An 80-Year Conjecture, a Genetic Algorithm, and Ag₁₀ versus Au₁₀
Some personal associations prompted by OpenAI’s unit-distance result.
Cite
(May 21, 2026). An 80-Year Conjecture, a Genetic Algorithm, and Ag₁₀ versus Au₁₀. AtomPub AP-2026-unit-distance-intuition. <>Yesterday OpenAI announced that their internal reasoning model had independently overturned a conjecture that Paul Erdős posed in 1946.
The problem is deceptively simple. Put n points on a plane. How many pairs of points can be at exactly unit distance from each other? For eighty years, the working answer was: approximately what you get from a square grid. The count grows no faster than n^{1+o(1)}. The grid, or something close to it, was assumed to be basically optimal.
OpenAI’s model found an infinite family of non-grid configurations where a fixed \delta > 0 exists such that the unit-distance count reaches n^{1+\delta} — directly beating the grid-optimal consensus. The model was handed the problem statement and produced 125 pages of chain-of-thought reasoning. No human mathematician supplied half the proof.
The obvious connection: 2D crystals
In 2D materials, atoms sit on a plane. Nearest-neighbor bond lengths are unit distances. The classic 2D lattices rank-ordered by coordination number:
- Square lattice — coordination 4
- Triangular lattice — coordination 6, densest planar packing
- Honeycomb (graphene) — coordination 3, a sublattice of the triangular
The unit-distance problem asks how many “nearest-neighbor bonds” a planar point set can have. The 80-year consensus was: periodic lattices are optimal. That consensus just broke.
The hard boundary
Before drawing the wrong conclusion: real material geometry is not a pure optimization of planar point configurations. Carbon sp² hybridization is three-coordinate because of electron structure, not because a better geometric arrangement has not been found. Thermodynamic stability, electronic constraints, and symmetry rules govern what configurations physically exist.
OpenAI’s new configuration family does not predict new materials. Worth being explicit about that.
But the quasicrystal parallel is real
In 1982, Dan Shechtman found diffraction patterns in a rapidly-cooled Al–Mn alloy that showed fivefold rotational symmetry — impossible under the crystallographic dogma that all long-range ordered solids must be periodic. The field’s response was initially rejection. Shechtman eventually won the 2011 Nobel Prize in Chemistry.
Quasicrystals exist because the assumption “ordered = periodic” turned out to be wrong. The unit-distance result is the same category of lesson: the intuition about what optimal planar arrangements look like had a systematic blind spot for eighty years.
Here’s where it gets personal
My undergraduate research was on gas-phase metal clusters: magnetron sputtering to generate clusters in the gas phase, photoelectron spectroscopy for experimental signals, and genetic algorithms to search potential energy surfaces (PES) for lowest-energy configurations. Theoretical spectra are generated from candidate geometries and compared against experiment; when they match, the geometry is confirmed as the structure.
In this workflow, the geometry is what is being solved for, not what is assumed at the start.
The case I keep coming back to is Ag₁₀ versus Au₁₀. Same group, same valence electron configuration, same nominal coordination chemistry. If structure were purely geometric, they should converge to the same lowest-energy configuration. They do not.
Gold’s relativistic effects — 6s orbital contraction, 5d orbital expansion, strong s–d hybridization — mean that Au clusters stay planar to significantly larger sizes than Ag clusters. Au₁₀ is essentially flat. Ag₁₀ has already transitioned to a 3D configuration. The genetic algorithm exploring the same geometric search space falls into different basins because the shape of the potential energy surface is different.
Geometry is downstream of electronic structure, not upstream.
What the unit-distance result actually means for structure search
The direct implication is limited. The unit-distance problem is a hard-constraint, potential-free, combinatorial geometry problem — categorically different from DFT global optimization on a potential energy surface.
The indirect implication, for me at least, is methodological. Genetic algorithms for structure search depend on priors about what “reasonable configurations” look like — how the initial population is seeded, how crossover and mutation operators are designed. Many implementations carry implicit preferences for high-symmetry offspring, which makes sense as a heuristic but is not obviously grounded in anything more principled than convention.
The cluster search community has been moving away from this for years: random seeding, basin-hopping, particle swarm optimization, machine-learning potentials for accelerated PES sampling. The unit-distance result is probably more supporting argument for an existing trend than paradigm shift. But it is a clarifying argument, and those are not always easy to find.
One sentence
OpenAI showed that the eighty-year intuition about optimal planar arrangements had a blind spot. For me, it was a useful occasion to think about how much geometric intuition is quietly embedded in the tools one reaches for most automatically — and how rarely that gets examined.
Note: OpenAI’s proof has not yet undergone formal peer review. Princeton’s Will Sawin has independently refined parts of the result, which is a good sign. Status should be monitored.
References
- Erdős, P. On sets of distances of n points. Amer. Math. Monthly 53, 248–250 (1946).
- Shechtman, D. et al. Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett. 53, 1951 (1984).
- Wang, L.-S. et al. Photoelectron spectroscopy of size-selected gold cluster anions. Phys. Rev. Lett. 91, 123401 (2003).
- OpenAI. Planar Unit Distance Conjecture (internal reasoning trace). Preprint (2026).