ChatGPT Overturns a 7‑Year Computational Geometry Challenge by Yao‑Class Legend Chen Lijie

On June 24, an arXiv preprint by UCSD researchers Barna Saha, Yinzhan Xu, and Christopher Ye proved that the classic computational‑geometry problem of finding the farthest pair of points in any super‑constant dimension requires time essentially quadratic in the number of points.

AI‑Generated Proof

The initial proof was produced by GPT‑5.5 Pro from a two‑sentence prompt that asked the model to adapt an existing proof idea to improve the known 2^{O(log* n)} bound. After several iterative rounds—incorporating feedback, using Codex for manuscript refinement, and consulting Claude Opus and Gemini—the model produced a complete, verifiable proof.

Background on the Problem

Chen Lijie, a renowned theoretical computer scientist, spent seven years advancing the farthest‑pair problem, pushing the lower‑bound exponent to 2^{Θ(log* n)} in a 2018 paper. His later work at OpenAI helped disprove the Erdős unit‑distance conjecture, introducing algebraic‑number‑theoretic techniques that later became the key to crossing the final barrier.

Why the Problem Is Hard

The task can be visualized as locating the two most distant people in a stadium where each person’s position is described by hundreds or thousands of coordinates, i.e., a high‑dimensional space. The best known algorithm runs in time roughly n^{2‑c/d} (where n is the number of points, d the dimension, and c a constant). Assuming the Strong Exponential Time Hypothesis (SETH), no algorithm can beat the near‑quadratic bound as the dimension grows, even extremely slowly (e.g., log log log log n).

Technical Barrier: Prime Density

Previous approaches split a long vector into L blocks of b bits and used b distinct primes to test each block via the Chinese Remainder Theorem. The smallest b primes already reach about b log b, and their product grows exponentially with b, making the encoding cost prohibitive when b is large.

Breakthrough via Number‑Field Prime Splitting

The new paper employs a more refined construction in a CM number field. In such a field, a modest‑size prime (≈√L) splits into Θ(b) prime ideals, effectively providing many “prime‑like” resources from a constant number of ordinary primes. For example, the integer 7, prime in ℤ, factors as (3+√2)(3‑√2) in ℚ(√2), illustrating how primes can “split” when the number system is extended.

Using this splitting, the reduction’s computational overhead shrinks to e O(b√L log L), which is sub‑quadratic for any super‑constant dimension, thereby establishing the claimed lower bound.

How AI Discovered the Proof

The paper’s sixth page reproduces the original prompt given to ChatGPT‑5.5 Pro: “Try to use this proof idea [link 1] to improve the 2^{O(log* n)} bound in [link 2].” The model failed on the first attempt, but after multiple back‑and‑forth interactions—prompt refinements, AI‑generated feedback, and manuscript repairs—it finally produced a valid proof.

Subsequent verification relied on formal tools: the authors used the Lean 4 theorem prover (via the Aristotle system) to formalize a critical lemma, which in turn depended on the Aleph Prover’s earlier formalization of the Erdős unit‑distance result.

Implications

The result bridges two rarely‑cited research communities—combinatorial geometry and fine‑grained complexity—by identifying a shared bottleneck (prime density) and transferring a breakthrough across fields. The new lower bound also impacts related problems such as bichromatic closest‑pair, maximum inner‑product search, and Hopcroft’s problem, and it informs theoretical limits of algorithms for Transformer attention and dynamic neuron‑trigger detection.

Beyond the specific theorem, the work illustrates a reproducible AI‑human collaboration model that may become a standard approach for future mathematical research.