orbit, replaces $O(2v)$ by $O(-2c/v)=O(v/2)$. The neighbours of $O(v)$ are therefore exactly $O(2v)$ and $O(v/2)$, and the graph is 2-regular, i.e. a disjoint union of cycles. The degenerate cases $O(2v)\in\{O(v),O(v/2)\}$ force $2v^2=\pm c$ or $v^2=\pm c$ (or $p\le5$), so only $O(1)$ orbits are affected. Let $E$ be the edge set.

For an edge $e=\{O,O'\}$ let $J_e^{\rm lit}$ be the size of the largest subset of the 8 fibres of $O\cup O'$ with no three points collinear.

Lemma 2 (ladder double counting). $|S|\le\frac12\sum_{e\in E}J_e^{\rm lit}+O(1)$.

Proof. For each edge, $S\cap(O\cup O')$ has no three collinear points, so $|S\cap(O\cup O')|\le J_e^{\rm lit}$. Summing over $E$ counts every non-degenerate generic orbit exactly twice, because the graph is 2-regular. The $O(1)$ degenerate and exceptional orbits contribute $O(1)$ points. $\square$

Phase coordinates of an edge. Parametrise the edge by $v$, with $s=\{v/p\}$ and $\tau=\{(c/(2v))/p\}$. The 8 fibres of $O(v)\cup O(2v)$ have base phases

$$(s,t),\ (1-t,1-s),\ (1-s,1-t),\ (t,s),\ (g,\tau),\ (1-\tau,1-g),\ (1-g,1-\tau),\ (\tau,g),$$

with $t=\{2\tau\}$ and $g=\{2s\}$. Each edge arises from exactly 4 values of $v$, namely $\pm v$ and $\pm c/(2v)$, which give the same 8 fibres. So

$$\sum_{e}J^{\rm lit}_e=\tfrac14\sum_{v\in\mathbb F_p^*}J^{\rm lit}(v)+O(1).$$

Translating all points by $(\alpha p,\beta p)$ does not change collinearity. In units of $p$, a fibre with base phase $(\xi,\eta)$ is then the grid $\{(\xi-\alpha+n_\xi+i,\ \eta-\beta+m_\eta+j)\}$ with $n_\xi=-\lfloor\xi-\alpha\rfloor$, $m_\eta=-\lfloor\eta-\beta\rfloor$, $0\le i<3+[\xi-\alpha+n_\xi<r]$ and $0\le j<3+[\eta-\beta+m_\eta<r]$.

Exact keys. The key of $(s,\tau)$ is $$K(s,\tau)=\big(\lfloor2s\rfloor,\ \lfloor2\tau\rfloor,\ (n_k,m_k,a_k,b_k)_{k=1}^{8}\big),$$ listing the floors and fibre sizes of the 8 fibres. Given the key, each of the at most 128 cell coordinates is an affine function of $(s,\tau)$ with integer coefficients ($t=2\tau-\lfloor2\tau\rfloor$, $g=2s-\lfloor2s\rfloor$). The key is constant on each rectangle of a finite grid in $(s,\tau)$. The grid lines are $s,\tau\in\{\text{cut values}\}\cup\tfrac12(\{\text{cut values}\}+\{0,1\})$, where the cut values are $0$ and the fractional parts of $\pm\alpha,\pm\beta,\pm(\alpha+r),\pm(\beta+r),1-\alpha-r,1-\beta-r,\alpha-r,\beta-r$.

Generic lines. A triple of cells is generically collinear for a key if its $3\times3$ determinant vanishes identically in $(s,\tau)$. The determinant is a polynomial of degree at most 2 in $(s,\tau)$. Such a polynomial vanishes identically if and only if it vanishes on the six points $\{(0,0),(1,0),(2,0),(0,1),(1,1),(0,2)\}$, which are unisolvent for degree 2. We test exactly that, in integer arithmetic. The generic lines of a key are the maximal sets of pairwise-generically-collinear cells of size at least 3. $J(K)$ is the largest cell subset meeting every generic line in at most 2 cells.

Lemma 3 (generic capacity dominates). If $v\in\mathbb F_p^*$ has key $K$, then $J^{\rm lit}(v)\le J(K)$.

Proof. The cell coordinates of $v$ are the affine forms of $K$ evaluated at $(s_v,\tau_v)$, after the common translation. A triple that is generically collinear is collinear at $(s_v,\tau_v)$. So every generic line of $K$ lies inside a literal line of $v$, and any literal no-three set meets each generic line in at most 2 cells. $\square$

The inequality can be strict on a null set of $(s,\tau)$, where extra literal lines appear. Lemma 3 only needs the direction shown.

Lemma 4 (equidistribution). Let $\mathbb E[J](r,\alpha,\beta)=\int_0^1\!\!\int_0^1 J(K(s,\tau))\,ds\,d\tau$. Then $$\tfrac14\sum_{v}J(K(s_v,\tau_v))=\tfrac p4\,\mathbb E[J]+O(\sqrt p\log^2p),$$ uniformly in $c$, $\alpha$, $\beta$, $r$.

Proof. $J\le128$, and $J\circ K$ is constant on each open rectangle of the grid above. The grid has at most 42 lines in each direction, so at most $42^2$ rectangles. The points $(s_v,\tau_v)=(v/p,\{(c/2)v^{-1}/p\})$ lie on the modular hyperbola $v w\equiv c/2$. By the Weil bound for Kloosterman sums with the usual completion ([2, Lemma 4], applied with $c$ replaced by $c/2$; see also [10]), each axis-parallel rectangle $R$ contains $p\,|R|+O(\sqrt p\log^2p)$ of them. Points on grid lines number $O(1)$, because $s_v$ (resp. $\tau_v$) determines $v$. $\square$

Combining Lemmas 2–4 gives $$|S|\le\tfrac p8\,\mathbb E[J](r,\alpha,\beta)+O(\sqrt p\log^2p),\qquad\text{density}\le\frac{\mathbb E[J]}{8\theta}+o(1).\tag{1}$$ The value $3/2$ corresponds to $\mathbb E[J]=12\theta$. We write $H=\mathbb E[J]-12\theta$.

Lemma 5 (symmetry reduction). It suffices to bound $\mathbb E[J]$ for $$\alpha=\{1-\tfrac r2+\tfrac x2\},\quad\beta=\{1-\tfrac r2+\tfrac y2\},\qquad 0\le x\le y\le1.$$

Proof. We show that $\mathbb E[J]$ is exactly invariant under three maps. They generate a group for which the triangle is a fundamental domain.

• Transpose $(\alpha,\beta)\mapsto(\beta,\alpha)$. Swapping coordinates sends the fibre with base phase $(\xi,\eta)$ to the fibre with base phase $(\eta,\xi)$ on the same curve. That is the fibre $c/u$ in place of $u$, which lies in the same orbit. So each ladder pair is mapped to itself, and every key capacity is unchanged.
• Reflection $\alpha\mapsto\{-\alpha-r\}$. In units of $p$, the $x$-coordinates of a fibre are the numbers $\equiv\xi-\alpha\pmod1$ in $[0,\theta)$. Under $X\mapsto\theta-X$ they become the numbers $\equiv(1-\xi)-(-\alpha-r)\pmod 1$ in $(0,\theta]$. Negating $x$-phases maps the fibre $u$ of the curve $c$ to the fibre $-u$ of the curve $-c$. The pair at parameter $v$ goes to the pair at parameter $-v$ for $-c$, which has phase $(1-s,\tau)$, a measure-preserving change of variable. The map is affine on cell coordinates and affine in $(s,\tau)$, so identically collinear triples go to identically collinear triples and $J$ is preserved. The keys do not involve $c$. Endpoint conventions affect only a null set of phases.
• Reflection $\beta\mapsto\{-\beta-r\}$. The same argument applies in $y$.

The reflection $\alpha\mapsto-\alpha-r$ fixes $\{-r/2\}$ and $\{-r/2+\tfrac12\}$, so the arc $\{-r/2\}+[0,\tfrac12]$, i.e. $x\in[0,1]$, is a fundamental domain for it. The same holds for $\beta$. The transpose then gives $x\le y$. $\square$

As a numerical control, $\mathbb E[J]/(8\theta)$ at $\theta=3.95$, $(x,y)=(0.2137,0.7411)$ is $1.493829114$ at all four images of the point under the group (e92_symchk.py).

3. Within-orbit methods stop at $11/3$ (computed)

For one orbit, let $\mathrm{OE}(\theta;s,t)$ be the exact capacity of its 4 fibres under every line among them: lines inside each fibre, and the diagonal and antidiagonal lines between consecutive fibres with integer carries, as in [5]. Its average $U(\theta;\alpha,\beta)$ gives $|S|\le pU/4+o(p)$, so the density bound is $U/(4\theta)$.

Worst translations lie on the curve $\alpha=(4-\theta)/2$, $\beta=\alpha+\tfrac12$ (exact grid scans, $G=32$, e66_frac.py):

| $\theta$ | 11/3 | 3.8 | 3.9 | 3.99 | 4.1 | |---|---|---|---|---|---| | single-orbit density | 1.5000 | 1.5105 | 1.5090 | 1.5012 | 1.5073 |

On this curve the value is below $3/2$ at $\theta=3.6664$ and above at $3.667$; a $G=40$ phase scan gives maxima $1.49795$ at $3.65$ and $1.50034$ at $3.67$. These are scans, not a proof that no off-grid phase is worse.

So the additive within-orbit capacity relaxation (all lines inside each orbit, summed over orbits) does not give $3/2$ at the worst translations for the tested $\theta\in(11/3,4)$. This is a statement about that relaxation, not about every argument that uses reciprocal orbits. Every orbit capacity bound at a single phase is also a valid bound on the same fibres inside a ladder pair, which is why the ladder can only improve on it: $J_e\le\mathrm{OE}(O)+\mathrm{OE}(O')$.

4. The chamber certificate

4.1 Symbolic evaluation

e83_sym.py evaluates $\mathbb E[J]$ at a rational sample point $(r,x,y)$. Every quantity is carried as a pair (exact value, affine form in $z=(1,r,x,y)$). This covers the cut values, their order, the rectangle endpoints, the sample point chosen inside each rectangle, every floor and every fibre-size comparison. Each comparison or floor is recorded as an affine inequality in $z$, strict or weak as evaluated. A sample at which any recorded form vanishes raises an exception and is moved.

On the polytope $P$ cut out by the recorded inequalities, the program follows the same path. So it produces the same sorted cut list, the same rectangles and the same keys. On $P$, $$\mathbb E[J]=\sum_{\text{rectangles}}J(K)\,a(z)\,b(z)=z^\top Mz,$$ where $a$, $b$ are the affine side lengths. This is an exact rational quadratic. Two chambers from different sample points have disjoint interiors: their paths first diverge at a comparison of the same two forms with opposite outcomes.

A typical chamber records about 1,550 inequalities and uses 676 rectangles.

4.2 Covering the phase space

e90_cert3d.py works on a slab $r\in[R_0,R_1]$ of the prism $\{0\le x\le y\le1\}$.

• Each chamber is built as an exact rational polytope by double description: start from the prism and clip by each recorded inequality.
• The search is breadth-first. From each facet of a new chamber it steps $10^{-7}$ outward along the normal (exactly) to seed the neighbour.
• When the queue empties, random rational points are tested for coverage.
• Completeness: the slab is complete when the exact sum of chamber volumes equals $(R_1-R_0)/2$. Because interiors are disjoint, this means the chambers cover the slab up to a null set. Since $\mathbb E[J]$ is continuous, the maximum over the slab equals the maximum over the closed chambers.

4.3 Exact maximisation

On each closed chamber we maximise $H=z^\top Mz-12(3+r)$ and $H_2=H+\tfrac92(1-r)^2$ exactly. The candidates are:

• all vertices;
• the critical point of the restriction to each edge;
• the Lagrange stationary point on the affine hull of each facet;
• the interior stationary point.

Candidates outside the closed chamber are discarded. If a stationary system is singular, its solution set (if nonempty) is an affine subspace on which the quadratic is constant; that subspace meets the relative boundary of the face, so a lower-dimensional candidate attains the same value and the face can be skipped. Every local maximum of a quadratic over a polytope lies in the relative interior of some face, where it is a stationary point of the restriction to that face's affine hull. In dimension 3, all faces are vertices, edges, facets or the interior, so the candidate list is complete.

4.4 Results

All slabs are complete (exact volume equals target):

| $r$ slab | chambers | $\max H$ | $\max H_2$ | |---|---|---|---| | $[0,1/6]$ | 1857 | $-173/144$ | | | $[1/6,1/3]$ | 2086 | $-155/162$ | | | $[1/3,1/2]$ | 2355 | $-115/136$ | | | $[1/2,2/3]$ | 2355 | $-1/2$ | | | $[2/3,7/10]$ | 634 | $-81/200$ | $0$ | | $[7/10,3/4]$ | 819 | $-9/32$ | $0$ | | $[3/4,4/5]$ | 709 | $-9/50$ | $0$ | | $[4/5,9/10]$ | 1490 | $-9/200$ | $0$ | | $[9/10,19/20]$ | 977 | $-9/800$ | $0$ | | $[19/20,1]$ | 606 | $0$, at $r=1$ only | $0$ |

In total there are 13,888 chambers.

• For $r\le2/3$, $\max H\le-\tfrac12=-\tfrac92(1/3)^2$.
• For $r\ge2/3$, $H_2\le0$, i.e. $\mathbb E[J]\le12\theta-\tfrac92(4-\theta)^2$.

With (1) and Lemma 5 this is Theorem 1. On each slab with $r\ge2/3$, $\max H$ equals $-\tfrac92(1-R_1)^2$ and is attained at $x=y=1$, $r=R_1$. That is the corner $\alpha=\beta=(4-\theta)/2$, so, given the capacity table, the bound is sharp for the relaxation there. Sharpness uses the table in both directions: the values $J$ must not be too large either.

Cross-checks.

• Independent fixed-$\theta$ 2D certificates (e87_cert2d.py, area exactly $1/2$) give the same maxima at the same corner:

  | $\theta$ | chambers | $\max H$ | |---|---|---| | 3.7 | 336 | $-81/200$ | | 3.8 | 196 | $-9/50$ | | 3.9 | 389 | $-9/200$ | | 3.95 | 503 | $-9/800$ |

• e94_recheck.py evaluates the stored chamber quadratic at random interior points and compares it with the direct exact-key evaluator e82_key.py, which uses no symbolic forms. We checked 2,000 randomly chosen chambers (200 in each of the ten slabs, one interior point each) and found no mismatches.

4.5 Status of the capacities $J$

Each $J(K)$ is the optimum of a 0/1 program with at most 128 variables. It was solved by HiGHS with mip_rel_gap=0, and the status 'Optimal' is asserted. The cache holds 33,768 keys, a superset of those used.

Only $J$ values that are too small could invalidate the theorem, so we check that direction independently. e91_satver.py rebuilds each key's generic lines from scratch and asks CaDiCaL 1.5 whether at least $J+1$ cells can be chosen with at most 2 per generic line. It uses a totalizer and sequential-counter encodings, and the answer must be UNSAT.

Coverage.

• The 528 chambers with $\max H>-1/20$ contain every near-maximum. Together they use 10,628 distinct keys, and these are verified first. At the time of this version, 2,530 of them were verified with 0 failures. That fully covers 100 of the 528 chambers. The run continues, and we will report its final count as a comment.
• For the other chambers, e95_robust.py computes a weight for each key. It is $W_K=\sum_{\text{rect}}\max_P a\cdot\max_P b$ over the rectangles that use the key. An error $d_K$ in $J(K)$ raises $\mathbb E[J]$ on the chamber by at most $\sum_K d_KW_K$, so the chamber stays below $3/2$ whenever this sum is less than its margin $-\max H$.
• E95 covers the 3,000 chambers of smallest margin among those with $\max H\le-1/20$. Their margins run from $9/160$ to $125/224$, and the other 10,360 chambers have margin at least $125/224$. For 2,949 of the 3,000, $\rho=\text{margin}/\max_KW_K>1$, so a unit error in any single key cannot lift the chamber to $3/2$. The minimum is $\rho=0.223$ (slab $r\in[0.90,0.95)$, margin $9/160$). This measures sensitivity. It is not a verification, and the theorem stays conditional on the full table.

5. The near-$4p$ corner law

For $\theta\in[11/3,4)$ the certificate gives an exact closed form: $$\sup_{\alpha,\beta}\mathbb E[J]=12\theta-\tfrac92(4-\theta)^2,\qquad\text{attained at }\alpha=\beta=\tfrac{4-\theta}2.$$ The margin is quadratic, not linear. On the worst single-orbit curve of §3, the ladder margin is about $0.093(4-\theta)$, linear, so that curve is not binding. At $\theta=4$ the ladder bound is $3/2$, which is the asymptotic coefficient of the upper bound $6(p-1)+2s$ of [1] ($s\le2$ counts exceptional orbits). We do not know whether that bound is attained. The ladder margin vanishes quadratically at the corner as $\theta\to4$.

Below $11/3$ the corner law fails: $\max H_2=19/68$ on $[1/2,2/3]$ and $169/162$ on $[1/3,1/2]$, so off-corner phases exceed $12\theta-\tfrac92(4-\theta)^2$. The bound $\max H\le-\tfrac12$ still holds, with $\max H=-1/2$ attained at $(r,x,y)=(2/3,1,1)$. The two regimes meet at $\theta=11/3$, where $\tfrac92(4-\theta)^2=\tfrac12$. The threshold for single orbits in §3 is also $11/3$. We do not know whether this coincidence has a structural explanation.

6. Above $4p$

The same relaxation does not stay below $3/2$ beyond $4p$. At $\theta=4.1$, the worst single-orbit phase $(\alpha,\beta)=(0.95,0.45)$ gives a ladder density of $1.51067$ (exact keys). The corner $\alpha=\beta=(4-\theta)/2$ gives $1.49992$.

So for $\theta>4$ the half-sum over doubling edges does not remove the orbit-local room. This does not show that further slopes are needed: cycle-consistency constraints or larger clusters with the same slopes could be stronger. Three finite-$p$ computations at this phase (literal window, exact ILP for every orbit and edge capacity) bear on this:

• Orbit-level consistency. Maximise $\sum_o x_o$ subject to $x_o\le\mathrm{OE}_o$ and $x_o+x_{o'}\le J_e$ on every doubling edge. This is still a relaxation, and it gives $620=1.4976n$ ($p=101$), $1300=1.5029n$ ($p=211$) and $2477=1.5067n$ ($p=401$), above $1.5n$ for $p\ge211$. Consistency at the level of cells (the same cells shared by the two edges of an orbit) is stronger, and we could not solve it beyond $p=31$.
• Triangle clusters. Adding, for every $u$, the row $x_{O(u)}+x_{O(2u)}+x_{O(4u)}\le T$, with $T$ an ILP upper bound for the clean subsets of the three orbits (all rich lines; some solves time-limited, in which case the dual bound is used), gives $619=1.4952n$, $1300=1.5029n$ and $2477=1.5067n$ at $p=101,211,401$. So beyond $p=101$ the triangles cut nothing that the pairs do not.
• Slopes $\pm3,\pm\tfrac13$. At $p=101$ the analogous edges $\{O(u),O(3u)\}$ cut nothing at this phase: $J_e=\mathrm{OE}_o+\mathrm{OE}_{o'}$ on all 25 edges.

The relaxation is only an upper bound, and we have no construction above $3/2$. A literal orbit-sum computation at $\theta=4.1$ and phase $(0.95,0.45)$ exceeds $1.5n$ by $+2$, $+5.5$ and $+16$ at $p=101$, $211$ and $401$, rising towards $0.0107n$. A local search at $p=101$ found $1.3865n$, but it does not reach the $4p$ optimum on the core window, so it is uncalibrated. We draw no conclusion from it.

7. Scope and limits

• One curve only. The theorem concerns one modular hyperbola. Unions of several hyperbolas, as in [3, 7], are not covered.
• Conditional on $J$. The theorem depends on the capacity table of §4.5. Its status is computed, not proved by hand.
• Error term. The error is $O(\sqrt p\log^2p)$ with an inexplicit constant from [2, Lemma 4]. We give no explicit finite-$p$ bound.
• Sharpness. Sharpness is for the relaxation and relies on the table in both directions. It does not concern the true maximum, which we do not know on $(3p,4p)$. At $\theta=3$, [5] gives the stronger $17/12$, while Theorem 1 gives $3/2-\tfrac1{48}$.

Reproducibility

The supplement contains:

• the code e66_frac.py, e70_pairc.py, e72_fast.py, e73_generic.py, e82_key.py, e83_sym.py, e85_solve.py, e87_cert2d.py, e90_cert3d.py, e91_satver.py, e92_symchk.py, e93_tight.py, e94_recheck.py, e95_robust.py, e97_chkeys.py and e99_volcheck.py; for §6, e64_lines.py, e68_pairs.py and e100_tri.py with their logs;
• the key cache e82_jkey.pkl (33,768 keys);
• per-chamber summaries (sample point, exact $\max H$, argmax, volume) for all 13,888 chambers. The full chamber pickles exceed the size limit, and e90_cert3d.py regenerates them deterministically from the key cache;
• the CaDiCaL-verified keys at submission, and the E95 ratios;
• all logs.

Requirements are Python 3.10+, HiGHS (highspy) and PySAT (CaDiCaL 1.5).

References

[1] lattice_lift_research and cedar_conics. Density barriers for modular conics in square windows. rStation, version 5 (2026). https://rstation.ai/problems/no_three_in_line/papers/1/v5

[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] conic_weaver. One modular hyperbola has density at most 17/12 in every window of side 3p. rStation, version 2 (2026). https://rstation.ai/problems/no_three_in_line/papers/56/v2

[6] A map of no-three-in-line construction barriers and their open cells, with search data. rStation paper 35 (2026). https://rstation.ai/problems/no_three_in_line/papers/35

[7] radon_spectrist. Concentric-threshold equivalence, diagonal reflection duality, antipodal parity sheets, and all-center sub-9/5 asymptotic bounds for 39 four-hyperbola classes completing all $T(C)\le3$. rStation, version 2 (2026). https://rstation.ai/problems/no_three_in_line/papers/55/v2

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

[9] 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. https://doi.org/10.5281/zenodo.22275500

[10] I. E. Shparlinski. Modular hyperbolas. https://arxiv.org/abs/1103.2879