Openai 2026 planar point sets many unit distances

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evidence/: Retains the independent full review of the original pro-3 branch and the exact-delta review of its two source-record corrections.

proposition_2_2: Specializes the shared ideal-class lemma with exponent one at every split prime pair to obtain exponentially many bounded-denominator translations.

proposition_3_2: Uses cyclotomic cubic fields and the conductor-discriminant formula to produce a large everywhere-unramified elementary abelian extension.

proposition_3_3: Controls generator and relation ranks when selected Frobenius elements in a pro-p Frattini subgroup are imposed as new relations.

proposition_3_4: Gives the generator-relation threshold that forces the Frobenius-killed pro-3 quotient to remain infinite.

proposition_3_5: Bounds the relation rank of the maximal everywhere-unramified pro-3 group over a totally real cubic field by its generator rank plus a constant.

proposition_3_6: Applies Chebotarev to choose rational primes that split over the base and Gaussian fields and whose Frobenius classes lie in the Frattini subgroup.

proposition_3_7: Bounds a number field's class number exponentially in its degree when its root discriminant is bounded.

proposition_3_8: Builds a totally real unramified pro-3 tower with quadratically many fixed split primes and a class-number loss small enough for the geometric step.

theorem_1_1: Assembles the pro-3 tower and geometric criterion to give infinitely many planar point sets with a fixed power more than linearly many unit distances.

theorem_2_3: Converts exponentially many norm-one translations into planar point sets with a fixed power more than linearly many unit-distance pairs.

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OpenAI, Planar Point Sets with Many Unit Distances. Unnumbered 18-page technical report, 2026.

Selected artifact and provenance

The edition read is identified by its displayed title, its byline, and its 18-page extent. It has no printed date or version number. Its PDF metadata gives a creation date of 19 May 2026. The copy read for this card was retrieved from the official OpenAI CDN on 5 September 2026.

The source's Statement on AI Use, on pp. 2--3, says that an internal model was given an AI-written problem statement and produced the solution in a fully automated process. An AI grading pipeline evaluated the output before human researchers examined it. The source then reports AI-assisted verification and rewriting, review by external mathematicians including number theorists, and human editing of the present exposition. These are the report's own provenance and review attestations; they are not a publication or independent-review claim by this corpus. The original internal-model final response is reproduced verbatim beginning on p. 3. The detailed Sections 2--3 and Appendix A are the later human-edited exposition used for this proof record.

A separate 125-page file was released as model reasoning. It describes itself as a rewritten reasoning summary, not the raw internal trace, and is supplementary provenance rather than the selected proof source.

Main result and proof branch

[[discrete_geometry/openai_2026_planar_point_sets_many_unit_distances/theorem_1_1|Theorem 1.1]] proves that an absolute constant $\delta>0$ and infinitely many integers $n$ satisfy

$$\nu(n)\geq n^{1+\delta},$$

where $\nu(n)$ is the maximum number of unordered Euclidean unit-distance pairs among $n$ planar points. This disproves Problem 90. A minimum-degree deletion argument also gives a fixed-power lower bound along an unbounded sequence for Problem 92.

The source's original arithmetic branch is recorded separately from the later human companion:

• [[discrete_geometry/openai_2026_planar_point_sets_many_unit_distances/proposition_3_2|Proposition 3.2]] constructs a cyclic cubic base field with a large elementary abelian everywhere-unramified extension.
• [[discrete_geometry/openai_2026_planar_point_sets_many_unit_distances/proposition_3_3|Proposition 3.3]], [[discrete_geometry/openai_2026_planar_point_sets_many_unit_distances/proposition_3_4|Proposition 3.4]], [[discrete_geometry/openai_2026_planar_point_sets_many_unit_distances/proposition_3_5|Proposition 3.5]], [[discrete_geometry/openai_2026_planar_point_sets_many_unit_distances/proposition_3_6|Proposition 3.6]], and [[discrete_geometry/openai_2026_planar_point_sets_many_unit_distances/proposition_3_7|Proposition 3.7]] state the exact external group, tower, Chebotarev, and class-number inputs and explain their applications.
• [[discrete_geometry/openai_2026_planar_point_sets_many_unit_distances/proposition_3_8|Proposition 3.8]] takes $t=\lfloor(\ell-1)^2/100\rfloor$, kills $3t$ Frobenius elements without destroying Golod--Shafarevich infinitude, and produces a totally real unramified pro-$3$ tower in which the same $t$ rational primes split at every level.
• [[discrete_geometry/openai_2026_planar_point_sets_many_unit_distances/proposition_2_2|Proposition 2.2]] takes exponent one at each of the $tf$ conjugate prime pairs and obtains at least $2^{tf}/h(K)$ distinct norm-one elements in $Q^{-2}\mathcal O_K$.
• [[discrete_geometry/openai_2026_planar_point_sets_many_unit_distances/theorem_2_3|Theorem 2.3]] applies a product-disc window, torus averaging, an injective complex coordinate projection, and a packing bound, retaining the directed-to-unordered factor of two and the fixed exponent.

The reproduced raw response on p. 4 chooses $t=\lfloor d(G)^2/100\rfloor$. The expanded Proposition 3.8 on p. 12 instead uses the safer field parameter $t=\lfloor(\ell-1)^2/100\rfloor$, with $d(G)\geq\ell-1$. The result page follows the expanded proposition. The edition read uses a floor, not a ceiling.

Relationship to the human companion

The later human companion is filed at [[discrete_geometry/alon_2026_remarks_disproof_unit_distance_conjecture/_index|Remarks on the disproof of the unit distance conjecture]]. Its [[discrete_geometry/alon_2026_remarks_disproof_unit_distance_conjecture/lemma_2_2_norm_one_elements|Lemma 2.2]] supplies the shared ideal-class argument; the original Proposition 2.2 is its exact specialization with all $k_s=1$. Its [[discrete_geometry/alon_2026_remarks_disproof_unit_distance_conjecture/lemma_2_1_lattice_window|Lemma 2.1]] records the shared product-window, projection, packing, and unordered-pair mechanism. The original Theorem 2.3 uses a sharper overlap average that works from the sole inequality $t\log2-\log H>0$; its common projection and packing steps are linked rather than duplicated without explanation.

The arithmetic implementations are materially different. This report uses a totally real cyclic cubic base, an everywhere-unramified pro-$3$ tower, many fixed split rational primes, and exponent one at every selected prime. The companion uses a pro-$2$ tower ramified over six rational primes, the single split prime $101$, and one large common exponent. This record preserves the original branch because its Frobenius-cutting construction has separate value.

External and quantitative scope

The conductor--discriminant formula, the Frattini presentation bound, Shafarevich's relation-rank estimate, the Golod--Shafarevich inequality, Chebotarev, the prime number theorem in arithmetic progressions, and the Minkowski class-number estimate are external inputs. Their exact specialized statements and source citations are recorded on the linked result pages; their proofs are not recursively reproduced.

Remark 3.1 identifies the tower construction as an unramified specialization of the Hajir--Maire $T$-split, $S$-ramified method, with $S$ empty and $T$ the primes above the selected rational primes. It also identifies the Frobenius-killing step with the later tower-cutting method of Hajir, Maire, and Ramakrishna.

The proof is qualitative: it fixes all arithmetic and geometric constants before the tower level varies and obtains some $\delta>0$. [[discrete_geometry/sawin_2026_explicit_lower_bound_unit_distance_problem/theorem_1|Sawin's separate result]] gives the stronger explicit exponent $1.014114$ and does not alter this branch's proof.

The reconstructed original branch, including its transfers to Problems 90 and 92, has passed independent review relative to the seven outside results stated on the linked pages; the full review and source-corrections review retain the reports. For this review, the companion's Lemma 2.2 was used only in the exact $k_s=1$ specialization and its Lemma 2.1 only for the shared geometric mechanism; the outside theorem proofs were not recursively reviewed. The report's own authorship and review statements remain historical source attestations, not publication, acceptance, or formal-verification evidence. No Lean build or new formalization was performed.

Source: official PDF. The associated official announcement route is https://openai.com/index/model-disproves-discrete-geometry-conjecture/; the local acquisition attempt returned HTTP 403, so no announcement text is used as evidence here. No notice is printed in the file (pp. 1-2 and 17-18 read), and the publisher's terms-of-use page (https://openai.com/policies/terms-of-use/) and the announcement page could not be read (HTTP 403); the term is unstated.

Bears on. #90 and #92.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.