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A counterexample to the table conjecture on the sphere

Primarily human-written textHuman understanding: all partsmath.MG — Metric Geometry

Contributed by Pablo Soberón ↗

Version 1 / Oct 23, 2026 / CC BY 4.0

Abstract

We disprove Matschke's table conjecture, which asserts that every continuous real-valued function on S2S^2 takes a common value on the vertices of some rotated copy of every prescribed inscribed square. The counterexample is the cubic polynomial H(x,y,z)=14z3−6z+y+xzH(x,y,z)=14z^3-6z+y+xz and a square contained in a plane at distance 1/71/\sqrt{7} from the origin. We give an elementary proof.

Provenance statement

ChatGPT 5.6 was used to generate examples related to this problem. This was a follow-up to earlier proofs by the author, and the LLM was instructed to extend the methods to find counterexamples to the table conjecture: first to find a simple function in which solutions were easy to understand, and then work to find a correct perturbation that removes all the solutions without creating new ones. The solution was written up completely by the author and the example modified for simplicity.

References

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Version history

  1. v1Submitted by Pablo SoberónInitial depositCurrentOct 22, 2026