A power improvement for the smallest Heilbronn triangle

Source and verification

This article explains family 191 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

🏷️ Maximizing the smallest triangle

For three planar points, define their triangle area by

$$\mathrm{Area}(pqr)=\frac{1}{2}|\det (q-p,r-p)|.$$

For an $n$-point set $P\subset [0,1{]}^2$, let $\Delta (P)$ be the smallest area among all triples of distinct points, and let $\Delta (n)$ be the largest possible value of $\Delta (P)$. A collinear triple gives minimum area zero.

A lower bound constructs one configuration in which every triangle is sufficiently large. An upper bound proves that every configuration contains a small triangle. These are opposite directions of the problem.

Manuscript claim

There are absolute $\eta ,c>0$ and $n_0$ such that

$$\Delta (n)\ge c{n}^{-2+\eta }(n\ge n_0).$$

The paper gives an explicit but extremely small admissible exponent in its final construction. (OpenAI, 2026)

This is a fixed power improvement over the $n^{-2}$ scale. It refutes the proposed bound $\Delta (n)=O_{\epsilon }(n^{-2+\epsilon })$ for every $\epsilon >0$: choose $\epsilon =\eta /2$, and the ratio of the claimed lower bound to $n^{-2+\eta /2}$ tends to infinity.

🏷️ The elementary arithmetic benchmark

Take a prime $p$ and the points

$$\left(\frac{i}{p},\frac{i^2\mathrm{mod}p}{p}\right),0\le i<p.$$

Three distinct labels have nonzero Vandermonde determinant modulo $p$, so their integer-coordinate lifts are not collinear. Before scaling, a nonzero triangle determinant is an integer of absolute value at least one. After scaling, every area is at least $1/(2p^2)$.

This supplies the baseline exponent two. To improve it, the manuscript must make determinants larger relative to the number of retained points, while controlling all triples, including those with repeated arithmetic labels.

The existing note Heilbronn’s triangle problem develops the upper-bound lineage and the strip-incidence interpretation. The present family concerns a new lower construction; it does not improve those upper bounds.

🏷️ Homogeneous coordinates

The proof samples integer columns $u=(u_1,u_2,u_3{)}^{\mathsf{T}}$ with

$$\begin{array}{c}0\le u_1,u_2<N,\\ N\le u_3<2N,\end{array}$$

representing unit-square points $\pi (u)=(u_1/u_3,u_2/u_3)$. If $A$ is the matrix of three such columns, then

$$\mathrm{Area}(\pi (u)\pi (v)\pi (w))=\frac{|\det A|}{2u_3{v}_3{w}_3}.$$

This follows by dividing each column by its last coordinate and using the determinant formula for area. Since the denominator is comparable to $N^3$, a lower bound on every surviving determinant becomes an area bound.

Proportional columns represent the same point and must also be excluded. Controlling nonzero determinants alone would leave both repeated points and collinear triples untreated.

🏷️ The first congruence condition

Assign each column a random label in a finite extension field. Three distinct labels give nonzero determinant for the columns $(1,\xi ,{\xi }^2{)}^{\mathsf{T}}$. Taking a field norm turns this condition into a polynomial relation over the base field.

The source expresses that relation as a sum of monomial determinants and encodes it in a designated coefficient of a base-$B$ determinant expansion. Large-base estimates prevent carries from obscuring the coefficient. Modulo a fixed power of $B$, distinct labels then exclude sufficiently small determinants.

A common random determinant-one matrix mixes the row residues without changing their determinant. Lattice counts control integer lifts of a fixed nonzero determinant, with the lattice index and residue-orbit size accounted for together. Repeated labels and singular residue matrices need separate conditional moment bounds; they cannot simply be discarded as impossible.

These are specialized arithmetic inputs. The geometric purpose is to make bad nonzero determinants require rare coincidences, giving a gain beyond uniform random sampling.

🏷️ The zero-determinant obstruction

An estimate for fixed nonzero determinants does not control determinant zero. Three rows can lie in one integer plane, producing a collinear projected triple even when other local statistics look favorable.

The manuscript adds an independent congruence condition at a second prime, using a small part of a quadratic surface with no three points on a line. Restricted translations exclude short integer relations, while a random invertible linear map permits uniform counting. The Chinese remainder theorem combines the two residue requirements before lifting to the integer box.

For a singular lift with distinct projected columns, a primitive integer null vector identifies a rank-two row lattice. Summing over these vectors, with separate treatment of coincident residues and low rank at the second prime, supplies the missing zero-determinant bound.

🏷️ Deletion and all large cardinalities

Once the expected numbers of proportional pairs and small-determinant triples are sufficiently small, delete one point from each remaining bad configuration. The improved distribution ensures that many points survive and that every triangle among them meets the target lower bound.

Passing from a sequence of construction sizes to every sufficiently large $n$ is another step. Taking subsets preserves a minimum-area lower bound, but uncontrolled gaps in the sequence could destroy its expression in terms of $n$. The source includes prime-interval estimates and parameter interpolation to control those gaps.

The documented formalization scope is narrower: it gives an unbounded sequence of sizes with a fixed power improvement, sufficient to refute the proposed universal almost-$n^{-2}$ upper bound. It does not document the manuscript’s all-sufficiently-large-cardinalities statement. No determinant-counting certificate or Lean build was run for this article.

🏷️ Key steps and proof outlines

Isolate the two probability estimates that deletion needs

The arithmetic construction supplies a distribution on integer columns in the projective box, with shared random residue data and conditionally independent samples. Let $r$ be the growing prime parameter; let $h,q,N$ be the two modulus-related scales and box scale, with $N=(hq{)}^{10}$; and let $\tau$ be the forbidden determinant cutoff. The construction fixes a field-extension parameter $d=41$.

The sampled-column estimates have the following usable form: a proportional pair has probability at most $C(hq{)}^6{N}^{-3}$, while a triple of pairwise distinct projections with $|\det A|\le \tau$ has probability at most

$$C(\log (2N){)}^2\frac{h}{N^3{r}^d}.$$

Constants may depend on the fixed construction parameters, but not on $r$. These estimates include different degeneracies; the triple estimate does not replace the proportional-pair estimate. (OpenAI, 2026)

The nonzero-determinant argument first conditions on residue data. Since $2\tau <h$, at most one integer determinant in $[-\tau ,\tau ]$ is compatible with the first congruence class. Lattice counting and residue-orbit averaging then control its lifts. Zero cannot simply be substituted into an estimate that used invertibility, so the manuscript proves a separate singular estimate.

A primitive relation organizes all singular triples

For three pairwise nonproportional columns with determinant zero, the rational rank is two. Choose a primitive integer null vector $x$ with $Ax=0$. Every coordinate of $x$ is nonzero: a relation using only two columns would make that pair proportional. A cross product of two independent rows gives $|x|\ll N^2$.

All rows now lie in the rank-two lattice perpendicular to $x$. Split possible $x$ into dyadic shells $R\le |x|<2R$. The first modulus gives a row-lattice covolume of the form $J_x=I|x|/g(x)$, where $I$ is the fixed index of the first row lattice and $g(x)$ records the corresponding divisibility factor. An extra index-$m$ row restriction bounds the number of rational-rank-two matrices in one shell, for fixed $x$, by

$$C\frac{N^{6}g(x{)}^3}{m^3{I}^3{R}^3}.$$

The second congruence separates three mechanisms. Affinely independent residues exclude short null vectors and permit a moment bound for $g(x{)}^3$. If two columns agree modulo $q$ but the null vector is not parallel modulo $q$ to their coordinate difference, row equality imposes an extra index-$q$ condition. In the remaining case, the null vector’s third nonzero coordinate is divisible by $q$, and its residue classes have only $O(R^3/q^2)$ representatives in a shell. When the reduced matrix also has rank at most one, rows lie on one of $q+1$ lines, each contributing a further lattice restriction.

Each case contributes at most $C{N}^6/I^2$ per shell after its sampling weight is included. Summing the $O(\log N)$ shells gives the singular estimate used in the triple bound. The extra congruence therefore pays for exactly the degeneracy that nonzero-determinant counting cannot see.

Spend the probability gain on more points

Set $a_r=r\sqrt{N^3/\tau }$ and $n_r=\lfloor a_r\rfloor$, and sample $2n_r$ columns. Let $Z$ count proportional pairs and distinct-projection triples with small determinant. Linearity of expectation gives

$$\frac{\mathbb{E}Z}{n_r}\ll n_r(hq{)}^6{N}^{-3}+n_r^2(\log (2N){)}^2\frac{h}{N^3{r}^{41}}.$$

No independence between bad events is needed. The first term tends to zero using $N=(hq{)}^{10}$. For the second, $n_r^2\le r^2{N}^3/\tau$, while the parameters give $h/\tau =O(r^{30})$ and $\log N=O(\log r)$. Hence it is $O((\log r{)}^2{r}^{-9})$.

Choose an outcome with $Z<n_r$. Mark one index in each violation and delete all marked indices. At least $n_r$ remain, with distinct projections and all determinants larger than $\tau$. Deleting points cannot create a new violating pair or triple. Since every third coordinate is below $2N$, every remaining area is at least $\tau /(16N^3)$. As $n_r\ge a_r/2$ eventually,

$$n_r^2\Delta (P_r)\ge r^2/64.$$

Convert the parameter gain to a cardinality exponent

For a fixed integer $k$ chosen by the construction, its parameter bounds give $A_k{r}^{\alpha }\le n_r\le D_k{r}^{\alpha }$, with $\alpha =45435k+16$. Therefore $r^2\ge D_k^{-2/\alpha }{n}_r^{2/\alpha }$, producing $\eta =2/\alpha >0$.

For arbitrary large $n$, choose a prime $r$ between $m$ and $2m$, where $m=\lceil (n/A_k{)}^{1/\alpha }\rceil$. The two-sided bounds ensure $n\le n_r\le C_{k}n$, without requiring $n_r$ to be monotone. Retaining any $n$ points preserves minimum area and yields the same power bound with a new constant. This final interpolation, developed in “Deletion and projective normalization,” is necessary for all large cardinalities and is broader than the documented formalization scope.

See Also

• The earlier Heilbronn note explains the complementary incidence-based upper bounds and the elementary area scale.
• Halving lines studies another planar extremal problem where a fixed power improvement requires uniform control of degeneracy.

🏷️ Manuscripts and verification scope

• A power improvement in the Heilbronn triangle lower bound — pinned manuscript and companion source.

The family scope page describes selected formalization coverage. The selected construction is an unbounded sequence of sizes. The manuscript’s assertion for every sufficiently large size is broader. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026. A power improvement in the Heilbronn triangle lower bound.