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Claim. With $f(d_1)\ge f(d_2)\ge\cdots$ the distance multiplicities of an $n$-point set $A\subset\mathbb R^2$, as in Problem 959, there is an absolute constant $c>0$ such that

$$\max_{|A|=n}\bigl(f(d_1)-f(d_2)\bigr)\ge n^{1+c}$$

for every sufficiently large $n$. The construction is the arithmetic point set behind OpenAI's fixed-power lower bound for unit distances, tuned so that one target shell carries more representations than every competing distance; the gap between the two most frequent distances is then a fixed power of $n$ above linear.

Theofil Xeff, posting under the forum account fefemath, submitted the claim to the site's proof-claims tab on 2026-07-21 as a partial claim, with the write-up linked above and no formalization. The posting names GPT 5.6 Sol and Fable 5 as the systems used.

Submission note. Posted to erdosproblems.com as a proof claim by Theofil Xeff (account fefemath) on 21 July 2026, giving "GPT 5.6 Sol, Fable 5" as the AI used:

We prove that $\max (f(d_1)-f(d_2)) \geq n^{1+c}$ for every sufficiently large $n$, where $c>0$ is an absolute constant. Interestingly, the proof reuses the arithmetic construction from OpenAI's recent solution to the unit-distance problem [90] (paper here). The construction is tuned so that one particular distance has more representations than every competing distance, by isolating a suitable target shell. Eventually, this produces a superlinear gap between the two most frequent distances. Notes: LLM GPT 5.6 Sol high found the main idea. Fable 5 helped to refine it. I am currently formalizing the proof in Lean, will update this when this is done.

Covers. A lower bound of order $n^{1+c}$ on the largest gap, which exceeds the $n^{1+c/\log\log n}$ that Clemen, Dumitrescu and Liu asked for (Problem 1.11) and that Snyder's claim asserts. The problem asks for an estimate; no upper bound is claimed, and the order of the gap remains open.

Depends on. [[problems/distance_problems/E0090/claims/2026_05_20_openai|OpenAI's fixed-power disproof of the unit distance conjecture]] supplies the point set the construction tunes.

Acceptance. None documented. The site's label was OPEN on 2026-10-06, the proof-claims thread carried no comment on the claim as of that date, and the site's page does not credit the result; nothing here has reviewed the write-up. The claim is therefore claimed.