Openai 2026 power improvement heilbronn triangle lower bound

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theorem_1_1: The claimed main result: n points in the unit square with every triangle of area at least c_1 n^(-2+eta), for every large n and one absolute eta > 0, so the almost-n^(-2) upper-bound formulation of Heilbronn's problem fails; unverified here, attributed by the release to an internal model at OpenAI.

OpenAI, A power improvement in the Heilbronn triangle lower bound, OpenAI Math Release preprint, September 25, 2026. Released under the Apache License 2.0 at https://github.com/openai/math (revision adc7f1241), folder preprints/A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026; the held PDF, main.pdf in the release, is retained as openai_2026_power_improvement_heilbronn_triangle_lower_bound.pdf, and the release's TeX bundle sits in the same folder.

@misc{OAI:A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026,
  author = {{OpenAI}},
  title = {{A power improvement in the Heilbronn triangle lower bound}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026/main.pdf}{OAI:A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026}},
  year = {2026}
}

Attestation, recorded from the source's own statements and not as this corpus's review: the release's root README says its manuscripts were "produced by an internal OpenAI model", that the collection "includes results at different stages of verification", that "Not all have accompanying Lean formalizations" and that "Some of the unformalized results could have issues". The manuscript's own README carries only the title, the author "OpenAI", the date September 25, 2026 and the citation block above; neither it nor the paper adds a statement on how the text was produced or checked. The paper names no author beyond "OpenAI", no arXiv identifier and no journal; its PDF metadata carries the title and the author "OpenAI" and no creation date (the TeX source suppresses the date fields). No refereed publication, arXiv version or independent review of the manuscript is recorded here and nothing on this card is independently reviewed.

Formalization, as the release lists it: the release's catalogue lean/formalization.yaml (its list of "papers with a formalized main result") does not name this manuscript. The release's Lean page for this family nevertheless describes a formalization: an unbounded sequence of sizes $n$ with point sets in the unit square whose every triangle has area at least $n^{-2+\eta}$ for one fixed $\eta>0$, and the refutation of the proposed $n^{-2+\varepsilon}$ upper bound for every $\varepsilon>0$; the page itself says the formalized sequence form is narrower than the paper's statement for every sufficiently large $n$. It names the comparator statement file lean/ComparatorChallenges/HeilbronnTriangle.lean, whose two statements OAI.Problem355.heilbronn_power_lower_bound and OAI.Problem355.almost_n_minus_two_refuted are printed there with sorry bodies (the comparator's challenge format), with the companion HeilbronnTriangle.json naming the solution module OAI.Geometry.HeilbronnTriangle.Main and the permitted axioms propext, Quot.sound and Classical.choice. The release tree holds that module under lean/OAI/Geometry/HeilbronnTriangle/ (242 files); its Main.lean closes both statements from the development, and a text search of the folder found no sorry and no axiom declaration. The comparator's exponent constant is $1/(100000\,k)$ with the paper's $k=T^2+1$, smaller than the paper's $2/(45435k+16)$, and the namespace label "Problem355" is the release's own numbering, not an Erdős number. All of this was read statically from the release's catalogue; not built, replayed or audited for fidelity in this repository. A Lean statement about a lower bound in the unit square is not a proof of the Erdős problem, which asks to estimate $\alpha(n)$ in the unit disk.

The release groups this manuscript alone in its family ("A power improvement in the Heilbronn triangle lower bound") and lists no companion manuscript, alternate proof or consequence paper for it.

Read status: claims checked for Theorem 1.1, the definitions of $\Delta(P)$ and $\Delta(n)$, the almost-$n^{-2}$ formulation (1.1) and the explicit exponent (8.7), read clause by clause in the TeX source (sections/01-introduction.tex, lines 1--55, and sections/08-alteration.tex, lines 97--137) on 2026-10-07; the proofs in sections/02-lattices.tex through sections/09-prime-interval.tex were read for their structure only and no step was checked; nothing here is independently reviewed.

Contents

The PDF has 24 pages; page numbers below are the PDF's. The title page (p. 1) carries the abstract and a table of contents.

• Section 1, Introduction (pp. 2--4). Defines $\operatorname{Area}(pqr)$ as half the absolute determinant of $q-p$ and $r-p$, $\Delta(P)$ as the least area over unordered triples of distinct points of a finite $P\subset[0,1]^2$ (so a collinear triple gives zero) and $\Delta(n)$ as the maximum of $\Delta(P)$ over $|P|=n$, noting why the maximum exists. Recounts the original conjecture $\Delta(n)=O(n^{-2})$ (Roth 1951), the two elementary constructions at scale $n^{-2}$ (Erdős's finite-field parabola, recorded in Roth's appendix, and random sampling with deletion), and the Komlós--Pintz--Szemerédi lower bound $\Delta(n)\gg(\log n)/n^2$ of 1982 by a hypergraph independence argument. States the almost-$n^{-2}$ formulation (1.1), discussed in Section 3 of Zakharov's survey: for every $\varepsilon>0$, $\Delta(n)\le C_\varepsilon n^{-2+\varepsilon}$ for $n\ge n_0(\varepsilon)$. States Theorem 1.1: absolute constants $\eta,c_1>0$ and $n_0\ge3$ with $\Delta(n)\ge c_1n^{-2+\eta}$ for every $n\ge n_0$; deduces $\Delta(n)\ge n^{-2+\eta/2}$ for large $n$, which contradicts (1.1) at $\varepsilon=\eta/2$. Lists the upper bounds: Roth's $o(n^{-1})$, Schmidt 1972, Roth 1972, Komlós--Pintz--Szemerédi 1981 ($n^{-8/7+\varepsilon}$), Cohen--Pohoata--Zakharov 2023 ($n^{-8/7-1/2000}$, their Theorem 1.1) and Cohen--Pohoata--Zakharov 2025 ($n^{-7/6+\varepsilon}$, Theorem 1.8 of the Inventiones paper). A paragraph names two earlier preprints claiming stronger power lower bounds and the defects it sees in their cited versions: Ellmann's version 12 (arXiv:1703.03297v12) acknowledges an unproved local uniformity assumption, which its deletion estimate uses, and calls its argument heuristic; Theorem 4.1 of Agama's version 13 (arXiv:2006.05269v13) counts a collinear triple (two endpoints of a diameter and the center); the manuscript says these objections concern the cited arguments and do not assert that the claimed bounds are false. Section 1.1 (p. 3) sketches the method: integer columns $u=(u_1,u_2,u_3)$ in the box $0\le u_1,u_2<N$, $N\le u_3<2N$ represent the points $(u_1/u_3,u_2/u_3)$, and three columns $A$ give a triangle of area $|\det A|/(2A_{31}A_{32}A_{33})$ (1.2). A first congruence condition at a prime power $h=B^k$ gives each column a label $\xi\in\mathbb F_{r^d}$ ($d$ fixed and odd, $r$ a growing prime), takes the field norm of the Vandermonde determinant of $(1,\xi,\xi^2)$ to $\mathbb F_r$, writes it as a sum of monomial determinants and places those in one designated coefficient of a base-$B$ expansion (carry control after Salem--Spencer and Behrend), so that distinct labels forbid a small determinant modulo $h$; a shared random special-linear matrix mixes rows, and lattice counts bound the lifts with a fixed nonzero determinant. A further congruence condition, modulo a second prime $q$ chosen independently, draws residues from a translated and linearly transformed cap of an elliptic quadric to exclude short integer relations and handle determinant zero. The Chinese remainder theorem combines the two, and a deletion step removes repeated points and small triangles. Section 1.2 (pp. 3--4) gives the organization and conventions (Euclidean lengths, primitive vectors, covolume in the real span, constants may depend on the fixed parameters but not on $r$ or the samples).
• Section 2, Lattice counts with a prescribed determinant (pp. 4--6). Lemma 2.1 (successive lengths: $\det L\le\prod\lambda_i\le s^{s/2}\det L$, a weaker form of Minkowski's second theorem, proved here; Henk cited for the classical statement); Lemma 2.2 (points of a rank-two lattice in a ball); Lemma 2.3 (covolume of $\Lambda\cap y^\perp$ for primitive $y$); Lemma 2.4 (unequal column bounds: for $R_1\ge R_2\ge R_3\ge1$, the number of integer triples with $|u_j|\le R_j$ and a fixed nonzero determinant is at most $C(R_1R_2R_3)^2(\log 2R_1)^2$, by summing over primitive normals in dyadic groups); Proposition 2.5 (for a lattice $\Lambda\subseteq\mathbb Z^3$ of index $I$ with third successive length at most $X$, where $X\ge2$: at most $C(\log 2X)^2X^6/I^2$ matrices with rows in $\Lambda$ of length at most $X$ and a fixed nonzero determinant).
• Section 3, A determinant obstruction from field norms (pp. 6--9). Fixes $d=41$, $M=\binom{4d-1}{d}$, $T=\binom M3$, $k=T^2+1$ (3.1); recalls the field $\mathbb F_{r^d}$ (Lidl--Niederreiter cited); defines the norm polynomial $\mathcal N$ (3.2), homogeneous of degree $d$ in each of three variable groups and alternating since $d$ is odd; Lemma 3.1 writes $\mathcal N$ as a sum of $T$ determinants of monomials; (3.4) is the Vandermonde nonvanishing for distinct labels. Lemma 3.2 (a prime in $(n,2n]$ for large $n$, proved in Appendix A) chooses a prime $B$ with $100k^2L^3<B\le200k^2L^3$ for $L=r^{10}$, and sets $h=B^k$ and $\tau=\lfloor B^{k-1}/2\rfloor$ (3.5); designated digit positions (3.6) with the matching property (3.7); the random digit column (3.8), digits in $\{1,\ldots,L\}$ with residues modulo $r$ prescribed by the label. Lemma 3.3 (determinant obstruction): three columns with distinct labels have $\det C$ modulo $h$ with no integer representative in $[-\tau,\tau]$; hence a lifted matrix with $|\det A|\le\tau$ forces a label coincidence, of probability at most $3r^{-d}$.
• Section 4, Residue orbits and a conditional divisor estimate (pp. 10--12). Lemma 4.1 (diagonal form $P\operatorname{diag}(1,B^b,B^e)Q$ over $\mathbb Z/B^k$, the prime-power Smith normal form with Smith 1861 cited; the row lattice $\Lambda_C$ has index $I=DE$ with $D=B^b$, $E=B^e$ and contains $E\mathbb Z^3$); Lemma 4.2 (the $\operatorname{SL}_3$ orbit of $C$ has at least $c_0h^8/(D^3E^2)$ elements and $GC$ is uniform on it); Lemma 4.3 (conditional digit estimate: $\Pr(b\ge j\mid\text{labels})\le(r/L)^{4j}$ and a bounded conditional expectation of $D$); with the chosen parameters $\mathbb E[D\mid\text{labels}]\le2$ (4.7) and $\mathbb E[D\mathbf 1_{\mathcal E}]\le6r^{-d}$ on the label-collision event (4.8).
• Section 5, An auxiliary cap and its inclusion probabilities (pp. 13--15). Parameters $H=h^2$, a prime $q$ with $h^{100}<q\le2h^{100}$ and $w=\lfloor q/(1000H^2)\rfloor$ (5.1); Lemma 5.1 (a set $S\subset\{0,\ldots,w-1\}^3$ with $|S|\ge w^3/q$ and no three collinear points, an affine part of the quadric $z=x^2-\nu y^2$ for a nonsquare $\nu$, with Barlotti cited); the allowed shifts $\mathcal A$ (5.3) and the auxiliary set $V=G_q(a+S)$ (5.4) for a uniform allowed shift $a$ and a uniform $G_q\in\operatorname{GL}_3(\mathbb F_q)$; Lemma 5.2 (at least $q^3/2$ allowed shifts; $V$ has no zero, no three collinear points and no short relation $x_1u^{(1)}+x_2u^{(2)}+x_3u^{(3)}=0$ with $0<|x|\le H$ when $x_1+x_2+x_3\ne0$, and none at all among affinely independent columns, hence among three distinct elements of $V$); the weight $W(A)$ (5.6) and Proposition 5.3 ($W\ll1$ for affinely independent residue columns, $W\ll q^3/s$ at rank at least two, $W\ll(q^3/s)^2$ always); the normalization (5.8).
• Section 6, Sampling integral columns (pp. 15--18). The box $N=(hq)^{10}$, $\mathcal B=([0,N)^2\times[N,2N))\cap\mathbb Z^3$ (6.1) and the sampling rule: a shared uniform $G_h\in\operatorname{SL}_3(\mathbb Z/h)$ and shared $V$, then per sample a label and digit column with $u\bmod h=G_hc$, a uniform residue in $V$ modulo $q$, and a uniform lift into $\mathcal B$. Lemma 6.1 (two samples project to the same point with probability $\ll(hq)^6N^{-3}$); Lemma 6.2 (exact lifting identity (6.3), expressing a conditional probability as a weighted sum over the orbit); the definition of a bad triple (pairwise distinct projections and $|\det A|\le\tau$); Proposition 6.3 (weighted count at a fixed determinant $\ll(\log 2N)^2N^6/I^2$, the nonzero case from Proposition 2.5 and the zero case from Proposition 7.1); Corollary 6.4 (a sampled triple is bad with probability $\ll(\log 2N)^2\,h/(N^3r^d)$).
• Section 7, Counting determinant-zero triples (pp. 18--21). Proposition 7.1 ($\sum W(A)\ll\log(2N)\,N^6/I^2$ over singular matrices with rows in $\Lambda$, columns in $\mathcal B$ and distinct projections); Lemma 7.2 (for a primitive null vector $x$: $g(x)\mid E$, $\det(\Lambda\cap x^\perp)=I|x|/g(x)$, $g(x)^3/I\le h^2$, surjectivity onto the plane modulo $q$, and the row count (7.3)); Lemma 7.3 (dyadic moment $\sum g(x)^3\ll R^3I$). The proof of Proposition 7.1 sums over the primitive null vector in dyadic shells and splits into affinely independent residues (shells beyond $H/2$ only), an equal residue pair with an extra index-$q$ row congruence, and an equal pair without one ($|x|\ge q$, with rank-two and rank-at-most-one subcases).
• Section 8, Deletion and projective normalization (pp. 21--23). The proof of Theorem 1.1: $a_r=r\sqrt{N^3/\tau}$, $n_r=\lfloor a_r\rfloor$, $2n_r$ samples, the count $Z$ of coincident pairs and bad triples with $\mathbb EZ=o(n_r)$ (8.2), deletion of one index per violation, the area identity (8.3) and $\Delta(P_r)\ge\tau/(16N^3)$ (8.4), hence $n_r^2\Delta(P_r)\ge r^2/64$ (8.5); the two-sided cardinality bound $A_kr^\alpha\le n_r\le D_kr^\alpha$ with $\alpha=45435k+16$ (8.6), the exponent $\eta=2/\alpha$ (8.7), $\Delta(P_r)\ge c_*n_r^{-2+\eta}$ (8.8) and the passage to every large $n$ by choosing a prime $r$ with Lemma 3.2 and discarding points. The text says the construction parameters were not optimized and that the earlier Komlós--Pintz--Szemerédi argument's hypergraph independence lemma is not needed.
• Appendix A, An elementary prime-interval estimate (p. 23). Proves Lemma 3.2 by the binomial-coefficient argument of Erdős 1932.
• References (pp. 23--24): Agama (arXiv:2006.05269v13), Barlotti 1956, Behrend 1946, Cohen--Pohoata--Zakharov 2023 (arXiv:2305.18253v1) and 2025 (Invent. Math. 240), Ellmann (arXiv:1703.03297v12), Erdős 1932, Henk (arXiv:math/0204158), Komlós--Pintz--Szemerédi 1981 and 1982, Lidl--Niederreiter 1994, Roth 1951 and 1972 (two papers), Salem--Spencer 1942, Schmidt 1972, Smith 1861, Zakharov 2026 (J. London Math. Soc. 113).

External inputs: the argument is presented as self-contained. The cited results are antecedents rather than premises: Minkowski's second theorem, Smith normal form, the elliptic-quadric cap and the Bertrand-type prime interval are each proved in the form used, and the finite-field facts are recalled with their short proofs. The manuscript flags nothing as unproved, numerical, computer-assisted or conditional; its one explicit caveat is that $\eta=2/(45435k+16)$ with $k=\binom{\binom{163}{41}}{3}^2+1$ is "extremely small" and unoptimized. The release folder holds no verification directory for this manuscript.

Bears on

• Problem 507: claimed partial answer, on the lower-bound side. The page asks to estimate $\alpha(n)$, the Heilbronn function for $n$ points in the unit disk. Theorem 1.1 claims $\Delta(n)\ge c_1n^{-2+\eta}$ for the unit square, a power improvement on the Komlós--Pintz--Szemerédi lower bound $(\log n)/n^2$ (the page cites their 1982 paper and has compiled no results yet). The claim says nothing about the upper bound, which stays at $n^{-7/6+\varepsilon}$ (Cohen, Pohoata and Zakharov), and the exponent gap remains. The claim is unverified here; the page's status rests on acceptance evidence, not on this card.
• Cohen, Pohoata and Zakharov 2023: comparison. The manuscript cites that paper's Theorem 1.1 for the bound $\Delta(n)\le n^{-8/7-1/2000}$, the earlier of the two Cohen--Pohoata--Zakharov upper bounds it lists; the manuscript's own Theorem 1.1, if it holds, supersedes the lower bound $\Omega(\log n/n^2)$ that card records as the known one. Unverified here; the card's own statements are untouched.
• Cohen, Pohoata and Zakharov 2024: comparison and bibliographic supply. The manuscript cites the published version, Inventiones mathematicae 240 (2025), pp. 1045--1118, Theorem 1.8, for the upper bound $\Delta(n)\ll_\varepsilon n^{-7/6+\varepsilon}$; that card holds the arXiv version and records the bound as the current record, which the manuscript does not contest. Unverified here.