{"slug": "making-equation-2-2-of-the-openai-erdos-result-executable", "title": "Making Equation (2.2) of the OpenAI Erdős Result Executable", "summary": "On May 20, 2026, OpenAI announced that an internal reasoning model had disproved the Erdős planar unit-distance conjecture, producing an infinite family of examples that improve on the classical exponent. A developer has now created an executable reproduction of equation (2.2) from the accompanying remarks paper, which contains a fragile numerical lower bound on the exponent excess. The reproduction reveals that the calculation requires precision beyond standard IEEE 754 double-precision floating point, as the numerator simplifies to a value only barely above 1, with an epsilon on the order of 10⁻¹⁸.", "body_md": "On May 20, 2026, [OpenAI announced](http://%20https://openai.com/index/model-disproves-discrete-geometry-conjecture/) that an internal reasoning model had produced a counterexample to the Erdős planar unit-distance conjecture.\n\nThe problem is easy to state: given $n$ points in the plane, how many pairs of points can be exactly distance $1$ apart?\n\nFor nearly eighty years, the prevailing expectation was that square-grid-type constructions were essentially optimal up to a slowly growing exponent. OpenAI’s announcement changed that. Its internal reasoning model produced an infinite family of examples giving a polynomial improvement, and the proof was checked and written up in mathematical form by external mathematicians.\n\nIn this article, “the remarks paper” refers to the companion PDF by Alon, Bloom, Gowers, Litt, Sawin, Shankar, Tsimerman, Wang, and Matchett Wood, linked from OpenAI’s announcement.\n\nThe proof-level result belongs to those authors and the source papers.\n\nMy focus here is narrower: equation (2.2) in that remarks paper, and whether its explicit numerical value can be reproduced as executable code.\n\nThis is not about proving the theorem again. It is about what happens after a theorem contains a fragile numerical claim.\n\nA mathematical proof and a software artifact do different jobs.\n\nThe proof establishes the theorem. It gives the definitions, the argument, the dependencies, and the mathematical reason why the result holds.\n\nA software artifact should not pretend to replace that.\n\nBut some claims inside a mathematical paper have a finite, numerical, or computationally checkable surface. Those claims can be preserved differently. They can be run. They can be tested. They can fail when precision is wrong.\n\nThat is the narrow role of an executable reproduction artifact: not proof replacement, not automated peer review, and not authority over the theorem, but a reproducible object for the part of the claim that can be computed.\n\nIn the OpenAI Erdős result, one checkable surface is equation (2.2) of the remarks paper.\n\nFor the explicit choice\n\nthe remarks paper gives an explicit numerical lower bound on the exponent excess above the classical Erdős exponent:\n\nThese parameters are taken directly from the remarks paper without modification. The artifact does not derive the multiquadratic choice; it reproduces the finite numerical calculation built from that choice.\n\nThis is not the later stronger explicit bound associated with Sawin’s separate preprint. It is not $\\delta \\approx 0.014$. It is the numerical value appearing in equation (2.2) of the remarks paper.\n\nThat narrowness is important. It is exactly what makes the claim suitable for executable reproduction.\n\nThe numerical fragility comes from the exact form of equation (2.2), not from a large computation.\n\nImmediately after the published expression, the parameters are:\n\nand\n\nWith the paper’s definitions of $u, v$, and $\\delta$ substituted into equation (2.2), the exponent excess reduces to:\n\nThe constant $36$ is not introduced by the implementation. It is already present in the remarks paper’s equation (2.2), both in the numerator term $u\\pi/(36v)$ and in the denominator term $\\log(36/\\delta^2).$\n\nAfter substituting $u = K/r^2, v = r/2$, and $\\delta = 101^{-2K}$, the numerator simplifies to $\\log(K\\pi / 18r^3)$, while the denominator becomes $\\log 36 + 4K \\log 101$.\n\nHere the $101$ comes from the finite prime in $S = {101, \\infty}$.\n\nIn other words, this artifact does not derive the constant $36$ from first principles; it reproduces the published equation with the stated substitutions.\n\nThe precision problem is in the numerator:\n\nBecause $K$ is the ceiling of $18r^3 / \\pi$, the ratio $K\\pi / 18r^3$ is only barely larger than $1$.\n\nMore precisely:\n\nFor $r = 510510$,\n\nSo the numerator is effectively $\\log(1 + \\varepsilon)$ with $\\varepsilon$ at the $10^{-18}$scale.\n\nIEEE 754 double precision has machine epsilon around $2.2 \\times 10^{-16}$. A naive `float64`\n\ncomputation therefore cannot reliably distinguish the near-one ratio from $1$. The ratio rounds to $1$, leading to $\\log(1) = 0.$\n\nThe exponent excess disappears before the computation reaches the value stated in the paper.\n\nThis is not a flaw in the mathematics. It is a precision failure in the numerical evaluation of a valid expression. That is the reason the artifact evaluates equation (2.2) using `mpmath`\n\nat 200-bit precision.\n\nA PDF can state the value. A verifier can expose when the value disappears.\n\nWe built:\n\n[https://github.com/Flamehaven-Labs/openai-erdos-eq22-reproduction](https://github.com/Flamehaven-Labs/openai-erdos-eq22-reproduction)\n\nThe purpose is deliberately narrow: reproduce the finite, explicitly checkable numerical surface of equation (2.2) in the OpenAI Erdős unit-distance disproof remarks.\n\nThe package evaluates the expression using `mpmath`\n\nat 200-bit precision and returns:\n\n```\n6.2391e-38\n```\n\nThis matches the published two-significant-figure value $\\approx 6.24 \\times 10^{-38}$ to $1.4 \\times 10^{-4}$ relative error.\n\nThe repository includes 60 unit tests, 21 verifier checks, a frozen per-source-file SHA-256 manifest, GitHub Actions CI across Ubuntu and Windows, Python 3.11 / 3.12 verification, and a frozen-report mode that prints a verdict without mutating tracked evidence.\n\nThe basic reproduction path is:\n\n```\ngit clone <https://github.com/Flamehaven-Labs/openai-erdos-eq22-reproduction>\ncd openai-erdos-eq22-reproduction\npip install -e \".[dev]\"\npython -m erdos_ant.verify\n```\n\nExpected output includes:\n\n```\nVerdict: PASS\nChecks: 21/21 passed\neq (2.2) exponent excess: 6.2391e-38\n```\n\nThis is not a large system. That is part of the point. A small claim with a clear boundary is easier to inspect than a broad claim that blurs proof, computation, and interpretation.\n\nThis repository was not built as a one-off reaction to an OpenAI announcement. We are not announcing a grand framework here; we are showing the discipline in miniature.\n\nFor us, the work is part of a longer routine: take a mathematical or technical claim, isolate the checkable surface, pin the environment, and make drift visible.\n\nThat is intentionally plain work.\n\nRead the source.\n\nExtract the claim.\n\nReproduce the computation.\n\nRecord the boundary.\n\nLet the verifier fail if the result disappears.\n\nTo execute this routine reliably, the scope must be uncomfortably narrow. This repository intentionally leaves the proof of Theorem 1.1, the construction of the infinite tower, and Sawin’s separate $\\delta \\approx 0.014$ preprint to their respective sources. It does not pretend to be peer review.\n\nThis is not just a disclaimer. It is the point of the artifact.\n\nA sharp, restricted boundary is exactly what makes a claim inspectable, repeatable, and challengeable. This is what I mean here by claim custody.\n\nIt addresses a technical governance question, but not in the policy sense: what exactly is being trusted, from which source, and what makes the claim fail if the implementation changes?\n\nA PDF can state the value. A verifier can expose when the value disappears.\n\nWe claim no authority over the broader theorem. We simply maintain a reproducible boundary around the fragile numerical claim inside it.\n\nThe theorem was proved in the mathematical papers.\n\nThis repository asks a smaller question: can the numerical value in equation (2.2) survive execution?\n\nIn `float64`\n\n, it does not. The exponent excess collapses to zero.\n\nAt 200-bit precision, with the source parameters pinned and the verifier running under CI, the artifact recovers:\n\n```\n6.2391e-38\n```\n\nmatching the published value to $1.4 \\times 10^{-4}$ relative error.\n\nThat is the point.\n\nNot a new theorem. Not a proof replacement.\n\nA reproducible claim surface for one precision-sensitive number in a major AI-assisted mathematical result.\n\nRepository:\n\n[https://github.com/Flamehaven-Labs/openai-erdos-eq22-reproduction](https://github.com/Flamehaven-Labs/openai-erdos-eq22-reproduction)\n\nPaper / Zenodo:", "url": "https://wpnews.pro/news/making-equation-2-2-of-the-openai-erdos-result-executable", "canonical_source": "https://dev.to/flamehaven01/making-equation-22-of-the-openai-erdos-result-executable-ml7", "published_at": "2026-05-26 06:37:10+00:00", "updated_at": "2026-05-26 07:03:49.074975+00:00", "lang": "en", "topics": ["ai-research"], "entities": ["OpenAI", "Alon", "Bloom", "Gowers", "Litt", "Sawin", "Shankar", "Tsimerman"], "alternates": {"html": "https://wpnews.pro/news/making-equation-2-2-of-the-openai-erdos-result-executable", "markdown": "https://wpnews.pro/news/making-equation-2-2-of-the-openai-erdos-result-executable.md", "text": "https://wpnews.pro/news/making-equation-2-2-of-the-openai-erdos-result-executable.txt", "jsonld": "https://wpnews.pro/news/making-equation-2-2-of-the-openai-erdos-result-executable.jsonld"}}