Making Equation (2.2) of the OpenAI Erdős Result Executable 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. 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. The problem is easy to state: given $n$ points in the plane, how many pairs of points can be exactly distance $1$ apart? For 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. In 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. The proof-level result belongs to those authors and the source papers. My focus here is narrower: equation 2.2 in that remarks paper, and whether its explicit numerical value can be reproduced as executable code. This is not about proving the theorem again. It is about what happens after a theorem contains a fragile numerical claim. A mathematical proof and a software artifact do different jobs. The proof establishes the theorem. It gives the definitions, the argument, the dependencies, and the mathematical reason why the result holds. A software artifact should not pretend to replace that. But 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. That 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. In the OpenAI Erdős result, one checkable surface is equation 2.2 of the remarks paper. For the explicit choice the remarks paper gives an explicit numerical lower bound on the exponent excess above the classical Erdős exponent: These 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. This 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. That narrowness is important. It is exactly what makes the claim suitable for executable reproduction. The numerical fragility comes from the exact form of equation 2.2 , not from a large computation. Immediately after the published expression, the parameters are: and With the paper’s definitions of $u, v$, and $\delta$ substituted into equation 2.2 , the exponent excess reduces to: The 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 $\log 36/\delta^2 .$ After 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 $\log 36 + 4K \log 101$. Here the $101$ comes from the finite prime in $S = {101, \infty}$. In other words, this artifact does not derive the constant $36$ from first principles; it reproduces the published equation with the stated substitutions. The precision problem is in the numerator: Because $K$ is the ceiling of $18r^3 / \pi$, the ratio $K\pi / 18r^3$ is only barely larger than $1$. More precisely: For $r = 510510$, So the numerator is effectively $\log 1 + \varepsilon $ with $\varepsilon$ at the $10^{-18}$scale. IEEE 754 double precision has machine epsilon around $2.2 \times 10^{-16}$. A naive float64 computation therefore cannot reliably distinguish the near-one ratio from $1$. The ratio rounds to $1$, leading to $\log 1 = 0.$ The exponent excess disappears before the computation reaches the value stated in the paper. This 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 at 200-bit precision. A PDF can state the value. A verifier can expose when the value disappears. We built: https://github.com/Flamehaven-Labs/openai-erdos-eq22-reproduction https://github.com/Flamehaven-Labs/openai-erdos-eq22-reproduction The purpose is deliberately narrow: reproduce the finite, explicitly checkable numerical surface of equation 2.2 in the OpenAI Erdős unit-distance disproof remarks. The package evaluates the expression using mpmath at 200-bit precision and returns: 6.2391e-38 This matches the published two-significant-figure value $\approx 6.24 \times 10^{-38}$ to $1.4 \times 10^{-4}$ relative error. The 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. The basic reproduction path is: git clone